{"query": "You are a highly experienced, conscientious, and fair academic reviewer, please help me review this paper. The review should be organized into nine sections: \n1. Summary: A summary of the paper in 100-150 words.\n2. Strengths/Weaknesses/Questions: The Strengths/Weaknesses/Questions of paper, which should be listed in bullet points, with each point supported by specific examples from the article where possible.\n3. Soundness/Contribution/Presentation: Rate the paper's Soundness/Contribution/Presentation, and match this score to the corresponding description from the list below and provide the result. The possible scores and their descriptions are: \n    1 poor\n    2 fair\n    3 good\n    4 excellent\n4. Rating: Give this paper an appropriate rating, match this rating to the corresponding description from the list below and provide the result. 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Paper Decision: It must include the Decision itself(Accept or Reject) and the reasons for this decision, based on the criteria of originality, methodological soundness, significance of results, and clarity and logic of presentation.\n\nHere is the template for a review format, you must follow this format to output your review result:\n**Summary:**\nSummary content\n\n**Strengths:**\n- Strength 1\n- Strength 2\n- ...\n\n**Weaknesses:**\n- Weakness 1\n- Weakness 2\n- ...\n\n**Questions:**\n- Question 1\n- Question 2\n- ...\n\n**Soundness:**\nSoundness result\n\n**Presentation:**\nPresentation result\n\n**Contribution:**\nContribution result\n\n**Rating:**\nRating result\n\n**Paper Decision:**\n- Decision: Accept/Reject\n- Reasons: reasons content\n\n\nPlease ensure your feedback is objective and constructive. The paper is as follows:\n\n# Improved Bayesian Regret Bounds for Thompson Sampling in Reinforcement Learning\n\n Ahmadreza Moradipari\\({}^{*}\\)\\({}^{\\dagger}\\) Mohammad Pedramfar\\({}^{*}\\) Modjtaba Shokrian Zini\\({}^{*}\\)\n\n\\({}^{\\ddagger}\\)\n\n&Vaneet Aggarwal\n\n\\({}^{\\ddagger}\\)Pardue University, West Lafayette, IN, USA, vaneet@purdue.eduThis work was supported in part by the National Science Foundation under grant CCF-2149588 and Cisco, Inc.\n\n###### Abstract\n\nIn this paper, we prove the first Bayesian regret bounds for Thompson Sampling in reinforcement learning in a multitude of settings. We simplify the learning problem using a discrete set of surrogate environments, and present a refined analysis of the information ratio using posterior consistency. This leads to an upper bound of order \\(\\widetilde{O}(H\\sqrt{d_{l_{1}}T})\\) in the time inhomogeneous reinforcement learning problem where \\(H\\) is the episode length and \\(d_{l_{1}}\\) is the Kolmogorov \\(l_{1}-\\)dimension of the space of environments. We then find concrete bounds of \\(d_{l_{1}}\\) in a variety of settings, such as tabular, linear and finite mixtures, and discuss how how our results are either the first of their kind or improve the state-of-the-art.\n\n## 1 Introduction\n\nReinforcement Learning (RL) is a sequential decision-making problem in which an agent interacts with an unknown environment typically modeled as a Markov Decision Process (MDP) [38, 8]. The goal of the agent is to maximize its expected cumulative reward. This problem has a variety of applications, including robotics, game playing, resource management, and medical treatments. The key challenge in RL is to balance the so-called exploration-exploitation trade-off efficiently: exploring unseen state-action pairs to gain more knowledge about the unknown environment or exploiting the current knowledge to maximize the expected cumulative reward. Two efficient approaches have been developed to control this trade-off: _optimism in the face of uncertainty_ (OFU) and _Thompson Sampling_ (TS) (or Posterior Sampling (PS)). OFU constructs a confidence set of statistically plausible MDPs that includes the true MDP with high probability and plays an optimistic policy according to the MDP with maximum gain from this set [5, 40]. TS samples a statistically plausible MDP from a posterior distribution and plays the optimistic policy of the sampled MDP [29, 31]. In this work, we focus on the latter, and by combining an information theoretical approach first introduced by [36] with analysis based on posterior consistency tools, we prove state-of-the-art Bayesian regret bounds in a variety of settings.\n\nIn this paper, we start by defining the Bayesian RL problem, where transition and reward functions are Bayesian and time inhomogeneous. The Bayesian RL problem we consider is more comprehensive than in previous works, as we allow for both Bayesian transition and Bayesian rewards, and do not make any assumption on their individual prior. To simplify the learning problem, we utilize the notion of surrogate environments, which is a discretization of the environments space, and its learning task and TS regret is a proxy to that of the main problem. The construction of the surrogate environments was first introduced by [18] with an incorrect proof, which is fixed in our work bydefining the surrogate environments through an optimization. Of main importance is the size of this new environment space. The Bayesian regret decomposes to the product of two terms, one being the cumulative mutual information of the environment and history traversed by the policy. By the well-known entropy estimation of the mutual information, this significant factor in the regret is connected to the \\(l_{1}-\\)dimensions (\\(d_{l_{1}}\\)) of the transition and reward functions space, which can be more succinctly interpreted as the \\(l_{1}-\\)dimension \\(d_{l_{1}}\\) of the environment space. The latter is in turn estimated by the size of the space of surrogate environments.\n\nThe information ratio, representing a trade-off of exploration/exploitation, is the other significant term in the decomposition of the TS Bayesian regret. In an improvement to [18], our novel analysis of this ratio based on posterior consistency tools, shows that this trade-off is bounded by \\(H^{3/2}\\), where \\(H\\) is the episode length. This bound is general and independent of the dimension of transition/reward function space at each step, which is is a key factor behind the advantage of our regret bound, such as the \\(\\sqrt{SA}\\) advantage in the tabular case compared to [18], or the lack of any restriction on the prior (e.g., Dirichlet prior) compared to [31]. Following a further refined approach, we finally estimate the TS Bayesian regret to be \\(\\widetilde{O}(\\lambda\\sqrt{d_{l_{1}}T})\\) for large enough \\(T\\) in the time inhomogeneous setting. Here, a new term 'value diameter' \\(\\lambda\\), which is the average difference of the optimal value functions at different states, is used in bounding the information ratio, where instead of \\(H^{3/2}\\), we have the smaller term \\(\\lambda H^{1/2}\\). Bounding the information ratio with \\(\\lambda\\) is a conceptual contribution of our work, which shows that the ratio is bounded by a _value-dependent_ term, which is in nature different from \\(H\\) but always \\(\\leq H+1\\). Further, there exists another bound for \\(\\lambda\\); in environments where states are reachable from one another in \\(D\\) steps, we have \\(\\lambda\\leq D+1\\). In 'well-connected' MDPs, one could have \\(D\\ll H\\), implying an improvement over the \\(H^{3/2}\\) information ratio bound.\n\nOur generic bound is abstract in terms of \\(d_{l_{1}}\\), so we estimate it in more explicit terms for useful applications. [18] have bounded \\(d_{l_{1}}\\) in the tabular and linear case without formalizing this notion, and while for tabular MDPs, \\(d_{l_{1}}\\) was bounded by \\(SAH\\), for linear MDPs with feature space dimension \\(d_{f}\\), we investigate their claim of the bound \\(d_{f}H\\). Detailed in Appendix G, we show a counterexample to their analysis, and we manage to find a correct estimate in this setting. We also introduce finite mixtures MDPs and are the first to prove a TS Bayesian regret of order \\(\\widetilde{O}(\\lambda\\sqrt{HmT})\\), where \\(m\\) is the number of mixtures.\n\nLastly, we note that our regret bound of order \\(\\widetilde{O}(\\lambda\\sqrt{d_{l_{1}}T})\\) is the first in the general nonlinear time inhomogeneous Bayesian RL setting for TS, and generalizing [31, Conj. 1], we conjecture it to be optimal if \\(\\lambda\\) can be replaced by \\(\\widetilde{O}(\\sqrt{H})\\).\n\nRelated work.Since the introduction of information ratio by [35, 36], a new line of research has emerged to provide tighter regret bounds for TS. The general approach involves factoring the Bayesian regret into two components: an information ratio that captures the trade-off between optimal action selection and information gain, and a cumulative information gain term that depends on the target environment and the history of previous observations. Then, both components are bounded separately using information theoretic tools.\n\nIn the bandit setting, this analysis has been used to bound Bayesian regret for TS [14, 9], as well as that of a new algorithm called information-directed sampling (IDS) [35, 26, 23, 19, 20]. This analysis has also been used in partial monitoring [25, 24] and RL with a specific Dirichlet prior and additional assumptions [28, 27] or when the true environment is too complicated to learn [4]. More recently, [18] studied the Bayesian regret of TS in RL without any prior assumptions for tabular MDP. This is the closest work to our paper and we discuss our generalization in detail in Section 5.\n\nThe Bayesian tabular MDP case has also been studied with the additional Dirichlet prior assumption in [31], where they achieve a regret bound matching ours. In an independent approach, the first non-linear Bayesian RL model was considered by [16] with a regret bound of \\(dH^{3/2}T^{1/2}\\) where \\(d\\) is a notion of dimension of their model, but their results were limited to Gaussian process settings with linear kernels. Finally, [11] considered general non-linear Bayesian RL models and introduced an algorithm that obtains \\(dH^{1+\\alpha/2}T^{1-\\alpha/2}\\) where \\(\\alpha\\) is a tuning parameter and \\(d\\) is the dimension of \\(\\mathcal{S}\\times\\mathcal{A}\\times\\mathcal{S}\\).\n\nIt is worth noting that there is another line of work that incorporates confidence regions into TS to achieve Bayesian regret bounds that can match the best possible frequentist regret bounds by UCB in both bandit settings [35] and RL [31; 30; 32; 12]. However, this technique often results in a sub-optimal Bayesian regret, as the best bound known for UCB itself is not optimal.\n\nWhile our work's emphasis is on theoretical guarantees for TS, we discuss here the experiments using this algorithm. Previous works on PSRL [35; 26; 23; 20; 31] come with extensive experiments on TS (and/or its variants), and discussions on computational efficiency of PSRL. In particular, experiments in [31] support the assertion that \"PSRL dramatically outperforms existing algorithms based on OFU\". In addition, PSRL with oracle access has been shown to be the most performant, esp. when compared to recent OFU based UCBVI/UCBVI-B, or even variants of PSRL such as Optimistic PSRL [39; Fig. 1.3]. However, an important limitation in experiments is the need for oracle access to an optimal policy, and that can not be always satisfied efficiently. Nevertheless, clever engineering can make TS work even in large scale Deep RL. Indeed, for general RL settings, the recent work [37] shows how to implement TS in Deep RL on the Atari benchmark and concludes that \"Posterior Sampling Deep RL (PSDRL) significantly outperforms previous state-of-the-art randomized value function approaches, its natural model-free counterparts, while being competitive with a state-of-the-art (model-based) reinforcement learning method in both sample efficiency and computational efficiency\". In summary, experiments in the literature provide enough support for the empirical performance of TS.\n\n## 2 Preliminaries\n\n### Finite-horizon MDP\n\nWe follow the literature's conventions in our notation and terminology to avoid confusion when comparing results. The environment is a tuple \\(\\mathcal{E}=(\\mathcal{S},\\mu_{\\mathcal{S}},\\mathcal{A},\\mu_{\\mathcal{A}},H,\\{ P_{h}\\}_{h=1}^{H},\\{r_{h}\\}_{h=1}^{H})\\), where \\(\\mathcal{S}\\) is the topological measurable state space, \\(\\mathcal{A}\\) is the topological measurable action space, \\(\\mu_{\\mathcal{S}}\\) and \\(\\mu_{\\mathcal{A}}\\) are base probability measures on \\(\\mathcal{S}\\) and \\(\\mathcal{A}\\) respectively, \\(H\\) is the episode length, \\(P_{h}:\\mathcal{S}\\times\\mathcal{A}\\rightarrow\\Delta_{\\mathcal{S},\\mu_{\\mathcal{ S}}}\\) is the transition probability kernel, and \\(r_{h}:\\mathcal{S}\\times\\mathcal{A}\\rightarrow\\Delta_{[0,1],\\mathrm{Lebesgue}}\\) is the reward function, where we fix the convention \\(r(s,a):=\\mathbb{E}_{x}[r(x|s,a)]=\\int_{0}^{1}xr(x|s,a)\\,\\mathrm{d}x\\) as we mostly deal with its mean value. Notice that \\(\\Delta_{X,\\mu}\\) is the set of probability distributions over \\(X\\) that are absolutely continuous with respect to \\(\\mu\\). We will use \\(\\Delta_{X}\\) when the base measure is clear from the context. We assume \\(\\mathcal{S}\\), \\(\\mathcal{A}\\) are known and deterministic while the transition probability kernel and reward are unknown and random. Throughout the paper, the implicit dependence of \\(P_{h}\\) and \\(r_{h}\\) on \\(\\mathcal{E}\\) should be clear from the context.\n\nLet \\(\\Theta_{h}^{P}\\) be the topological function space of \\(P_{h}\\) and \\(\\Theta^{P}=\\Theta_{1}^{P}\\times\\cdots\\times\\Theta_{H}^{P}\\) be the full function space. The space \\(\\Theta_{h}^{P}\\) is assumed to be separable and equipped with prior probability measure \\(\\rho_{h}^{P}\\) yielding the product prior probability measure \\(\\rho^{P}=\\rho_{1}^{P}\\otimes\\cdots\\otimes\\rho_{H}^{P}\\) for \\(\\Theta^{P}\\). The exact same definition with similar notations \\(\\Theta_{h}^{R},\\rho_{h}^{R},\\rho^{R},\\Theta^{R}\\) applies for the reward function. Notice the explicit assumption of time inhomogeneity in these definitions, with all 'layers' \\(h\\) being independent. The two sets define the set of all environments parametrized by \\(\\Theta=\\Theta_{1}\\times\\cdots\\times\\Theta_{H}\\) where \\(\\Theta_{h}=\\Theta_{h}^{P}\\times\\Theta_{h}^{R}\\). Note that the prior is assumed to be known to the learner. This setting implies that an environment \\(\\mathcal{E}\\) sampled according to the prior \\(\\rho=\\rho^{P}\\otimes\\rho^{R}\\) is essentially determined by its transition and reward functions pair \\(\\{(P_{h},r_{h})\\}_{h=1}^{H}\\). We simplify the notation to view \\(\\Theta\\) as the set of all environments, i.e., saying\n\n\\begin{table}\n\\begin{tabular}{|c|c|c|c|c|} \\hline Reference & Tabular & Linear & General & Comments \\\\ \\hline\n[29] & \\(\\sqrt{H^{3}S^{2}AL}\\) & - & - & - \\\\ \\hline\n[30] & - & - & \\(L^{*}\\sqrt{d_{K}d_{E}HL}\\) & \n\\begin{tabular}{c} Uses Eluder dimension \\\\ Lipschitz assumption \\\\ \\end{tabular} \\\\ \\hline\n[31] & \\(\\sqrt{H^{3}SAL}\\) & - & - & Dirichlet prior \\\\ \\hline\n[28] & \\(\\sqrt{H^{3}SAL}\\) & - & - & Assumptions on prior \\\\\n**[12]** & \\(L^{*}\\sqrt{H^{3}S^{2}A^{2}L}\\) & - & \\(L^{*}\\gamma\\sqrt{HL}\\) & \n\\begin{tabular}{c} Assumptions on regularity \\& noise \\\\ Lipschitz assumption \\\\ \\end{tabular} \\\\ \\hline \\hline\n[18] & \\(\\sqrt{H^{4}S^{2}A^{2}L}\\) & - & - & - \\\\ \\hline This paper & \\(\\lambda\\sqrt{H^{2}SAL}\\) & \\(\\lambda\\sqrt{d_{l_{1}}HL}\\) & \\(\\lambda\\sqrt{d_{l_{1}}HL}\\) & \n\\begin{tabular}{c} Assumptions 1 \\& 2 \\\\ Holds in the limit \\(L\\rightarrow\\infty\\) \\\\ \\end{tabular} \\\\ \\hline \\end{tabular}\n\\end{table}\nTable 1: Bayesian regret bounds for TS (i.e. PSRL)\\(\\mathcal{E}\\in\\Theta\\) should be viewed as \\(\\{(P_{h},r_{h})\\}_{h=1}^{H}\\in\\Theta\\). The space of all possible real-valued functions \\(\\{(P_{h},r_{h})\\}_{h=1}^{H}\\) has a natural vector space structure. Therefore it is meaningful to discuss the notion of the convex combination of environments. We assume that \\(\\Theta\\) is a convex subspace of the space of all possible environments. This assumption is not restrictive, since we may replace any environment space with its convex hull. Note that we do not assume that the support of the prior is convex.\n\n_Remark 1_.: The case of joint prior may be of interest, but to our knowledge all prior works also take \\(\\rho^{P},\\rho^{R}\\) to be independent.\n\nAgent, policy and history.An agent starts at an initial state \\(s^{\\ell}_{1}\\), which is fixed for all episodes \\(\\ell\\). It observes a state \\(s^{\\ell}_{h}\\) at layer \\(h\\) episode \\(\\ell\\), takes action \\(a^{\\ell}_{h}\\), and receives reward \\(r^{\\ell}_{h}\\). The environment changes to the next random state \\(s^{\\ell}_{h+1}\\) with probability \\(P_{h}(s^{\\ell}_{h+1}|s^{\\ell}_{h},a^{\\ell}_{h})\\). The agent stops acting at \\(s_{H+1}\\) and the environment is reset to its initial state.\n\nWe define \\(\\mathcal{H}_{\\ell,h}\\) as the history \\((s^{\\ell}_{1},a^{\\ell}_{1},r^{\\ell}_{1},\\ldots,s^{\\ell}_{h},a^{\\ell}_{h},r^{ \\ell}_{h})\\). Denote by \\(\\mathcal{D}_{\\ell}=\\left(\\mathcal{H}_{1,H},\\ldots,\\mathcal{H}_{\\ell-1,H}\\right)\\) the history up to episode \\(\\ell\\), where \\(\\mathcal{D}_{1}:=\\emptyset\\). Finally, let \\(\\Omega_{h}=\\prod_{i=1}^{h}(\\mathcal{S}\\times\\mathcal{A}\\times[0,1])\\) be the set of all possible histories up to layer \\(h\\).\n\nA policy \\(\\pi\\) is represented by stochastic maps \\((\\pi_{1},\\ldots,\\pi_{H})\\) where each \\(\\pi_{h}:\\Omega_{h-1}\\times\\mathcal{S}\\rightarrow\\Delta_{\\mathcal{A},\\mu_{ \\mathcal{A}}}\\). Let \\(\\Pi_{S}\\) denote the entire stationary policy class, stationary meaning a dependence only on the current state and layer and let \\(\\Pi\\subseteq\\Pi_{S}\\).\n\nValue and state occupancy functions.Define the value function \\(V^{\\mathcal{E}}_{h,\\pi}\\) as the value of the policy \\(\\pi\\) interacting with \\(\\mathcal{E}\\) at layer \\(h\\):\n\n\\[V^{\\mathcal{E}}_{h,\\pi}(s):=\\mathbb{E}^{\\mathcal{E}}_{\\pi}\\left[\\sum_{h^{\\prime }=h}^{H}r_{h^{\\prime}}(s_{h^{\\prime}},a_{h^{\\prime}})\\bigg{|}s_{h}=s\\right]\\,, \\tag{1}\\]\n\nwhere \\(\\mathbb{E}^{\\mathcal{E}}_{\\pi}\\) denotes the expectation over the trajectory under policy, transition, and reward functions \\(\\pi,P_{h},r_{h}\\). The value function at step \\(H+1\\) is set to null, \\(V^{\\mathcal{E}}_{H+1,\\pi}(\\cdot):=0\\). We assume there is a measurable function \\(\\pi^{*}_{\\mathcal{E}}:\\Theta\\rightarrow\\Pi\\) such that \\(V^{\\mathcal{E}}_{h,\\pi^{*}_{\\mathcal{E}}}(s)=\\max_{\\pi\\in\\Pi}V^{\\mathcal{E}}_{ h,\\pi}(s),\\ \\forall s\\in\\mathcal{S},h\\in[H]\\). The optimal policy \\(\\pi^{*}\\) is a function of \\(\\mathcal{E}\\), making it a random variable in the Bayesian setting. Lastly, let the _state-action occupancy probability measure_ be \\(\\mathbb{P}^{\\mathcal{E}}_{\\pi}(s_{h}=s,a_{h}=a)\\), also known as the state occupancy measure under policy \\(\\pi\\) and environment \\(\\mathcal{E}\\). It follows from the definitions that this measure is absolutely continuous with respect to \\(\\mu_{\\mathcal{S}\\times\\mathcal{A}}:=\\mu_{\\mathcal{S}}\\times\\mu_{\\mathcal{A}}\\). Let \\(d^{\\mathcal{E}}_{h,\\pi}(s,a)\\) denote the Radon-Nikodym derivative so that we have \\(d^{\\mathcal{E}}_{h,\\pi}(s,a)\\,\\mathrm{d}\\mu_{\\mathcal{S}\\times\\mathcal{A}}= \\mathrm{d}\\mathbb{P}^{\\mathcal{E}}_{\\pi}(s_{h}=s,a_{h}=a)\\). We will assume throughout the paper that this density \\(d^{\\mathcal{E}}_{h,\\pi}(s,a)\\) is measurable and upper bounded for all \\(\\pi,\\mathcal{E},s,a,h\\). The upper bound is a reasonable assumption, and it happens trivially in the tabular case (\\(d^{\\mathcal{E}}_{h,\\pi}(s,a)\\leq SA\\)). This also happens, e.g., when one assumes that the maps \\((\\mathcal{E},s,a,s^{\\prime},h)\\mapsto P^{\\mathcal{E}}_{h}(s^{\\prime}|s,a)\\) and \\((\\pi,s,a,h)\\mapsto\\pi_{h}(a|s)\\) are continuous and \\(\\Theta\\), \\(\\mathcal{S}\\), \\(\\mathcal{A}\\) and the set of all optimal policies (as a subset of \\(\\Pi\\)) are compact.\n\n### Bayesian regret\n\nWe formulate the expected regret over \\(L\\) episodes and \\(T=LH\\) total steps in an environment \\(\\mathcal{E}\\) as\n\n\\[\\mathfrak{R}_{L}(\\mathcal{E},\\pi)=\\mathbb{E}\\left[\\sum_{\\ell=1}^{L}\\left(V^{ \\mathcal{E}}_{1,\\pi^{\\mathcal{E}}_{\\mathcal{E}}}(s^{\\ell}_{1})-V^{\\mathcal{E}}_ {1,\\pi^{\\mathcal{E}}}(s^{\\ell}_{1})\\right)\\right]\\,, \\tag{2}\\]\n\nwhere the expectation is over the randomness of \\(\\pi=\\{\\pi^{\\ell}\\}_{\\ell}\\). The Bayesian regret is \\(\\mathfrak{B}\\mathfrak{R}_{L}(\\pi)=\\mathbb{E}[\\mathfrak{R}_{L}(\\mathcal{E},\\pi)]\\). For Thompson Sampling (TS), the algorithm selects the optimal policy of a given sample \\(\\mathcal{E}_{\\ell}\\) picked from the posterior \\(\\mathcal{E}_{\\ell}\\sim\\mathbb{P}(\\mathcal{E}\\in\\cdot|\\mathcal{D}_{\\ell})\\):\n\n\\[\\pi^{\\ell}_{\\text{TS}}=\\text{argmax}_{\\pi\\in\\Pi}V^{\\mathcal{E}_{\\ell}}_{1,\\pi}(s ^{\\ell}_{1})\\,. \\tag{3}\\]\n\nImportantly, the law of TS aligns with the posterior, i.e., \\(\\mathbb{P}(\\mathcal{E}|\\mathcal{D}_{\\ell})=\\mathbb{P}(\\pi^{\\ell}_{\\text{TS}}= \\pi^{*}_{\\mathcal{E}}|\\mathcal{D}_{\\ell})\\).\n\n_Remark 2_.: Note that \\(\\mathbb{P}(\\pi^{\\ell}_{\\text{TS}}=\\pi^{*}_{\\mathcal{E}}|\\mathcal{D}_{\\ell})\\) is a probability for a specific measure on the space of optimal policies. To ensure that \\(\\int_{\\Pi}\\mathbb{P}(\\pi^{*}|\\mathcal{D}_{\\ell})\\mathrm{d}\\rho_{\\Pi^{*}}=1\\), we need an appropriate measure \\(\\rho_{\\Pi^{*}}\\) on \\(\\Pi^{*}\\). Given the law of TS, the natural choice for this measure is the push-forward of the prior measure \\(\\rho\\) under the map \\(star:\\Theta\\rightarrow\\Pi^{*}\\), where \\(star(\\mathcal{E})=\\pi^{*}_{\\mathcal{E}}\\).\n\n### Notations\n\nFor Bayesian RL, conditional expressions involving a given history \\(\\mathcal{D}_{\\ell}\\) are widely used. We adopt the notation in [18] to refer to such conditionals; let \\(\\mathbb{P}_{\\ell}(\\cdot):=\\mathbb{P}(\\cdot|\\mathcal{D}_{\\ell})\\), \\(\\mathbb{E}_{\\ell}[\\cdot]:=\\mathbb{E}[\\cdot|\\mathcal{D}_{\\ell}]\\). We can rewrite the Bayesian regret as\n\n\\[\\mathfrak{B}\\mathfrak{R}_{L}(\\pi)=\\sum_{\\ell=1}^{L}\\mathbb{E}\\left[\\mathbb{E}_ {\\ell}\\left[V^{\\mathcal{E}}_{1,\\pi^{\\mathcal{E}}_{\\ell}}(s^{\\ell}_{1})-V^{ \\mathcal{E}}_{1,\\pi}(s^{\\ell}_{1})\\right]\\right] \\tag{4}\\]\n\nand define the conditional mutual information \\(\\mathbb{I}_{\\ell}(X;Y):=D_{\\text{KL}}(\\mathbb{P}((X,Y)\\in\\cdot|\\mathcal{D}_{ \\ell})||\\mathbb{P}(X\\in\\cdot|\\mathcal{D}_{\\ell})\\otimes\\mathbb{P}(Y\\in\\cdot| \\mathcal{D}_{\\ell}))\\). For a random variable \\(\\chi\\) and random policy \\(\\pi\\), the following will be involved in the information ratio:\n\n\\[\\mathbb{I}_{\\ell}^{\\pi}(\\chi;\\mathcal{H}_{\\ell,h}):=\\mathbb{I}_{\\ell}(\\chi; \\mathcal{H}_{\\ell,h}|\\pi)=\\mathbb{E}_{\\pi}[D_{\\text{KL}}(\\mathbb{P}_{\\ell}(( \\chi,\\mathcal{H}_{\\ell,h})\\in\\cdot|\\pi)||\\mathbb{P}_{\\ell}(\\chi\\in\\cdot|\\pi) \\otimes\\mathbb{P}_{\\ell}(\\mathcal{H}_{\\ell,h}\\in\\cdot|\\pi))]\\,, \\tag{5}\\]\n\nNote that \\(\\mathbb{E}[\\mathbb{I}_{\\ell}(X;Y)]=\\mathbb{I}(X;Y|\\mathcal{D}_{\\ell})\\). To clarify, \\(\\mathbb{P}_{\\ell}(\\mathcal{H}_{\\ell,h}\\in\\cdot|\\pi)\\) is the probability of \\(\\mathcal{H}_{\\ell,h}\\) being generated under \\(\\pi\\) within some environment. Given that the histories under consideration are generated by the TS algorithm, they are always generated in the true environment \\(\\mathcal{E}\\) under an optimal policy \\(\\pi^{*}_{\\mathcal{E}^{\\prime}}\\). For \\(\\pi=\\pi^{\\ell}_{\\text{TS}}\\), this can be computed as \\(\\mathbb{P}_{\\ell}(\\mathcal{H}_{\\ell,h}|\\pi)=\\int_{\\mathcal{E}}P(\\mathcal{H}_{ \\ell,h}|\\pi,\\mathcal{E})\\,\\mathrm{d}\\mathbb{P}_{\\ell}(\\mathcal{E})\\), where \\(P(\\mathcal{H}_{\\ell,h}|\\pi,\\mathcal{E})\\) is an expression in terms of transition and reward functions of \\(\\mathcal{E}\\) and \\(\\pi\\).\n\nFinally, we define \\(\\bar{\\mathcal{E}}_{\\ell}\\) as the mean MDP where \\(P^{\\bar{\\mathcal{E}}_{\\ell}}_{h}(\\cdot|s,a)=\\mathbb{E}_{\\ell}[P^{\\bar{ \\mathcal{E}}}_{h}(\\cdot|s,a)]\\) is the mean of posterior measure, and similarly for \\(r^{\\bar{\\mathcal{E}}_{\\ell}}_{h}(\\cdot|s,a)=\\mathbb{E}_{\\ell}[r^{\\bar{ \\mathcal{E}}}_{h}(\\cdot|s,a)]\\). We note that under the independence assumption across layers, the same is given for the state-occupancy density \\(d^{\\bar{\\mathcal{E}}_{\\ell}}_{h,\\pi}=\\mathbb{E}_{\\ell}[d^{\\bar{\\mathcal{E}}}_{ h,\\pi}]\\).\n\n## 3 Bayesian RL problems\n\n**Definition 1**.: A Bayesian RL in this paper refers to the time-inhomogeneous finite-horizon MDP with independent priors on transition and reward functions, as described in Section 2.1.\n\nThe Bayesian RL _problem_ is the task of finding an algorithm \\(\\pi\\) with optimal Bayesian regret as defined in Eq. (4). Below we list the variations of this problem. A setting considered by most related works such as [31, 16] is the following:\n\n**Definition 2**.: The **time (reward) homogeneous** Bayesian RL refers to the Bayesian RL setting where the prior \\(\\rho^{P}\\) (\\(\\rho^{R}\\)) is over the space \\(\\Theta^{P}\\) (\\(\\Theta^{R}\\)) containing the single transition (reward) function \\(P\\) (\\(r\\)) defining \\(\\mathcal{E}\\), i.e., all layers have the same transition (reward) functions.\n\n**Definition 3**.: The **tabular** Bayesian RL is a Bayesian RL where \\(\\mathcal{S},\\mathcal{A}\\) are finite sets.\n\n**Definition 4** (Linear MDP [41, 22]).: Let \\(\\phi^{P}:\\mathcal{S}\\times\\mathcal{A}\\rightarrow\\mathbb{R}^{d^{P}_{\\ell}}, \\phi^{R}:\\mathcal{S}\\times\\mathcal{A}\\rightarrow\\mathbb{R}^{d^{R}_{\\ell}}\\) be feature maps with bounded norm \\(\\|\\phi^{P}(s,a)\\|_{2},\\|\\phi^{R}(s,a)\\|_{2}\\leq 1\\). The **linear** Bayesian RL is a Bayesian RL where for any \\(\\mathcal{E}=\\{(P^{\\mathcal{E}}_{h},r^{\\mathcal{E}}_{h})\\}_{h=1}^{H}\\in\\Theta\\), there exists vector-valued maps \\(\\psi^{P,\\mathcal{E}}_{h}(s),\\psi^{R,\\mathcal{E}}_{h}(s)\\) with bounded \\(l_{2}-\\)norm such that for any \\((s,a)\\in\\mathcal{S}\\times\\mathcal{A}\\),\n\n\\[P^{\\mathcal{E}}_{h}(\\cdot|s,a)=\\langle\\phi^{P}(s,a),\\psi^{P,\\mathcal{E}}_{h}( \\cdot)\\rangle\\,,\\;\\;r^{\\mathcal{E}}_{h}(\\cdot|s,a)=\\langle\\phi^{R}(s,a), \\psi^{R,\\mathcal{E}}_{h}(\\cdot)\\rangle \\tag{6}\\]\n\nA restricted version of the finite mixtures called linear mixture was first considered in [6] in the frequentist setting. Here, we consider the general setting.\n\n**Definition 5**.: The **finite mixtures** Bayesian RL is a Bayesian RL where for any \\(h\\in[H]\\) there exists fixed conditional distributions \\(\\{Z^{P}_{h,i}:\\mathcal{S}\\times\\mathcal{A}\\rightarrow\\Delta_{S}\\}_{i=1}^{m^{P}_ {h}}\\) and \\(\\{Z^{R}_{h,i}:\\mathcal{S}\\times\\mathcal{A}\\rightarrow\\Delta_{[0,1]}\\}_{i=1}^{m^{ R}_{h}}\\), such that for any environment \\(\\mathcal{E}\\) given by \\(\\{(P^{\\mathcal{E}}_{h},r^{\\mathcal{E}}_{h})\\}_{h=1}^{H}\\), there exists parametrized probability distributions \\(\\mathbf{a}^{P,\\mathcal{E}}_{h}:\\mathcal{S}\\times\\mathcal{A}\\rightarrow\\Delta_{m^{ P}_{h}},\\mathbf{a}^{R,\\mathcal{E}}_{h}:\\mathcal{S}\\times\\mathcal{A}\\rightarrow\\Delta_{m^{ R}_{h}}\\) such that\n\n\\[P^{\\mathcal{E}}_{h}(\\cdot|s,a)=\\sum_{i=1}^{m^{P}_{h}}a^{P,\\mathcal{E}}_{h,i}(s,a )Z^{P}_{h,i}(\\cdot|s,a),\\;\\;r^{\\mathcal{E}}_{h}(\\cdot|s,a)=\\sum_{i=1}^{m^{R}_{h }}a^{R,\\mathcal{E}}_{h,i}(s,a)Z^{R}_{h,i}(\\cdot|s,a) \\tag{7}\\]Surrogate learning\n\nNext, we define the discretized surrogate learning problem, and bound the size of the surrogate environments space, a significant term in the regret. To do so, we need to first define the Kolmogorov dimension of a set of parametrized distributions, esp. working out the case of \\(l_{1}-\\)distance. In the definitions below, we implicitly assume any required minimal measurability assumptions on the involved sets.\n\n**Definition 6**.: Given a set \\(\\mathcal{F}\\) of \\(\\mathcal{O}-\\)parametrized distributions \\(P:\\mathcal{O}\\to\\Delta(\\mathcal{S})\\) over a set \\(\\mathcal{S}\\) where both \\(\\mathcal{O},\\mathcal{S}\\) are measurable. Let \\(\\mathcal{M}(\\cdot,\\cdot):\\mathcal{F}\\times\\mathcal{F}\\to\\mathbb{R}^{\\geq 0}\\) be a _distance_, i.e., \\(\\mathcal{M}(P,Q)\\geq 0\\stackrel{{\\leftrightarrow}}{{\\rightleftarrows}}P=Q\\). Then its right \\(\\varepsilon-\\)covering number is the size \\(K_{\\mathcal{M}}(\\varepsilon)\\) of the smallest set \\(\\mathcal{C}_{\\mathcal{M}}(\\varepsilon)=\\{P_{1},\\ldots,P_{K_{\\mathcal{M}}( \\varepsilon)}\\}\\subset\\mathcal{F}\\) such that\n\n\\[\\forall P\\in\\mathcal{F},\\ \\exists P_{j}\\in\\mathcal{C}_{\\mathcal{M}}( \\varepsilon):\\ \\mathcal{M}(P,P_{j})\\leq\\varepsilon\\,. \\tag{8}\\]\n\nThe potential asymmetry of \\(\\mathcal{M}\\) (e.g., KL-divergence) requires the notion of left/right covering number. The right covering number will be the default, so covering number will always refer to that.\n\n**Definition 7**.: Let \\(d_{\\mathcal{M}}(\\varepsilon)=\\log(K_{\\mathcal{M}}(\\varepsilon))\\). Define the Kolmogorov \\(\\mathcal{M}-\\)dimension \\(d_{\\mathcal{M}}\\) of \\(\\mathcal{F}\\) as\n\n\\[d_{\\mathcal{M}}=\\limsup_{\\varepsilon\\to 0}\\frac{d_{\\mathcal{M}}( \\varepsilon)}{\\log(\\frac{1}{\\varepsilon})}. \\tag{9}\\]\n\nFor \\(l_{1}(P,Q):=\\sup_{o\\in\\mathcal{O}}||P(\\cdot|o)-Q(\\cdot|o)||_{1}\\), applying Definition 6 to the sets \\(\\Theta_{h}^{P},\\Theta_{h}^{R}\\) with \\(\\mathcal{O}=\\mathcal{S}\\times\\mathcal{A}\\), and denote the respective covering numbers by \\(L_{h}^{P}(\\varepsilon),L_{h}^{R}(\\varepsilon)\\) corresponding to covering sets \\(\\mathcal{C}_{h}^{P}(\\varepsilon),\\mathcal{C}_{h}^{R}(\\varepsilon)\\). Similarly applying Eq. (9) and denote the corresponding \\(l_{1}-\\)dimensions by \\(d_{l_{1}h}^{P}(\\varepsilon),d_{l_{1}h}^{R}(\\varepsilon),d_{l_{1}h}^{P}( \\varepsilon),d_{l_{1}h}^{P}(\\varepsilon),d_{l_{1}h}^{P},d_{l_{1}h}^{R}\\) and \\(d_{l_{1}}^{P}:=\\sum_{h}d_{l_{1}h}^{P},d_{l_{1}}^{R}:=\\sum_{h}d_{l_{1}h}^{R}\\). The sums \\(d_{l_{1},h}:=d_{l_{1}h}^{P}+d_{l_{1}h}^{R},d_{l_{1}}:=d_{l_{1}}^{P}+d_{l_{1}h}^{R}\\) can be interpreted as the \\(l_{1}-\\)dimension of \\(\\Theta_{h}\\) and \\(\\Theta\\), i.e., the environment space.\n\n_Remark 3_.: We can also apply this framework to the KL-divergence, by \\(\\mathcal{M}_{\\mathrm{KL}}(P,Q):=\\sup_{o\\in\\mathcal{O}}D_{\\mathrm{KL}}(P(\\cdot| o)||Q(\\cdot||o))\\). This was implicitly used by [18] to prove their regret bound in the tabular case. Note that Pinsker's lemma (Lemma 9) implies that the KL-divergence is larger than the squared total variance, and the latter is trivially larger than the \\(l_{1}\\) distance. Therefore, \\(l_{1}-\\)dimension is smaller than \\(d_{\\mathcal{M}_{\\mathrm{KL}}}\\), allowing for tighter regret bounds.\n\nWe now revisit the definition of \\(\\varepsilon-\\)value partitions and show their existence is guaranteed by finite \\(l_{1}-\\)covering numbers. These partitions are the origins of surrogate environments.\n\n**Definition 8**.: Given \\(\\varepsilon>0\\), an \\(\\varepsilon-\\)value partition for a Bayesian RL problem is a partition \\(\\{\\Theta_{k}\\}_{k=1}^{K}\\) over \\(\\Theta\\) such that for any \\(k\\in[K]\\) and \\(\\mathcal{E},\\mathcal{E}^{\\prime}\\in\\Theta_{k}\\),\n\n\\[V_{1,\\pi_{\\mathcal{E}}^{\\prime}}^{\\mathcal{E}}(s_{1}^{\\ell})-V_{1,\\pi_{ \\mathcal{E}}^{\\prime}}^{\\mathcal{E}^{\\prime}}(s_{1}^{\\ell})\\leq\\varepsilon\\,. \\tag{10}\\]\n\nA _layered_\\(\\varepsilon-\\)value partition is one where the transition functions are independent over layers after conditioning on \\(k\\). Throughout this paper, we will only consider layered \\(\\varepsilon-\\)value partition. We define \\(K_{\\mathrm{surr}}(\\varepsilon)\\) as the minimum \\(K\\) for which there exists a layered \\(\\varepsilon-\\)value partition.\n\nInspired by Eq. (9), we define the surrogate dimension as \\(d_{\\mathrm{surr}}=\\limsup_{\\varepsilon\\to 0}\\frac{K_{\\mathrm{surr}}( \\varepsilon)}{\\log(1/\\varepsilon)}\\).\n\n**Lemma 1**.: _Given a Bayesian RL, we have \\(K_{\\mathrm{surr}}(\\varepsilon)\\leq\\prod_{h}L_{h}^{P}(\\varepsilon/(2H)^{2}) \\times L_{h}^{R}(\\varepsilon/(4H))\\). This implies \\(d_{\\mathrm{surr}}\\leq d_{l_{1}}\\)._\n\nThe above is proved in Appendix B. It is hard to find \\(d_{\\mathrm{surr}}\\), but one can estimate \\(d_{l_{1}}\\), and according to the above, this acts as a proxy for \\(K_{\\mathrm{surr}}\\). This is useful as the regret relates to \\(K_{\\mathrm{surr}}\\). But to show this, we need to construct _surrogate environments_ inside each partition, and show that learning those is almost equivalent to the original problem. Let \\(\\zeta\\) be a discrete random variable taking values in \\(\\{1,\\cdots,K_{\\mathrm{surr}}(\\varepsilon)\\}\\) that indicates the partition \\(\\mathcal{E}\\) lies in, such that \\(\\zeta=k\\) if and only if \\(\\mathcal{E}\\in\\Theta_{k}\\).\n\n**Lemma 2**.: _For any \\(\\varepsilon-\\)value partition and any \\(\\ell\\in[L]\\), there are random environments \\(\\tilde{\\mathcal{E}}_{\\ell}^{*}\\in\\Theta\\) with their laws only depending on \\(\\zeta,\\mathcal{D}_{\\ell}\\), such that_\n\n\\[\\mathbb{E}_{\\ell}\\left[V_{1,\\pi_{\\mathcal{E}}^{*}}^{\\mathcal{E}}(s_{1}^{\\ell} )-V_{1,\\pi_{\\mathcal{E}}^{\\prime}}^{\\mathcal{E}}(s_{1}^{\\ell})\\right]- \\mathbb{E}_{\\ell}\\left[V_{1,\\pi_{\\mathcal{E}}^{*}}^{\\mathcal{E}^{*}}(s_{1}^{ \\ell})-V_{1,\\pi_{\\mathcal{E}}^{*}}^{\\mathcal{E}^{*}}(s_{1}^{\\ell})\\right]\\leq \\varepsilon\\,. \\tag{11}\\]_The expectation in both equations is over \\(\\mathcal{E}\\) and \\(\\pi_{\\mathcal{F}}^{\\ell}\\in\\{\\pi_{\\mathcal{E}^{\\prime}}^{*}\\}_{\\mathcal{E}^{\\prime }\\in\\Theta}\\), with both sampled independently \\(\\sim\\mathbb{P}_{\\ell}(\\cdot)\\), and the \\(K\\) different values of \\(\\tilde{\\mathcal{E}}_{t}^{*}\\). The second expectation over \\((\\tilde{\\mathcal{E}}_{t}^{*},\\mathcal{E})\\) is over pairs that are in the same partition, i.e., \\(\\tilde{\\mathcal{E}}_{\\ell}^{*},\\mathcal{E}\\) are independent only after conditioning on \\(\\zeta\\)._\n\nWe note that the proof in [18, App. B.1] contains the use of a lemma that does not apply to construct the law of the environment \\(\\tilde{\\mathcal{E}}_{\\ell}^{*}\\). More details is provided in Appendix C, where we find \\(\\tilde{\\mathcal{E}}_{\\ell}^{*}\\) by minimizing an expected value of \\(\\pi_{\\text{TS}}^{\\ell}\\).\n\n## 5 Bayesian regret bounds for Thompson Sampling\n\n### General Bayesian regret bound\n\nWe start by introducing the notion of value diameter.\n\n**Definition 9**.: Given the environment \\(\\mathcal{E}\\), its value diameter is defined as\n\n\\[\\lambda_{\\mathcal{E}}:=\\max_{1\\leq h\\leq H}(\\sup_{s}V_{h,\\pi_{\\mathcal{E}}^{*} }^{\\mathcal{E}}(s)-\\inf_{s}V_{h,\\pi_{\\mathcal{E}}^{*}}^{\\mathcal{E}}(s))+\\max _{1\\leq h\\leq\\mathcal{E},s\\in\\mathcal{S},a\\in\\mathcal{A}}(r_{h}^{\\sup}(s,a)-r _{h}^{\\inf}(s,a)),\\]\n\nwhere \\(r_{h}^{\\sup}(s,a)\\) (and \\(r_{h}^{\\inf}(s,a)\\)) is the supremum (and infimum) of the set of rewards that are attainable under the distribution \\(r_{h}(s,a)\\) with non-zero probability. As a special case, if rewards are deterministic, then we have \\(r_{h}^{\\sup}(s,a)=r_{h}^{\\inf}(s,a)\\) for all \\(s,a\\). The (average) value diameter over \\(\\Theta\\) is denoted by \\(\\lambda:=\\mathbb{E}_{\\mathcal{E}\\sim\\rho}[\\lambda_{\\mathcal{E}}^{2}]^{1/2}\\).\n\nAs the value function is between \\(0\\) and \\(H\\), we have \\(\\lambda_{\\mathcal{E}}\\leq H+1\\) implying \\(\\lambda\\leq H+1\\). Note that value diameter is closely related to the notion of diameter commonly defined in finite RL problems. Strictly speaking, for a time-homogeneous RL, it is straightforward to see that the value diameter is bounded from above by one plus the diameter [33].\n\nWe now discuss the assumptions surrounding our results. The main technical assumption of this paper is the existence of consistent estimators, which as we will see in Appendix K, is closely related to the notion of posterior consistency:\n\n**Assumption 1**.: _There exists a strongly consistent estimator of the true environment given the history._\n\nRoughly speaking, we assume that with unlimited observations under TS, it is possible to find the true environment. For this assumption to fail, we need to have two environments that produce the same distribution over histories under TS and are therefore indistinguishable from the point of view of TS. The precise description of this assumption is detailed in Appendix K.\n\nAnother necessary technical assumption is that almost all optimal policies visit almost all state action pairs in their respective environment.\n\n**Assumption 2**.: _For almost every environment \\(\\mathcal{E}\\in\\Theta\\) and almost every \\((s,a)\\in\\mathcal{S}\\times\\mathcal{A}\\) and every \\(h\\in[H]\\), we have_\n\n\\[d_{h,\\pi_{\\mathcal{E}}^{*}}^{\\mathcal{E}}(s,a)\\neq 0.\\]\n\nRecall that, for any environment \\(\\mathcal{E}\\in\\Theta\\), the policy \\(\\pi_{\\mathcal{E}}^{*}\\) is the optimal policy of \\(\\mathcal{E}\\) within the policy class \\(\\Pi\\). Therefore, one example of how the above assumption holds is when \\(\\Pi\\) is the set of \\(\\varepsilon\\)-greedy algorithms and transition functions of environments assign non-zero probability to every state. Under these assumptions, we discuss our main result and its corollaries.\n\n**Theorem 3**.: _Given a Bayesian RL problem, for all \\(\\varepsilon>0\\), we have_\n\n\\[\\mathfrak{B}\\mathfrak{R}_{L}(\\pi_{\\text{TS}})\\leq 2\\lambda\\sqrt{\\log(K_{\\text{ surr}}(\\varepsilon))T}+L\\varepsilon+T_{0} \\tag{12}\\]\n\n_where \\(T_{0}\\) does not depend on \\(T\\). This can be further upper bounded by_\n\n\\[\\mathfrak{B}\\mathfrak{R}_{L}(\\pi_{\\text{TS}})\\leq\\widetilde{O}(\\lambda\\sqrt{d_ {l_{1}}T})\\,. \\tag{13}\\]\n\n_for large enough \\(T\\). Given a homogeneous \\(l_{1}\\) dimension \\(d_{\\mathrm{hom}}=d_{l_{1},h},\\forall h\\), this simplifies to_\n\n\\[\\mathfrak{B}\\mathfrak{R}_{L}(\\pi_{\\text{TS}})\\leq\\widetilde{O}(\\lambda\\sqrt{Hd _{\\mathrm{hom}}T})\\,. \\tag{14}\\]\n\n_Remark 4_.: For all regret bounds, we will replace \\(\\lambda\\leq H+1\\) to compare our result. For the case of homogeneous dimensions, we obtain \\(\\widetilde{O}(H^{3/2}\\sqrt{d_{\\mathrm{hom}}T})\\). Crucially, our main result shows a new conceptual understanding of the information ratio by bounding it by two terms of different nature: \\(H\\) and \\(\\lambda\\), where the latter can be bounded by either the largest diameter of the environments or \\(H\\).\n\n_Remark 5_.: Despite not impacting the asymptotics, the impact of \\(T_{0}\\) can be large depending on the structure of the RL problem, and could be dominant even for large \\(T\\)s in practice.\n\n_Remark 6_.: Considering time as a part of the state observation, one could apply this regret analysis to particular time-homogeneous settings. However, this mapping of time-inhomogeneous RLs to homogeneous ones is not surjective, hence the result above does not readily extend to time-homogeneous settings.\n\nWhile [16] were the first to consider a nonlinear Bayesian RL model, their bound is limited to the Gaussian process (with linear kernel) setting, while ours in the nonlinear time inhomogeneous setting makes no assumptions on the prior and is the first such bound. Our novel analysis allow us to upper bound the information ratio by \\(\\lambda\\sqrt{H}\\) instead of, for example \\(H^{3/2}\\sqrt{SA}\\) ([18]) in the tabular case, improving the regret bound by a square root relevant to the dimension \\(d\\) of the problem.\n\nThe detailed proof is given in Appendix D. Following [18], the regret (4) is rewritten using Lemma 2 to reduce the problem into its surrogate, and we use the well-known information-ratio trick by multiplying and dividing by the mutual information. We follow that with a Cauchy-Schwarz, summarized below\n\n\\[\\mathfrak{B}\\mathfrak{R}_{L}(\\pi_{\\text{TS}}) \\leq\\mathbb{E}\\left[\\sum_{\\ell=1}^{L}\\frac{\\mathbb{E}_{\\ell}\\left[ V_{1,\\pi_{\\ell}}^{\\mathcal{E}_{\\epsilon}^{*}}(s_{1}^{\\ell})-V_{1,\\pi_{\\text{TS}} ^{\\mathcal{E}_{\\epsilon}^{*}}}^{\\mathcal{E}_{\\epsilon}^{*}}(s_{1}^{\\ell}) \\right]}{\\sqrt{\\mathbb{I}_{\\ell}^{\\pi_{\\text{TS}}^{\\mathcal{E}_{\\epsilon}^{*} }}(\\tilde{\\mathcal{E}}_{\\ell}^{*};\\mathcal{H}_{\\epsilon,H})}}\\right]+L\\varepsilon \\tag{15}\\] \\[\\leq\\sqrt{\\mathbb{E}\\left[\\sum_{\\ell=1}^{L}\\frac{\\left(\\mathbb{E}_ {\\ell}\\left[V_{1,\\pi_{\\ell}}^{\\mathcal{E}_{\\epsilon}^{*}}(s_{1}^{\\ell})-V_{1, \\pi_{\\text{TS}}^{\\mathcal{E}_{\\epsilon}^{*}}}^{\\mathcal{E}_{\\epsilon}^{*}}(s_{ 1}^{\\ell})\\right]\\right)^{2}}{\\mathbb{I}_{\\ell}^{\\pi_{\\text{TS}}^{\\mathcal{E} _{\\epsilon}^{*}}}(\\tilde{\\mathcal{E}}_{\\ell}^{*};\\mathcal{H}_{\\epsilon,H})} \\right]}\\mathbb{E}\\left[\\sum_{\\ell=1}^{L}\\mathbb{I}_{\\ell}^{\\pi_{\\text{TS}}^{ \\mathcal{E}_{\\epsilon}^{*}}}(\\tilde{\\mathcal{E}}_{\\ell}^{*};\\mathcal{H}_{ \\epsilon,H})}\\right]+L\\varepsilon \\tag{16}\\]\n\nNote the cost \\(\\varepsilon\\) at each episode (Lemma 2) in the first inequality, yielding the overall error \\(L\\varepsilon\\). Then, we can bound the mutual information appearing in the regret term by \\(\\mathbb{E}\\left[\\sum_{\\ell=1}^{L}\\mathbb{I}_{\\ell}^{\\pi_{\\text{TS}}^{\\mathcal{ E}_{\\epsilon}^{*}}}(\\tilde{\\mathcal{E}}_{\\ell}^{*};\\mathcal{H}_{\\epsilon,H}) \\right]=I_{\\ell}^{\\pi_{\\text{TS}}^{\\mathcal{E}_{\\epsilon}^{*}}}(\\tilde{ \\mathcal{E}}_{\\ell}^{*};\\mathcal{D}_{\\ell})\\leq I_{\\ell}^{\\pi_{\\text{TS}}^{ \\mathcal{E}_{\\epsilon}^{*}}}(\\zeta;\\mathcal{D}_{\\ell})\\leq\\log(K_{\\text{surr}} (\\varepsilon))\\), where we used the mutual information chain rule, followed by data processing inequality to substitute \\(\\tilde{\\mathcal{E}}_{\\ell}^{*}\\rightarrow\\zeta\\), and finally used the trivial bound by the entropy. But the main novelty of our approach lies in our control of the first term\n\n\\[\\Gamma_{\\ell}(\\pi_{\\text{TS}}^{\\ell}):=\\frac{\\left(\\mathbb{E}_{\\ell}\\left[V_{1,\\pi_{\\ell}}^{\\mathcal{E}_{\\epsilon}^{*}}(s_{1}^{\\ell})-V_{1,\\pi_{\\text{TS}}^{ \\mathcal{E}_{\\epsilon}^{*}}}^{\\mathcal{E}_{\\epsilon}^{*}}(s_{1}^{\\ell})\\right] \\right)^{2}}{\\mathbb{I}_{\\ell}^{\\pi_{\\text{TS}}^{\\mathcal{E}_{\\epsilon}^{*}}}( \\tilde{\\mathcal{E}}_{\\ell}^{*};\\mathcal{H}_{\\epsilon,H})} \\tag{17}\\]\n\ncalled the information ratio. In our analysis, we have the following bound on its expectation.\n\n\\[\\mathbb{E}[\\Gamma_{\\ell}(\\pi_{\\text{TS}}^{\\ell})\\mid\\mathcal{E}_{0}]\\leq \\mathbb{E}\\left[\\sum_{h}\\int\\frac{\\mathbb{E}_{\\ell}\\left[(\\lambda_{\\mathcal{E} }d_{h,\\pi^{*}}^{\\mathcal{E}_{\\epsilon}^{\\prime}}(s,a))^{2}\\right]}{\\mathbb{E}_{ \\ell}\\left[d_{h,\\pi^{*}}^{\\mathcal{E}_{\\epsilon}}(s,a)\\right]}\\mu_{\\mathcal{S }\\times\\mathcal{A}}\\mid\\mathcal{E}_{0}\\right],\\]\n\nwhere the average is taken over all histories \\(\\mathcal{D}_{\\ell}\\) that are generated from running TS on the true environment \\(\\mathcal{E}_{0}\\), and we have introduced the smaller term \\(\\lambda_{\\mathcal{E}}\\) instead of \\(H\\) in [18]. While [18] essentially bound the above only in the tabular setting with \\(SAH^{3}\\), we manage to generally bound the above with a more precise bound using Doob's consistency theorem. Assumption 1 allows us to use Doob's consistency theorem to conclude that for almost every environment \\(\\mathcal{E}_{0}\\), almost every infinite sequence of histories \\((\\mathcal{D}_{\\ell})_{\\ell=1}^{\\infty}\\) sampled from \\(\\mathcal{E}_{0}\\), and every integrable function \\(f\\), the posterior mean \\(\\mathbb{E}_{\\ell}[f(\\mathcal{E})]=\\mathbb{E}[f(\\mathcal{E})\\mid\\mathcal{D}_{ \\ell}]\\) converges to \\(f(\\mathcal{E}_{0})\\). In particular, we conclude that \\(\\mathbb{E}[\\Gamma_{\\ell}(\\pi_{\\text{TS}}^{\\ell})\\mid\\mathcal{E}_{0}]\\) tends to \\(\\lambda_{\\mathcal{E}_{0}}^{2}H\\) in the limit, allowing us to claim that for large enough \\(\\ell\\), the expected information ratio \\(\\mathbb{E}[\\Gamma_{\\ell}(\\pi_{\\text{TS}}^{\\ell})]\\) is uniformly bounded by \\(2\\mathbb{E}[\\lambda_{\\mathcal{E}}^{2}]H=2\\lambda^{2}H\\). As there are \\(L\\) many such ratios, the two bounds together yield \\(2\\sqrt{\\lambda^{2}HL}\\cdot\\sqrt{\\log(K_{\\text{surr}}(\\varepsilon))}+L\\varepsilon\\). This bound is true for large enough \\(\\ell\\), giving the additional additive term \\(T_{0}\\) in the theorem. Since this term is additive, applying Lemma 1 to bound \\(\\log(K_{\\text{surr}}(\\varepsilon))\\), we have successfully shown the asymptotic behavior of the regret, independent of the prior, is of order \\(\\widetilde{O}(H\\sqrt{d_{l_{1}}T})\\).\n\n### Applications\n\nIn each application below, the challenge is to bound \\(d_{l_{1}}\\) using the specifics of the model, and except for the case of tabular Bayesian RL, such analysis has not been carried out rigorously. We formalize the corollaries and show they are state-of-the-art compared to the literature.\n\nTabular RL.The result below follows from Theorem 3; the main contribution comes from our new information ratio bound, followed by the estimate \\(\\widetilde{O}((\\frac{1}{\\varepsilon})^{SAH})\\) of \\(K_{\\mathrm{surr}}(\\varepsilon)\\) ([18]).\n\n**Corollary 4**.: _Given a tabular Bayesian RL problem, for large enough \\(T\\),_\n\n\\[\\mathfrak{B}\\mathfrak{R}_{L}(\\pi_{\\mathsf{TS}})\\leq\\widetilde{O}(\\lambda\\sqrt{ HSAT})\\,, \\tag{18}\\]\n\n_where the polylogarithmic terms are explicitly in terms of \\(H,S,A,L\\)._\n\nWe observe that our result matches [31] when their result in the time homogeneous setting (Definition 2) is extended to time inhomogeneous. However, in that paper, the authors assume a Dirichlet based prior which we do not.\n\nLinear RL.A previous state-of-the-art \\(\\widetilde{O}(d_{f}H^{3/2}\\sqrt{T})\\) was claimed by [18] to hold for linear Bayesian RLs with deterministic reward. We note:\n\n* As in the previous cases, their proof in bounding their information ratio includes a factor of \\(d_{f}\\), which ours avoids.\n* We show that the proof bounding \\(K_{\\mathrm{surr}}(\\varepsilon)\\) in [18, App. B.4] is incorrect, starting with a wrong application of Cauchy-Schwarz and a wrong mutual information in their definition of information ratio. We provide counterexamples for the estimates found therein to substantiate our claim (see Appendix G.1).\n\nTo state our own corollary in this case, we need to define a few notions. Let \\(d^{f}_{l_{1}}=d^{P,f}_{l_{1}}+d^{R,f}_{l_{1}}\\) be the sum of the \\(l_{1}-\\)dimensions of the feature map space \\(\\{\\psi^{P,\\mathcal{E}}_{h}\\}_{\\mathcal{E}\\in\\Theta},\\{\\psi^{R,\\mathcal{E}}_{h}\\} _{\\mathcal{E}\\in\\Theta}\\) where the \\(l_{1}-\\)distance between feature maps is defined as \\(l_{1}(\\psi^{\\mathcal{E}}_{h},\\psi^{\\mathcal{E}^{\\prime}}_{h})=\\int_{s}\\|\\psi^{ \\mathcal{E}}_{h}-\\psi^{\\mathcal{E}^{\\prime}}_{h}\\|_{1}\\mu_{\\mathcal{S}}\\). Our corollary also provides a concrete bound in the case of _mixture_ linear Bayesian RL where the feature maps are themselves a sum of finitely many **fixed** feature maps. This means for all \\(\\mathcal{E}\\in\\Theta\\), we have\n\n\\[\\psi^{P,\\mathcal{E}}_{h}=\\sum_{i=1}^{m_{h}^{P}}a^{P,\\mathcal{E}}_{h,i}\\Psi^{P} _{h,i}(s),\\ \\ \\psi^{R,\\mathcal{E}}_{h}=\\sum_{i=1}^{m_{h}^{R}}a^{R,\\mathcal{E}}_{h,i} \\Psi^{R}_{h,i}(s) \\tag{19}\\]\n\nwhere \\(\\{\\Psi^{P}_{h,i}(s)\\}_{i=1}^{m_{h}^{P}},\\{\\Psi^{R}_{h,i}(s)\\}_{i=1}^{m_{h}^{R}}\\) are finitely many fixed feature maps and \\(\\forall\\mathcal{E},h:\\sum_{i}|a^{P,\\mathcal{E}}_{h,i}|^{2},\\sum_{i}|a^{R, \\mathcal{E}}_{h,i}|^{2}\\leq C_{a}\\) for some constant \\(C_{a}>0\\). Let \\(M=M^{P}+M^{R}=\\sum_{h}m_{h}^{P}+\\sum_{h}m_{h}^{R}\\).\n\n**Corollary 5**.: _For a linear Bayesian RL, for large enough \\(T\\),_\n\n\\[\\mathfrak{B}\\mathfrak{R}_{L}(\\pi_{\\mathsf{TS}})\\leq\\widetilde{O}(\\lambda\\sqrt{ d^{f}_{l_{1}}T}). \\tag{20}\\]\n\n_Given a linear Bayesian RL with finitely many states and total feature space dimension \\(d_{f}=d^{P}_{f}+d^{R}_{f}\\), we have \\(d_{l_{1}}\\leq 2d_{f}HS\\), yielding for large enough \\(T\\),_\n\n\\[\\mathfrak{B}\\mathfrak{R}_{L}(\\pi_{\\mathsf{TS}})\\leq\\widetilde{O}(\\lambda\\sqrt{ Hd_{f}ST}). \\tag{21}\\]\n\n_Given a mixture linear Bayesian RL, for large enough \\(T\\),_\n\n\\[\\mathfrak{B}\\mathfrak{R}_{L}(\\pi_{\\mathsf{TS}})\\leq\\widetilde{O}(\\lambda\\sqrt{ MT})\\,, \\tag{22}\\]\n\nThe proof is given in Appendix G. The fact that \\(d_{l_{1}}\\) appears instead of \\(d_{f}\\) in the general bound is not counter-intuitive, as we should expect the complexity of the feature map space \\(\\{\\psi^{P,\\mathcal{E}}_{h}(s)\\}_{\\mathcal{E}\\in\\Theta,h\\in[H]},\\{\\psi^{R, \\mathcal{E}}_{h}(s)\\}_{\\mathcal{E}\\in\\Theta,h\\in[H]}\\) to play a role in the regret, especially as this space can be very complex, and model very different environments that can not be grouped in the same \\(\\varepsilon-\\)value partition.\n\nTherefore, opposite to the claim made by [18], this complexity can not be captured by simply \\(d_{f}\\) except maybe in degenerate cases, such as when \\(\\mathcal{S}\\) is finite, which is our second statement. More generally, if each feature map \\(\\psi^{P,\\mathcal{E}}_{h}(s),\\psi^{R,\\mathcal{E}}_{h}(s)\\) can be characterized with a vector of uniformly bounded norm \\(\\mathbf{a}^{P,\\mathcal{E}}_{h}\\in\\mathbb{R}^{m_{h}^{P}},\\mathbf{a}^{R,\\mathcal{E}}_{h }\\in\\mathbb{R}^{m_{h}^{R}}\\), then we can bound the regret in terms of \\(m_{h}^{P},m_{h}^{R}\\)s, as is done in Eq. (22) (the finite state case corresponds to \\(m_{h}^{P}=d^{P}_{f}S,m_{h}^{R}=d^{R}_{f}S\\)).\n\nFinite mixtures RL.To state our finite mixtures model result, we need to set the following notations. Let \\(d_{l_{1}}^{m_{1}}=d_{l_{1}}^{m,P}+d_{l_{1}}^{m,R}=\\sum_{h}d_{l_{1},h}^{m,P}+\\sum_{ h}d_{l_{1},h}^{m,R}\\) correspond to the total \\(l_{1}-\\)dimension of the space of mixtures coefficient maps \\(\\{\\mathbf{a}_{h}^{P,\\mathcal{E}}(s,a)\\}_{\\mathcal{E}\\in\\Theta},\\{\\mathbf{a}_{h}^{R, \\mathcal{E}}(s,a)\\}_{\\mathcal{E}\\in\\Theta}\\) with \\(l_{1}-\\)distance defined as \\(l_{1}(\\mathbf{a}_{h}^{\\mathcal{E}},\\mathbf{a}_{h}^{\\mathcal{E}^{\\prime}})=\\sup_{s,a}\\| \\mathbf{a}_{h}^{\\mathcal{E}}(s,a)-\\mathbf{a}_{h}^{\\mathcal{E}^{\\prime}}(s,a)\\|_{1}\\). Define also the restricted finite mixtures model where \\(\\mathbf{a}_{h}^{P,\\mathcal{E}},\\mathbf{a}_{h}^{R,\\mathcal{E}}\\) are vectors in \\(\\mathbb{R}^{m_{h}},\\mathbb{R}^{m_{h}^{R}}\\) independent of \\((s,a)\\) and let \\(M=M^{P}+M^{R}=\\sum_{h}m_{h}^{P}+\\sum_{h}m_{h}^{R}\\).\n\n**Corollary 6**.: _Given a finite mixtures Bayesian RL problem, for large enough \\(T\\),_\n\n\\[\\mathfrak{BV}_{L}(\\pi_{\\text{TS}})\\leq\\widetilde{O}(\\lambda\\sqrt{d_{l_{1}}^{m }T}). \\tag{23}\\]\n\n_Assuming the restricted finite mixtures model, for large enough \\(T\\),_\n\n\\[\\mathfrak{BV}_{L}(\\pi_{\\text{TS}})\\leq\\widetilde{O}\\left(\\lambda\\sqrt{MT} \\right). \\tag{24}\\]\n\n_which, given a uniform dimension \\(m=m_{h}^{P}=m_{h}^{R}\\), yields \\(\\widetilde{O}(\\lambda\\sqrt{HmT})\\)._\n\nWe prove the above in Appendix H, deriving it from our generic bound, after relating the \\(l_{1}-\\)dimension \\(d_{l_{1}}\\) of the environment space to that of the mixtures coefficients. To the best of our knowledge, this is the first bound for finite mixtures Bayesian RL problems. We note that in a previous work ([6]), a restricted version of finite mixtures, like in Eq. (24), was considered in the frequentist setting.\n\nWe finish this section by proposing the following conjecture, in line with [31, Conj. 1].\n\n**Conjecture 7**.: _For the Bayesian RL, the following is true and optimal for **all**\\(T\\):_\n\n\\[\\mathfrak{BV}_{L}(\\pi_{\\text{TS}})\\leq O\\left(\\inf_{\\varepsilon>0}(\\sqrt{H\\log (K_{\\operatorname{surr}}(\\varepsilon))T}+L\\varepsilon)\\right). \\tag{25}\\]\n\n_where the constant factor is independent of the prior. This means there exists a Bayesian RL problem such that \\(\\mathfrak{BV}_{L}(\\pi_{\\text{TS}})=\\widetilde{O}(\\sqrt{Hd_{\\operatorname{surr }}T})\\). All polylogarithmic terms are in terms of \\(H,d_{\\operatorname{surr}},T\\)._\n\nNote that the above coincides with the lower bound for the (model-based) time inhomogeneous frequentist setting; see e.g., [21] for the proven lower bound for the tabular case. This is also \\(\\sqrt{H}\\) higher (this factor being baked in \\(d_{\\operatorname{surr}}\\)) than that of the time homogeneous frequentist setting, which is expected, according to [21, App. D]. Note that in this conjecture, the \\(\\lambda\\) in our bound is replaced by \\(\\sqrt{H}\\), and the conjecture is not for \\(T\\) large enough, but for all \\(T\\). Supporting this conjecture requires experiments where TS can be exactly implemented assuming access to an oracle which provides the optimal policy for a query environment. Simulations have been performed for the similar [31, Conj. 1] in the time homogeneous case. Our conjecture is similar but with the additional expected factor of \\(\\sqrt{H}\\) due to time inhomogeneity, thus their simulation also supports the above.\n\n## 6 Conclusions\n\nIn this paper, we have addressed the Bayesian Reinforcement Learning (RL) problem in the context of time inhomogeneous transition and reward functions. By considering both Bayesian transition and Bayesian rewards without prior assumptions, we have extended the scope of previous works, making our formulation more comprehensive. To simplify the learning problem, we have introduced surrogate environments, which discretize the environment space. We have established a connection between the size of this new environment space and the \\(l_{1}\\)-dimensions of the transition and reward functions space, providing insights into the \\(l_{1}\\)-dimension of the environment space denoted by \\(d_{l_{1}}\\). We have employed posterior consistency tools to analyze the information ratio, which captures the trade-off between exploration and exploitation. We conjecture that (at least a weakened version of) our posterior consistency assumption should hold in general, which is left for future work. Our analysis has resulted in a refined approach to estimate the Bayesian regret in Thompson Sampling (TS), yielding a regret bound of \\(\\widetilde{O}(\\lambda\\sqrt{d_{l_{1}}T})\\) for large enough time steps \\(T\\). The result is specialized to linear, tabular, and finite mixtures MDPs.\n\n**Limitations:** While the paper provides asymptotic generic regret bound for TS in a generalized setup which improve the state of the art results, finding lower bounds, esp. one dependent on \\(\\lambda\\), are left open. In addition, the issue of prior misspecificity is not discussed and left for future studies.\n\n## References\n\n* [1] Amirhesam Abedsoltan, Mikhail Belkin, and Parthe Pandit. Toward large kernel models. _arXiv preprint arXiv:2302.02605_, 2023.\n* [2] Alekh Agarwal, Yujia Jin, and Tong Zhang. VOQL: Towards optimal regret in model-free rl with nonlinear function approximation. _arXiv preprint arXiv:2212.06069_, 2022.\n* [3] Shipra Agrawal and Randy Jia. Optimistic posterior sampling for reinforcement learning: worst-case regret bounds. _Advances in Neural Information Processing Systems_, 30, 2017.\n* [4] Dilip Arumugam and Benjamin Van Roy. Deciding what to model: Value-equivalent sampling for reinforcement learning. 35:9024-9044, 2022.\n* [5] Peter Auer, Thomas Jaksch, and Ronald Ortner. Near-optimal regret bounds for reinforcement learning. _Advances in neural information processing systems_, 21, 2008.\n* [6] Alex Ayoub, Zeyu Jia, Csaba Szepesvari, Mengdi Wang, and Lin Yang. Model-based reinforcement learning with value-targeted regression. In _International Conference on Machine Learning_, pages 463-474. PMLR, 2020.\n* [7] Mohammad Gheshlaghi Azar, Ian Osband, and Remi Munos. Minimax regret bounds for reinforcement learning. In _International Conference on Machine Learning_, pages 263-272. PMLR, 2017.\n* [8] Dimitri Bertsekas and John N Tsitsiklis. _Neuro-dynamic programming_. Athena Scientific, 1996.\n* [9] Sebastien Bubeck and Mark Sellke. First-order bayesian regret analysis of thompson sampling. In _Algorithmic Learning Theory_, pages 196-233. PMLR, 2020.\n* [10] Qi Cai, Zhuoran Yang, Chi Jin, and Zhaoran Wang. Provably efficient exploration in policy optimization. In _International Conference on Machine Learning_, pages 1283-1294. PMLR, 2020.\n* [11] Souradip Chakraborty, Amrit Bedi, Alec Koppel, Pratap Tokekar, Furong Huang, and Dinesh Manocha. Posterior coreset construction with kernelized stein discrepancy for model-based reinforcement learning. In _NeurIPS 2022 Workshop on Score-Based Methods_, 2022.\n* [12] Sayak Ray Chowdhury and Aditya Gopalan. Online learning in kernelized markov decision processes. In _Proceedings of the Twenty-Second International Conference on Artificial Intelligence and Statistics_, pages 3197-3205. PMLR, 2019.\n* [13] Christoph Dann, Mehryar Mohri, Tong Zhang, and Julian Zimmert. A provably efficient model-free posterior sampling method for episodic reinforcement learning. _Advances in Neural Information Processing Systems_, 34:12040-12051, 2021.\n* [14] Shi Dong and Benjamin Van Roy. An information-theoretic analysis for thompson sampling with many actions. _Advances in Neural Information Processing Systems_, 31, 2018.\n* [15] Yaqi Duan, Mengdi Wang, and Martin J Wainwright. Optimal policy evaluation using kernel-based temporal difference methods. _arXiv preprint arXiv:2109.12002_, 2021.\n* [16] Ying Fan and Yifei Ming. Model-based reinforcement learning for continuous control with posterior sampling. In _International Conference on Machine Learning_, pages 3078-3087. PMLR, 2021.\n* [17] Subhashis Ghosal and Aad Van der Vaart. _Fundamentals of nonparametric Bayesian inference_, volume 44. Cambridge University Press, 2017.\n* [18] Botao Hao and Tor Lattimore. Regret bounds for information-directed reinforcement learning. _arXiv preprint arXiv:2206.04640_, 2022.\n* [19] Botao Hao, Tor Lattimore, and Wei Deng. Information directed sampling for sparse linear bandits. _Advances in Neural Information Processing Systems_, 34:16738-16750, 2021.\n\n* [20] Botao Hao, Tor Lattimore, and Chao Qin. Contextual information-directed sampling. In _International Conference on Machine Learning_, pages 8446-8464. PMLR, 2022.\n* [21] Chi Jin, Zeyuan Allen-Zhu, Sebastien Bubeck, and Michael I Jordan. Is q-learning provably efficient? _Advances in neural information processing systems_, 31, 2018.\n* [22] Chi Jin, Zhuoran Yang, Zhaoran Wang, and Michael I Jordan. Provably efficient reinforcement learning with linear function approximation. In _Conference on Learning Theory_, pages 2137-2143. PMLR, 2020.\n* [23] Johannes Kirschner, Tor Lattimore, Claire Vernade, and Csaba Szepesvari. Asymptotically optimal information-directed sampling. In _Conference on Learning Theory_, pages 2777-2821. PMLR, 2021.\n* [24] Tor Lattimore and Andras Gyorgy. Mirror descent and the information ratio. In _Conference on Learning Theory_, pages 2965-2992. PMLR, 2021.\n* [25] Tor Lattimore and Csaba Szepesvari. An information-theoretic approach to minimax regret in partial monitoring. In _Conference on Learning Theory_, pages 2111-2139. PMLR, 2019.\n* [26] Fang Liu, Swapna Buccapatnam, and Ness Shroff. Information directed sampling for stochastic bandits with graph feedback. In _Proceedings of the AAAI Conference on Artificial Intelligence_, volume 32, 2018.\n* [27] Xiuyuan Lu. _Information-directed sampling for reinforcement learning_. Stanford University, 2020.\n* [28] Xiuyuan Lu and Benjamin Van Roy. Information-theoretic confidence bounds for reinforcement learning. _Advances in Neural Information Processing Systems_, 32, 2019.\n* [29] Ian Osband, Daniel Russo, and Benjamin Van Roy. (more) efficient reinforcement learning via posterior sampling. _Advances in Neural Information Processing Systems_, 26, 2013.\n* [30] Ian Osband and Benjamin Van Roy. Model-based reinforcement learning and the eluder dimension. _Advances in Neural Information Processing Systems_, 27, 2014.\n* [31] Ian Osband and Benjamin Van Roy. Why is posterior sampling better than optimism for reinforcement learning? In _International conference on machine learning_, pages 2701-2710. PMLR, 2017.\n* [32] Ian Osband, Benjamin Van Roy, Daniel J Russo, Zheng Wen, et al. Deep exploration via randomized value functions. _J. Mach. Learn. Res._, 20(124):1-62, 2019.\n* [33] Martin L Puterman. _Markov decision processes: discrete stochastic dynamic programming_. John Wiley & Sons, 2014.\n* [34] Daniel Russo. Worst-case regret bounds for exploration via randomized value functions. _Advances in Neural Information Processing Systems_, 32, 2019.\n* [35] Daniel Russo and Benjamin Van Roy. Learning to optimize via information-directed sampling. _Advances in Neural Information Processing Systems_, 27, 2014.\n* [36] Daniel Russo and Benjamin Van Roy. An information-theoretic analysis of thompson sampling. _The Journal of Machine Learning Research_, 17(1):2442-2471, 2016.\n* [37] Remo Sasso, Michelangelo Conserva, and Paulo Rauber. Posterior sampling for deep reinforcement learning. _arXiv preprint arXiv:2305.00477_, 2023.\n* [38] Richard S Sutton and Andrew G Barto. _Reinforcement learning: An introduction_. MIT press, 2018.\n* [39] Daniil Tiapkin, Denis Belomestny, Daniele Calandriello, Eric Moulines, Remi Munos, Alexey Naumov, Mark Rowland, Michal Valko, and Pierre Menard. Optimistic posterior sampling for reinforcement learning with few samples and tight guarantees. _Advances in Neural Information Processing Systems_, 35:10737-10751, 2022.\n\n* [40] Aristide Tossou, Debabrota Basu, and Christos Dimitrakakis. Near-optimal optimistic reinforcement learning using empirical bernstein inequalities. _arXiv preprint arXiv:1905.12425_, 2019.\n* [41] Lin Yang and Mengdi Wang. Sample-optimal parametric q-learning using linearly additive features. In _International Conference on Machine Learning_, pages 6995-7004. PMLR, 2019.\n* [42] Andrea Zanette, David Brandfonbrener, Emma Brunskill, Matteo Pirotta, and Alessandro Lazaric. Frequentist regret bounds for randomized least-squares value iteration. In _International Conference on Artificial Intelligence and Statistics_, pages 1954-1964. PMLR, 2020.\n* [43] Zihan Zhang, Yuan Zhou, and Xiangyang Ji. Almost optimal model-free reinforcement learningvia reference-advantage decomposition. _Advances in Neural Information Processing Systems_, 33:15198-15207, 2020.\n* [44] Dongruo Zhou, Quanquan Gu, and Csaba Szepesvari. Nearly minimax optimal reinforcement learning for linear mixture markov decision processes. In _Conference on Learning Theory_, pages 4532-4576. PMLR, 2021.", "response": "## Review\n\n### Summary\n\nThis paper investigates the performance of Thompson sampling in the context of Bayesian reinforcement learning, where both the transition and reward functions are sampled from a known prior distribution. It establishes Bayesian regret bounds applicable to various settings, including tabular, linear, and finite mixture MDPs. The analysis utilizes the Kolmogorov dimension, leading to results that match or improve upon existing state-of-the-art findings. While the general approach is valuable and opens up new avenues for exploration, there are concerns regarding the absence of lower bounds and the need for more explicit definitions of assumptions in theorems. Overall, the paper presents significant theoretical contributions to the understanding of Thompson sampling in reinforcement learning.\n\n### Strengths\n\n- The paper provides a comprehensive analysis of Thompson sampling in MDPs, claiming to offer the most general results to date.\n- It generalizes previous state-of-the-art regret bounds in both tabular and linear settings.\n- The writing is clear and easy to follow, with a thorough discussion of previous works.\n- The authors present proofs for their claims and provide corollaries for specialized settings.\n\n### Weaknesses\n\n- The paper lacks lower bounds to substantiate the tightness of the upper bounds.\n- Some theorem statements could be more explicit, particularly regarding assumptions.\n- The weak notion of Bayesian regret is seen as a limitation, given existing results in the frequentist setting.\n- There is no empirical validation of the derived bounds, limiting practical applicability.\n\n### Questions\n\n- Is the claim that the law of Thompson sampling aligns with the true posterior distribution valid under all conditions?\n- What is the relationship between the Kolmogorov dimension and the concept of covering dimensions?\n- Can the authors elaborate on how the proposed bounds compare to existing literature, especially in terms of frequentist regret?\n- What assumptions underlie the results, and how do they affect the generalizability of the findings?\n\n### Soundness\n\n**Score:** 3\n\n**Description:** 3 = good; the theoretical results are sound and well-supported, but some assumptions need clearer definitions.\n\n### Presentation\n\n**Score:** 2\n\n**Description:** 2 = fair; while the writing is mostly clear, the presentation would benefit from a more structured overview of results and additional clarity in some sections.\n\n### Contribution\n\n**Score:** 3\n\n**Description:** 3 = good; the paper makes significant contributions to the theoretical understanding of Thompson sampling in reinforcement learning, though some limitations are noted.\n\n### Rating\n\n**Score:** 6\n\n**Description:** 6 = Weak Accept; the paper is technically solid and has moderate-to-high impact potential, but it needs further improvements, particularly regarding empirical validation.\n\n### Paper Decision\n\n**Decision:** Accept (poster)\n\n**Reasons:** The decision to accept is based on the originality and significance of the paper's contributions to Bayesian reinforcement learning. The theoretical results are strong, and the potential for high impact in the field is evident, though the absence of empirical results and lower bounds is a concern. Overall, the clarity and depth of the analysis provide a solid foundation for future work.\n"}
{"query": "You are a highly experienced, conscientious, and fair academic reviewer, please help me review this paper. The review should be organized into nine sections: \n1. Summary: A summary of the paper in 100-150 words.\n2. Strengths/Weaknesses/Questions: The Strengths/Weaknesses/Questions of paper, which should be listed in bullet points, with each point supported by specific examples from the article where possible.\n3. Soundness/Contribution/Presentation: Rate the paper's Soundness/Contribution/Presentation, and match this score to the corresponding description from the list below and provide the result. The possible scores and their descriptions are: \n    1 poor\n    2 fair\n    3 good\n    4 excellent\n4. Rating: Give this paper an appropriate rating, match this rating to the corresponding description from the list below and provide the result. The possible Ratings and their descriptions are: \n    1 strong reject\n    2 reject, significant issues present\n    3 reject, not good enough\n    4 possibly reject, but has redeeming facets\n    5 marginally below the acceptance threshold\n    6 marginally above the acceptance threshold\n    7 accept, but needs minor improvements \n    8 accept, good paper\n    9 strong accept, excellent work\n    10 strong accept, should be highlighted at the conference   \n5. Paper Decision: It must include the Decision itself(Accept or Reject) and the reasons for this decision, based on the criteria of originality, methodological soundness, significance of results, and clarity and logic of presentation.\n\nHere is the template for a review format, you must follow this format to output your review result:\n**Summary:**\nSummary content\n\n**Strengths:**\n- Strength 1\n- Strength 2\n- ...\n\n**Weaknesses:**\n- Weakness 1\n- Weakness 2\n- ...\n\n**Questions:**\n- Question 1\n- Question 2\n- ...\n\n**Soundness:**\nSoundness result\n\n**Presentation:**\nPresentation result\n\n**Contribution:**\nContribution result\n\n**Rating:**\nRating result\n\n**Paper Decision:**\n- Decision: Accept/Reject\n- Reasons: reasons content\n\n\nPlease ensure your feedback is objective and constructive. The paper is as follows:\n\n# Group Robust Classification\n\nWithout Any Group Information\n\nChristos Tsirigotis\n\nUniversite de Montreal, Mila, ServiceNow Research &Joao Monteiro\n\nServiceNow Research &Pau Rodriguez\n\nServiceNow Research &David Vazquez\n\nServiceNow Research &Aaron Courville\n\nUniversite de Montreal, Mila, CIFAR CAI Chair\n\nWork done during internship at ServiceNow Research. Author correspondence at tsirigoc@mila.quebec.This work was also funded, in part, from A. Courville's Sony Research Award. Courville also acknowledges support from his Canada Research Chair.\n\n###### Abstract\n\nEmpirical risk minimization (ERM) is sensitive to spurious correlations in the training data, which poses a significant risk when deploying systems trained under this paradigm in high-stake applications. While the existing literature focuses on maximizing group-balanced or worst-group accuracy, estimating these accuracies is hindered by costly bias annotations. This study contends that current bias-unsupervised approaches to group robustness continue to rely on group information to achieve optimal performance. Firstly, these methods implicitly assume that all group combinations are represented during training. To illustrate this, we introduce a systematic generalization task on the MPI3D dataset and discover that current algorithms fail to improve the ERM baseline when combinations of observed attribute values are missing. Secondly, bias labels are still crucial for effective model selection, restricting the practicality of these methods in real-world scenarios. To address these limitations, we propose a revised methodology for training and validating debiased models in an entirely bias-unsupervised manner. We achieve this by employing pretrained self-supervised models to reliably extract bias information, which enables the integration of a logit adjustment training loss with our validation criterion. Our empirical analysis on synthetic and real-world tasks provides evidence that our approach overcomes the identified challenges and consistently enhances robust accuracy, attaining performance which is competitive with or outperforms that of state-of-the-art methods, which, conversely, rely on bias labels for validation.\n\n## 1 Introduction\n\nSupervised learning algorithms typically rely on the empirical risk minimization (ERM) paradigm - minimizing the average loss on a training set. The ERM paradigm operates under the assumption that the training data is a representative sample of the true data distribution [67]. Consequently, models that achieve low expected loss can still be _unfair_ when the model is tasked to predict outcomes for underrepresented groups [38, 10, 66], and prone to _rely on spurious correlations_ between target annotations and generative attributes of the training data [32, 62] that do not hold up under more general testing conditions. As automated predictors are deployed in failure-critical applications [24] or interact with society, where fairness must be guaranteed [13, 36], robustness and fairness become requirements that standard learning strategies do not satisfy.\n\nThese limitations have motivated the search for alternatives that perform uniformly across different data subgroups [69, 61] or, equivalently, that rely less on spurious features [2] or \"shortcuts\" [28]. Part of solving this problem includes making the correct assumptions about data, which should consider spurious statistical correlations due to bias in its generative process. A popular example of this issue is the _cow vs. camel_ classification problem [2].\n\nAs one might expect, pictures of cows very often contain a grass background, while camels are usually depicted in a desert. As such, a binary classifier that predicts whether there is grass in the background of an image could achieve a high prediction accuracy on top of natural images of the two animals. However, such a classifier would fail whenever the background changes from the typical occurrences.\n\nWhile progress has been made, existing methods to improve robustness on the least frequent attribute combinations (_e.g._, cow on sand), require knowledge about the bias attributes (bias-supervision) either during training [6, 61], or validation [57, 48, 50]. This limits their applicability in practical scenarios where bias annotations could be too costly or impossible to obtain. Even in cases where they can be obtained somewhat efficiently, human annotators might be biased themselves or sensitive annotations may not be readily available due to privacy concerns [76].\n\nMoreover, access to group information is rarely satisfiable on natural data, where a _curse of generative dimensionality_ implies that given the high number of attributes controlling the generative process, no realistic finite sample can cover all possible - exponential in number - combinations of such attributes. In fact, many group robustness algorithms that rely on data rebalancing or loss reweighting techniques [61, 48] assume that all group combinations are present during training. However, this condition is not satisfied in systematic generalization tasks, where certain data subgroups that are present in test data, are absent from training data. This motivates us to study the extent that training algorithms can successfully operate using group information implicitly present in the training dataset.\n\nSpecifically, in the case of these challenging systematically split datasets, we study the generalization of models on unseen combinations of attributes that were independently observed during training. This combinatorial type of generalization draws its name from the cognitive property of systematicity [26]. In deep learning, this question was introduced for the first time by Lake and Baroni [46], who studied the systematicity of RNN models in sequence prediction tasks. Here, we raise the same question in classification. To do so, we introduce a benchmark consisting of systematic splits from the MPI3D (thereby named sMPI3D) image dataset [29].In particular, we use the '_real_' split of MPI3D which consists of photographs of a robotic arm that has a colored rigid object attached to its end effector. The images are captured in a way that controls their generative attributes, such as the shape of the rigid object or the position of the robotic arm. We use it to ask the following question: given a'shape' classifier that has been trained on objects of all possible shapes and colors but only a subset of their combinations (e.g., red cubes and blue spheres), how would it perform for a new color-shape combination (e.g., blue cubes)? In this example, the 'color' attribute plays the role of a _bias_, obstructing view of all possible colors that cubes could have. In Appendix D.1 we describe in more detail the construction of the sMPI3D task, while in Figure 1(d) we illustrate an example of a systematic split.\n\nAs discussed above, existing state-of-the-art methods suffer from two limitations. The first one, as demonstrated by Asgari et al. [3, Table 1], is that the robust accuracy of many recent training algorithms, which do not demand any bias annotations during training, degrades severely when there is no access to bias labels during model selection. This highlights that robust algorithms should prescribe a way of performing bias-unsupervised validation, assuming the same access to i.i.d. data\n\nFigure 1: **Unsupervised Logit Adjustment (uLA). We train a linear classifier on top of an SSL pre-trained model to obtain biased predictions. These predictions are then leveraged to train a debiased model. No bias information is used during training or cross-validation.**\n\nresources as training. The second, revealed by our study on sMPI3D (Section 4.2), is that in most cases these methods fail to improve over the ERM baseline, which fails to systematically generalize.\n\nTo address these issues, our approach, summarized in Figure 1, is based on pretraining a base encoder using self-supervised learning [7] to extract a proxy for the missing bias labels and to provide an initialization point for finetuning a debiased model. Pretraining a proxy for the bias variable enables us to use this network for two purposes: first, to train a group-robust network and, second, to define a validation criterion for robust model selection. The debiasing training algorithm is based on the logit adjustment paradigm [54]. Our entirely bias-unsupervised methodology to group robustness using logit adjustment, which we call uLA, is able to compete with or outperform state-of-the-art counterparts, that otherwise utilize bias labels during model selection, in synthetic and real-world benchmarks. At the same time, it is the only method to consistently offer improvements over the ERM baseline in the sMPI3D systematic generalization task; thus effectively tackling the identified challenges about explicit or implicit use of group information.\n\n## 2 Preliminaries\n\n**Problem Formulation.** Consider a multi-class classification task, where \\(X\\subset\\mathcal{X}\\) is the input variable and \\(Y\\subset\\mathcal{Y}\\), the categorical target variable with \\(|\\mathcal{Y}|=K\\) classes. We have access to a dataset of observations \\(\\mathcal{D}\\coloneqq\\{(x_{n},y_{n})\\}_{n=1}^{N}\\) sampled i.i.d. from an underlying data distribution \\(p_{\\text{data}}(X,Y)\\) over \\(\\mathcal{X}\\times\\mathcal{Y}\\). The setting above may become problematic once we consider that the deployment data \\(\\mathcal{D}_{\\text{test}}\\) are sampled from a different testing distribution: \\(p_{\\text{test}}\\neq p_{\\text{data}}\\). In other words, we assume that there are two data generating processes; one which generates development data (\\(\\mathcal{D}\\) and \\(\\mathcal{D}_{\\text{valid}}\\), where \\(\\mathcal{D}_{\\text{valid}}\\) is used for validation) according to \\(p_{\\text{data}}\\), and one which generates deployment data (\\(\\mathcal{D}_{\\text{test}}\\)) according to \\(p_{\\text{test}}\\).\n\nIn further detail, we focus on a particular transfer learning problem from \\(p_{\\text{data}}\\) to \\(p_{\\text{test}}\\), which is due to a distribution shift in attributes which participate in the generative process of the input variable \\(X\\). Our study considers anti-causal prediction tasks for which the target variable \\(Y\\) is one of the generative attributes of \\(X\\), and \\(Z\\subset\\mathcal{Z}\\) another (possibly unobserved) categorical generative attribute with \\(|\\mathcal{Z}|=L\\) classes. \\(Z\\) is marginally independent to \\(Y\\) under \\(p_{\\text{test}}\\), but it might not be under \\(p_{\\text{data}}\\). For this reason, we say that the variables \\(Y\\) and \\(Z\\) are _spuriously correlated_ in training and validation data, and \\(Z\\) will also be referred to as the _bias attribute_. Under this setting, a group \\((y,z)\\in[K]\\times[L]\\) is defined as a combination of target and bias attribute values, and group robustness can be formulated as a transfer learning problem from \\(p_{\\text{data}}\\) to \\(p_{\\text{test}}\\) under the following two assumptions relating the two joint distributions over \\(X\\), \\(Y\\) and \\(Z\\):\n\n\\[p_{\\text{test}}(y,z)\\propto\\mathbb{1}_{\\operatorname{Supp}p_{ \\text{data}}(Y)\\times\\operatorname{Supp}p_{\\text{data}}(Z)}(y,z), \\tag{1}\\] \\[\\text{and }p_{\\text{test}}(x|y,z)=p_{\\text{data}}(x|y,z), \\tag{2}\\]\n\nwhere \\(1_{S}(s)=1\\) if \\(s\\in S\\) else \\(0\\), the characteristic function of a set \\(S\\), and \\(\\operatorname{Supp}p\\) denotes the support set of a distribution \\(p\\). Relation 1 asserts that, during test time, the target and bias variables\n\nFigure 2: **Left**: Example tasks. Circled in dashed lines are samples from training split, exhibiting statistically major attribute groups. Those outside are example test samples, where all groups are equally considered. A classifier trained on a biased training set may misclassify a bias-conflicting test sample - recognizing a ‘red three’ as ‘zero’ (2a) or a ‘blue sphere’ as a ‘cone’ (2b) - or be unfair to the sensitive gender attribute when tasked to classify a facial attribute, e.g. hair color (2c). **Right**: An example of a systematic split. \\(C\\) is the number of color values per shape. ‘Crosses’ represent groups used to sample the training and validation splits, while ‘circles’ are entirely out-of-distribution.\n\nare distributed uniformly over the product of their respective marginal supports under \\(p_{\\text{data}}\\). In other words, all combinations of observed attributes values are considered equally. This also implies that \\(Y\\) and \\(Z\\) are marginally independent under \\(p_{\\text{test}}\\)3. On the other hand, relation 2 assumes the invariance of mechanism, which is an assumption typically found in _disentangled causal process_ literature [64]. These two assumptions underlie the group robustness literature, in that they are equivalent to the evaluation of classifiers under a popular robust performance criterion which is described below.\n\nFootnote 3: Generally, the variables, which are uniform over a support which factorizes into a cartesian product of their marginal supports, are independent: \\(c\\mathbb{1}_{\\mathcal{X}\\times\\mathcal{Y}}(x,y)=c\\mathbb{1}(x,y\\in\\mathcal{X} \\times\\mathcal{Y})=c\\mathbb{1}(x\\in\\mathcal{X})\\mathbb{1}(y\\in\\mathcal{Y})\\).\n\n**Performance Criteria.** Let \\(\\hat{y}(x;f)\\coloneqq\\operatorname*{argmax}_{y}f(x)_{y}\\) be the predictions of a scoring function \\(f:\\mathcal{X}\\rightarrow\\mathbb{R}^{K}\\). The accuracy of \\(f\\) under \\(p_{\\text{test}}\\) corresponds to the _group-balanced accuracy_\n\n\\[\\operatorname{Acc}_{Y|X}(f;p_{\\text{test}})=\\underset{x\\sim p_{\\text{data}}( \\cdot|y,z)}{\\mathbb{E}}\\mathbb{1}\\big{(}y=\\hat{y}(x;f)\\big{)}=\\frac{1}{KL} \\sum_{y,z}\\underset{x\\sim p_{\\text{data}}(\\cdot|y,z)}{\\mathbb{E}}\\mathbb{1} \\big{(}y=\\hat{y}(x;f)\\big{)}, \\tag{3}\\]\n\nwhich is a frequently used performance metric in literature [61, 48, 50]. Essentially, this performance criterion first separately computes the accuracy for samples \\(x\\) of each group \\((y,z)\\) and then averages them. The averaging operation per individual group accuracy directly stems from the uniformity assumption in \\(p_{\\text{test}}\\) (rel. 1) and implements a group fairness notion: we care equally about performing well in all groups. Another popular group robustness criterion is the _worst-group accuracy_[61] which substitutes the average accuracy over individual groups with the minimum (worst) accuracy.\n\nAs described in Section 5, there exist different approaches to improve group fairness [61, 48, 50] that depend on knowing biases for cross-validation. Here we focus on logit adjustment techniques like the one proposed by Liu et al. [50], which co-train two models where one corrects the biases of the other. In the following sections, we show how, by decoupling the training of the biased model from the bias-corrected model, it is possible to obtain a proxy criterion that can be used for cross-validation without knowing the bias attribute.\n\n**Logit Adjustment** was originally developed as a technique for supervised learning under class-imbalanced [25, 53] or long-tailed [54] data. For clarity of presentation, we will assume for the moment that we have access to bias labels. We describe sLA, a bias-supervised training process with logit adjustment. Let \\(h_{\\theta}:\\mathcal{X}\\rightarrow\\mathbb{R}^{|\\mathcal{Y}|}\\) be the model that we want to train for group robustness, such as a neural network parameterized by \\(\\theta\\) implementing a function from samples in \\(\\mathcal{X}\\) to the unnormalized logit space of \\(Y\\). We then train by minimizing the average cross-entropy loss for the following _logit adjusted model_\n\n\\[p_{\\theta}(y|x,z)\\propto\\exp(h_{\\theta}(x)_{y}+\\log\\hat{p}_{\\text{data}}(y|z)). \\tag{4}\\]\n\nWe estimate the conditional \\(\\hat{p}_{\\text{data}}(y|z)\\) directly from the available training data, for example via empirical frequencies of the finite number of groups \\((y,z)\\in[K]\\times[L]\\). Finally, during inference, we cancel out the contribution of the logit bias term \\(\\hat{p}_{\\text{data}}(y|z)\\) and predict only according to the optimized neural network \\(h_{\\theta^{*}}\\).\n\nNote that, since \\(Y\\) is a categorical variable, \\(\\log\\hat{p}_{\\text{data}}(y|z)\\) takes non-positive values. For this reason, we can intuitively interpret logit adjustment as a soft masking operation for outputs of \\(h_{\\theta}\\) which are unlikely in the training data when we have observed \\(z\\). In this way, we account for the dependency of \\(Y\\) to \\(Z\\) which spuriously exists in the training distribution. By fitting the cross-entropy objective under logit adjustment, the network \\(h_{\\theta}\\) has to model the remaining relations for \\(Y\\mid X\\) that are not spurious since those are already accounted for. An expressive enough model class in Equation (4) can achieve this, assuming further that the likelihood ratio \\(\\frac{p_{\\text{data}}(x|y,z)}{p_{\\text{data}}(x|z)}=\\frac{p_{\\text{data}}(y|x,z)}{p_{\\text{data}}(y|z)}\\) is independent of \\(z\\). In appendix A, we derive sLA from first principles, bridging the gap between the well-studied application of this technique to the class-imbalance problem with its application to group robustness. Moreover, we demonstrate that sLA is a well-justified procedure by proving the following proposition.\n\n**Proposition 2.1** (sLA optimizes the group-balanced accuracy).: _Under the assumption that the hypothesis class \\(p_{\\theta}(y|x,z)\\) (eq. 4) contains \\(p_{\\text{data}}(y|x,z)\\), the minimizer network \\(h_{\\theta^{*}}\\) of the cross-entropy loss maximizes a lower bound to the group-balanced accuracy \\(\\operatorname{Acc}_{Y\\mid X}(h_{\\theta^{*}};p_{\\text{test}})\\)._\n\n**Systematic Generalization.** We expect sLA models to systematically generalize, as they optimize for the transfer learning problem defined by Equations (1) and (2). As Figure 1(d) suggests, systematic generalization is just an extreme case of such problems, for which samples from some combinationshave \\(p_{\\text{data}}(y,z)=0\\) during training, however they populate the test set with \\(p_{\\text{test}}(y,z)>0\\). As a counterexample, the setting by Ahmed et al. [1] does not correspond to a systematic generalization task, as we define it. In what they refer to as systematic splits of colored MNIST, all possible combinations of color and digit are exposed to the model during its training, while in our sMPI3D some color and shape combinations are only revealed during test-time. This detail makes our setting significantly more challenging, as the conditions for applying importance sampling are not met, as \\(\\operatorname{Supp}p_{\\text{data}}(Y,Z)\\not\\supseteq\\operatorname{Supp}p_{ \\text{test}}(Y,Z)\\)[4, Chapter 5]. This means that a simple re-weighting of the per-sample loss with \\(\\frac{1}{p_{\\text{data}}(y,z)}\\), in order to estimate \\(\\mathbb{E}_{p_{\\text{test}}}[l_{\\text{ce}}(y,h_{\\theta}(x))]\\), is not appropriate for the task.\n\n**Self-Supervised Learning (SSL)** refers to a collection of methods [7] in unsupervised representation learning where unlabeled data provides the supervision by defining tasks where a model is asked to predict some part of the input from other parts, for example by contrasting independently transformed versions of data samples [14; 35; 15]. Methods like SimCLR [14], MoCo [35], BYOL [30] and Barlow Twins [73] provide solutions for low-data generalization, robustness, as well as transferability of learnt representations for image classification. Although self-supervised learning methods are generally more robust to distribution shifts than purely supervised methods [49], they can still be affected by significant shifts, for example in long-tailed data learning [5; 63]. In this work, we explore the limits and utility of SSL in group robust classification. An extended preliminary discussion about self-supervised algorithms can be found in Appendix B.\n\n## 3 ULA: Bias-unsupervised Logit Adjustment\n\nHere, we introduce bias-unsupervised logit adjustment (uLA), a logit correction approach which improves on the work of Liu et al. [50] by _removing dependency on explicit bias annotations both during training and validation_. For this reason, we create a proxy variable for the bias via the predictions of a pretrained network using SSL. The advantage of pretraining, over co-training a bias network [57; 17; 50], is two-fold: First, we can reuse the fixed bias proxy network to define a validation criterion for group robustness. This enables us to perform hyperparameter search, but also informs us about when to stop training the debiased model, which is critical for optimal performance [48]. Second, recent literature [39; 43] has demonstrated, in the case of training with bias-supervised data, that a linear classifier on top of pretrained base models provides with substantial group-robustness improvements. Here, we initialize the debiased model from the pretrained base model and finetune it using logit adjustment. Figure 1 provides a summary of our approach and its pseudocode can be found in Appendix C. Further training details are discussed in the following.\n\n### Bias-unsupervised Training\n\n**Biased network: pretrain with SSL.** We start by training a _base model_, \\(f_{\\text{base}}\\), using an SSL method on the unlabeled data of the training set \\(\\mathcal{D}\\). We decide on the hyperparameters of the SSL algorithm (learning rate, weight decay, temperature, augmentations and potentially others) by maximizing the i.i.d. validation accuracy of an online linear classifier, which probes the representations of the base model. Afterwards, we train a linear classifier \\(g_{\\phi}\\) on top of a frozen \\(f_{\\text{base}}\\) and against target variable labels \\(Y\\) using a vanilla cross-entropy loss, in order to derive a proxy for the bias variable. Finally, we retrieve the composite neural network \\(h_{\\text{bias}}=g_{\\phi}\\circ f_{\\text{base}}\\), and use its predictions \\(\\hat{y}(x;h_{\\text{bias}})\\) as a proxy for the missing bias variable observations. For the same purpose, Nam et al. [57] trained a bias-extracting network by employing Generalized Cross Entropy [75], an objective which provides robustness to noisy labels. Since the goal of the proxy network is to predict the spurious attribute, bias-conflicting or minority samples can be perceived as mislabeled data points. Here, we follow these observations and leverage the representations learnt with SSL which, when composed with a low-capacity (linear) classifier, provide with a model that is more robust to label-noise [71]. In addition to deriving a bias proxy, we will also use \\(f_{\\text{base}}\\) as an initialization point in the parameter space for training the debiased model.\n\nIn this paper, we have chosen to use MoCoV2+ [15] as the SSL algorithm for the image classification tasks we consider. We make this choice since contrastive learning algorithms, like the MoCo family [35; 15], offer relatively stable training since they explicitly prevent representation collapse in their loss function (see Appendix B). However, as we demonstrate in Section 4.3, our method is not restrained to the use of a particular SSL algorithm.\n\n**Debiased network: logit adjustment.** In the absence of bias labels during training, we need a substitute for the estimate \\(\\hat{p}_{\\text{data}}(Y,Z)\\) in eq. 4. We use the predictions \\(y_{\\text{bias}}\\) of the bias proxy network \\(h_{\\text{bias}}\\) to that end. The resulting joint distribution between the target variable and the biased network's predictions, \\(\\hat{p}_{\\text{data}}(y,y_{\\text{bias}})\\), can be thought of as a soft confusion matrix of \\(h_{\\text{bias}}\\) and can be computed using the available training data with\n\n\\[\\hat{p}_{\\text{data}}(y,y_{\\text{bias}})=\\frac{1}{|\\mathcal{D}|}\\sum_{x^{ \\prime},y^{\\prime}\\in\\mathcal{D}}p_{\\text{bias}}(y_{\\text{bias}}\\,|\\,x^{ \\prime})\\mathbb{1}(y=y^{\\prime}), \\tag{5}\\]\n\nwhere \\(p_{\\text{bias}}(y_{\\text{bias}}\\,|\\,x)\\propto\\exp(h_{\\text{bias}}(x)_{y_{\\text {bias}}}/\\tau)\\) is the biased model conditional. Note that \\(p_{\\text{bias}}(y_{\\text{bias}}\\,|\\,x)\\) is post-hoc calibrated by a temperature hyperparameter \\(\\tau\\). As we rely on the biased network to approximate the spurious correlation structure described in \\(p_{\\text{data}}(Y,Z)\\), it is crucial that the predicted conditional probabilities of the biased network are calibrated correctly [31, 56, 53]. Afterwards, we are ready to begin training a debiased network with logit adjustment as in Section 2; using only the predictions of \\(h_{\\text{bias}}\\) as the bias variable. The debiased network is initialized at the composition of a random linear classifier with the pretrained network \\(f_{\\text{base}}\\), and during training we finetune it while adjusting its output logits by\n\n\\[p_{\\theta}(y\\,|\\,x)\\propto\\exp\\Big{(}h_{\\theta}(x)_{y}+\\eta\\,\\log\\hat{p}_{ \\text{data}}\\big{(}y\\,|\\,\\hat{y}(x;h_{\\text{bias}})\\Big{)}, \\tag{6}\\]\n\nwhere \\(\\eta\\geq 0\\) is a hyperparameter controlling the strength of the additive logit bias. Notice that for \\(\\eta=0\\) we fall back to ERM training. By tuning \\(\\eta\\) we can mitigate calibration errors of the debiased model [53], similar to what \\(\\tau\\) does for the bias proxy. Selected hyperparameter configurations can be found in Appendix E.\n\n### Bias-unsupervised Validation\n\nWe re-purpose the pretrained biased classifier \\(h_{\\text{bias}}\\) so that training no longer requires bias-annotated validation data for model selection. Our bias-unsupervised validation criterion calculates a balanced accuracy across pairs \\((y,y_{\\text{bias}})\\in[K]\\times[K]\\) of true labels and biased classifier predictions. In practice, we compute\n\n\\[\\overline{\\text{BalAcc}}(f;h_{\\text{bias}})\\coloneqq\\frac{1}{K^{2}}\\sum_{y,y_ {\\text{bias}}}\\frac{1}{|S_{y,y_{\\text{bias}}}|}\\sum_{x_{i},y_{i}\\in S_{y,y_{ \\text{bias}}}}\\mathbb{1}\\big{(}y_{i}=\\hat{y}(x_{i};f)\\big{)} \\tag{7}\\]\n\n\\[S_{y,y_{\\text{bias}}}\\coloneqq\\left\\{(x_{i},y_{i})\\in\\mathcal{D}_{\\text{valid }}\\,|\\,y_{i}=y\\text{ and }\\hat{y}(x_{i};h_{\\text{bias}})=y_{\\text{bias}}\\right\\},\\]\n\nwhere \\(S_{y,y_{\\text{bias}}}\\) are partitions of \\(\\mathcal{D}_{\\text{valid}}\\) based on the value of predictions of \\(h_{\\text{bias}}\\) on a sample \\(x_{i}\\) and its ground-truth target label \\(y_{i}\\). This corresponds to a form of group-balanced accuracy. Alternatively, we could also calculate a form of worst-group accuracy by taking the minimum across \\(S_{y,y_{\\text{bias}}}\\). We find that worst-case validation is more suitable for tasks with small number of classes \\(K\\).\n\nDuring training, we evaluate models at every epoch and we select the one that maximizes our validation score across the duration of a training trial. In addition, we use this criterion to tune hyperparameters. In particular, for each task we tune learning rate, weight decay, logit adjustment strength coefficient \\(\\eta\\), calibration temperature \\(\\tau\\) and, in addition, the number of pretraining steps for the SSL backbone - whenever it is applicable - and for the linear classification probe of the bias proxy network. Fig. 1 depicts our bias-unsupervised training and validation procedures.\n\n## 4 Experiments\n\n**Datasets.** The tasks we consider are all specific instances of the setup above (see Fig. 2 and Section 2). This spans group robustness challenges like colored MNIST [57, cMNIST], corrupted CIFAR10 [37, cCIFAR10] and Waterbirds[62], fair classification benchmarks like CelebA [51], and systematic generalization tasks such as the contributed sMPI3D. Details about their construction can be found in Appendix D.\n\n**Training Setup.** For cMNIST, we train a 3-hidden layer MLP, while we use a ResNet18 [34] for cCIFAR10 and sMPI3D, and a ResNet50 for Waterbirds and CelebA. For all datasets except Waterbirds, we pretrain the base model with the MoCoV2+ [15] process, while training of the linear probe for the bias network and finetuning for the logit adjusted debiased network happen withAdamW [52] optimizer. For Waterbirds instead, we leverage a base model which was pretrained on Imagenet [60], following baselines in the literature for fair comparison, and we finetune it using SGD. Finally, for cMNIST, cCIFAR10 and sMPI3D we use our group-balanced bias-unsupervised validation criterion, whereas for Waterbirds and CelebA the worst-group version. Further details are described in Appendix E.\n\n**Baselines.** We compare uLA with vanilla ERM and a diverse set of group robustness techniques described in Section 5. GroupDRO [61] provides with a fully bias-supervised baseline, while LrF [57], JtT [48], LC [50] and DFA [17] are bias-unsupervised during training although they require bias annotations during validation to achieve robust optimal performance. We also consider two fully bias-unsupervised methods: Bardenhagen et al. [9] propose early stopping networks to derive a proxy for bias-unsupervised validation, and MaskTune [3] which provide competitive results under fully bias-unsupervised benchmarks without performing any validation procedure.\n\n### Results on Benchmarks\n\nIn Table 1, we report the group-balanced accuracy on cMNIST and cCIFAR10, across different percentages of bias-conflicting examples in the training set. For cMNIST, we observe that our method performs overall competitively against LrF, DFA and LC, even though these baselines use bias annotations during model selection. On the other hand, for cCIFAR10, we observe a significantly improved group robust performance for 3/4 difficulty levels. The highest difference is observed at the 1.0% task, where our method outperforms GroupDRO by about 24% absolute increase in group-balance accuracy.\n\nIn Table 2, we observe worst-group accuracy results in the more challenging Waterbirds and CelebA datasets. In both cases, our approach performs again competitively among bias-unsupervised training algorithms that leverage bias information during validation, falling slightly behind LC, which is the best out of the ones considered. Notably, our approach is still the best performing fully bias-unsupervised method, outperforming the supervised learning pretraining validation scheme of Bardenhagen et al. [9], and performing on par with MaskTune on Waterbirds and better than it on CelebA by \\(\\approx\\)8% absolute worst-group accuracy.\n\n\\begin{table}\n\\begin{tabular}{l c c c c c c c c c c} \\hline \\hline  & \\multicolumn{2}{c}{Bias Labels} & \\multicolumn{3}{c}{cCIFAR10} \\\\ \\cline{2-11}  & Train & Val & \\(0.5\\%\\) & \\(1.0\\%\\) & \\(2.0\\%\\) & \\(5.0\\%\\) & \\(0.5\\%\\) & \\(1.0\\%\\) & \\(2.0\\%\\) & \\(5.0\\%\\) \\\\ \\hline\n[61] GroupDRO\\({}^{\\dagger}\\) & ✓ & ✓ & \\(63.12\\) & \\(68.78\\) & \\(76.30\\) & \\(84.20\\) & \\(33.44\\) & \\(38.30\\) & \\(45.81\\) & \\(57.32\\) \\\\ \\hline\n[57] LrF\\({}^{*}\\) & ✗ & ✓ & \\(52.50_{\\pm 0.23}\\) & \\(61.89_{\\pm 0.47}\\) & \\(70.13_{\\pm 0.24}\\) & \\(80.57_{\\pm 3.88}\\) & \\(28.74_{\\pm 3.0}\\) & \\(33.07_{\\pm 0.77}\\) & \\(39.14_{\\pm 0.30}\\) & \\(50.27_{\\pm 1.56}\\) \\\\\n[17] DFA\\({}^{*}\\) & ✗ & ✓ & \\(65.22_{\\pm 4.41}\\) & \\(81.73_{\\pm 2.34}\\) & \\(84.79_{\\pm 0.88}\\) & \\(89.66_{\\pm 1.0}\\) & \\(29.95_{\\pm 1.9}\\) & \\(36.49_{\\pm 1.9}\\) & \\(17.82_{\\pm 5.1}\\) & \\(51.13_{\\pm 1.18}\\) \\\\\n[50] LC\\({}^{\\dagger}\\) & ✗ & ✓ & \\(71.53_{\\pm 1.87}\\) & \\(82.52_{\\pm 1.81}\\) & \\(\\mathbf{86.21_{\\pm 1.09}}\\) & \\(91.61_{\\pm 0.87}\\) & \\(\\mathbf{34.56_{\\pm 0.37}}\\) & \\(37.34_{\\pm 0.60}\\) & \\(47.81_{\\pm 2.00}\\) & \\(54.55_{\\pm 1.56}\\) \\\\ \\hline ERM\\({}^{*}\\) & ✗ & ✗ & \\(35.19_{\\pm 4.90}\\) & \\(52.09_{\\pm 2.88}\\) & \\(65.86_{\\pm 3.29}\\) & \\(82.17_{\\pm 0.74}\\) & \\(23.08_{\\pm 1.25}\\) & \\(25.82_{\\pm 0.33}\\) & \\(30.06_{\\pm 0.71}\\) & \\(39.42_{\\pm 0.64}\\) \\\\ uLA (ours) & ✗ & ✗ & \\(\\mathbf{75.13_{\\pm 0.78}}\\) & \\(81.80_{\\pm 1.41}\\) & \\(84.79_{\\pm 1.90}\\) & \\(\\mathbf{92.79_{\\pm 0.85}}\\) & \\(34.39_{\\pm 1.14}\\) & \\(\\mathbf{62.49_{\\pm 0.74}}\\) & \\(\\mathbf{63.88_{\\pm 1.07}}\\) & \\(\\mathbf{74.19_{\\pm 0.58}}\\) \\\\ \\hline \\hline \\end{tabular}\n\\end{table}\nTable 1: Results using datasets from Chu et al. [17] for various % of bias-conflicting examples in the training set. We report avg. group-balanced test accuracy (%) and std. dev. over 5 seeds. \\({}^{*}\\)Results from Chu et al. [17]. \\({}^{\\dagger}\\)Results from Liu et al. [50].\n\n\\begin{table}\n\\begin{tabular}{l c c c c c c} \\hline \\hline  & \\multicolumn{2}{c}{Bias Labels} & \\multicolumn{2}{c}{Waterbirds} & \\multicolumn{2}{c}{CelebA} \\\\ \\cline{2-7}  & Train & Val & i.i.d. & worst group & i.i.d. & worst group \\\\ \\hline\n[61] GroupDRO\\({}^{*}\\) & ✓ & ✓ & \\(93.5\\) & \\(91.4\\) & \\(92.9\\) & \\(88.9\\) \\\\ \\hline\n[9] ERM & ✗ & ✓ & \\(97.6\\) & \\(86.7\\) & \\(93.1\\) & \\(77.8\\) \\\\\n[57] LfF\\({}^{*}\\) & ✗ & ✓ & \\(97.5\\) & \\(75.2\\) & \\(86.0\\) & \\(77.2\\) \\\\\n[48] JtT\\({}^{*}\\) & ✗ & ✓ & \\(93.6\\) & \\(86.0\\) & \\(88.0\\) & \\(81.1\\) \\\\\n[50] LC & ✗ & ✓ & - & \\(90.5_{\\pm 1.1}\\) & - & \\(88.1_{\\pm 0.8}\\) \\\\ \\hline ERM & ✗ & ✗ & \\(97.3\\) & \\(72.6\\) & \\(95.6\\) & \\(47.2\\) \\\\ Bardenhagen et al. [9] & ✗ & ✗ & \\(97.5\\) & \\(78.5\\) & \\(88.0\\) & \\(78.9\\) \\\\\n[3] MaskTune & ✗ & ✗ & \\(93.0_{\\pm 0.7}\\) & \\(86.4_{\\pm 1.9}\\) & \\(91.3_{\\pm 0.1}\\) & \\(78.0_{\\pm 1.2}\\) \\\\ uLA (ours) & ✗ & ✗ & \\(91.5_{\\pm 0.7}\\) & \\(86.1_{\\pm 1.5}\\) & \\(93.9_{\\pm 0.2}\\) & \\(86.5_{\\pm 3.7}\\) \\\\ \\hline \\hline \\end{tabular}\n\\end{table}\nTable 2: Results on Waterbirds and CelebA. We report avg. test accuracy (%) and std. dev. over 5 seeds. \\({}^{*}\\)Results from Liu et al. [48].\n\n### Systematic Generalization\n\nsMPI3D is our contributed task which we use to study combinatorial systematicity in classifiers. With this benchmark, we aim to study the ability of classifiers to generalize to samples generated from novel combinations of observed generative attributes values, which under \\(p_{\\text{data}}\\) have \\(0\\) probability. The target task is to classify the _shape_ of an object, which is spuriously correlated with its _color_. We devise 4 difficulty levels for this task which we denote by \\(C\\), the number of color values present in the training set per shape. In Fig. 1(d), we display a possible split between in-distribution and o.o.d. combinations of attributes for \\(C=4\\). Details about its construction are given in Appendix D.\n\n**Results.** In Table 3, we further validate our approach on our contributed systematic generalization task. Under this setting, all bias-unsupervised approaches are evaluated fairly since their models are validated with exactly the same data resources; access to bias labels cannot give model selection advantage to any algorithm since there are no o.o.d. samples in the validation (just like in the training) split. Our method is the only one which consistently offers group-balanced accuracy improvements across difficulty levels, demonstrating generalization to o.o.d. samples. On the other hand, GroupDRO, vanilla ERM, JT and LC are not able to increase group-balanced accuracy over the percentage of i.i.d. samples present in the balanced test set.\n\n### Ablation Studies\n\nWe perform a set of studies to understand better the efficacy of our approach. In Fig. 3, we ablate the choice of training paradigm and the influence of a pretrained base model using SSL to the robust performance of a trained model to the target task. The paradigms we choose are among vanilla cross-entropy minimization, uLA and sLA (bias-supervised logit adjustment - see Section 2). For this ablation, a bias-supervised validation procedure was used for comparison against the fully bias-supervised sLA baseline. We find that finetuning the pretrained base model gives the best performance across training paradigms and tasks. Second, form CelebA and cCIFAR10, vanilla cross-entropy finetuning does not offer stark performance improvement over vanilla training from scratch. Only when we apply a logit adjustment training procedure, we are able to take significant advantage of the learnt representation space. At the same time, the accuracy gaps between uLA and\n\n\\begin{table}\n\\begin{tabular}{l c c c c c} \\hline \\hline  & \\multicolumn{2}{c}{Bias Labels} & \\multicolumn{4}{c}{sMPI3D (various \\(C\\) colors / shape)} \\\\ \\cline{2-6}  & Train \\& Val & \\(2\\) & \\(3\\) & \\(4\\) & \\(5\\) \\\\ \\hline \\% i.i.d. samples & - & \\(33.33\\) & \\(50.00\\) & \\(66.66\\) & \\(83.33\\) \\\\ \\hline\n[61] GroupDRO & ✓ & \\(31.23_{\\pm 1.88}\\) & \\(46.01_{\\pm 4.13}\\) & \\(69.68_{\\pm 7.82}\\) & \\(82.18_{\\pm 8.39}\\) \\\\ \\hline ERM & ✗ & \\(31.94_{\\pm 1.67}\\) & \\(47.68_{\\pm 4.06}\\) & \\(71.94_{\\pm 7.71}\\) & \\(83.10_{\\pm 8.64}\\) \\\\\n[48] JT & ✗ & \\(31.89_{\\pm 0.88}\\) & \\(48.93_{\\pm 2.04}\\) & \\(67.78_{\\pm 3.14}\\) & \\(83.31_{\\pm 3.51}\\) \\\\\n[50] LC\\({}^{*}\\) & ✗ & \\(31.29_{\\pm 0.96}\\) & \\(45.01_{\\pm 2.31}\\) & \\(61.67_{\\pm 5.06}\\) & \\(94.62_{\\pm 0.88}\\) \\\\ uLA (ours) & ✗ & \\(\\mathbf{59.58}_{\\pm 3.12}\\) & \\(\\mathbf{80.53}_{\\pm 3.16}\\) & \\(\\mathbf{91.11}_{\\pm 3.52}\\) & \\(\\mathbf{98.05}_{\\pm 0.64}\\) \\\\ \\hline \\hline \\end{tabular}\n\\end{table}\nTable 3: Results on sMPI3D for various numbers of \\(C\\) colors per shape value in the training set (see Appendix D). We report avg. group-balanced accuracy (%) and std. dev. over 5 seeds & dataset generations. \\({}^{*}\\)GroupMix augmentation was not used for fairness of comparison.\n\nFigure 3: Ablations on the influence of SSL pretraining and on the use of Vanilla process, sLA, or uLA for training against the downstream task. For cCIFAR10 (1%) and sMPI3D (\\(C=3\\)) we report the group-balanced test accuracy, whereas for CelebA the worst-group. We report avg. accuracies and std. dev. over 5 seeds.\n\nsLA are small which indicates that we are able to recover correctly the bias attribute with the proxy network. On the other hand, for sMPI3D the gaps between uLA and sLA are large. This shows that, in the systematic generalization case, bias extraction remains an open challenge for future work, and that improvement over baseline procedures in Table 3 is due to pretraining with SSL.\n\nIn addition, we study the impact of the choice of SSL method that we use to pretrain the backbone. For this reason, we perform an ablation experiment on cCIFAR10 (1%) by changing the pretraining strategy from MoCoV2+ to BYOL [30] or Barlow Twins[73]. We find that, while BYOL performs within the error margin of the best MoCoV2+ setting, Barlow Twins underperforms. Barlow Twins seeks to match the empirical (i.i.d.) cross-correlation between features to the identity. Arguably, we expect that the cross-correlation is different under the shifted test set. In any case, uLA significantly outperforms the non-uLA baselines with any of the considered SSL methods.\n\nFinally, we study how the choice of pretraining paradigm for the bias network influences the quality of hyperparameter search using our proposed validation criterion. In Fig. 4, we present two separate searches on the same space of hyperparameters using two different pretraining approaches. In red we see our approach of pretraining an SSL base model for the bias network, and in blue we see a baseline approach where we pretrain with purely supervised learning. We observe that SSL pretraining enables stronger correlation between the proposed bias-unsupervised group-balanced validation criterion and the corresponding test accuracy on cCIFAR10. It is more difficult to tune hyperparameters with a bias network pretrained with supervised learning, because it may fit the training set entirely. In that case, the validation criterion collapses to the in-distribution test accuracy which is not indicative of the group-balanced test accuracy. On the contrary, classification with a linear probe on top of SSL representations prevents from fitting the training set entirely, having small generalization gaps in-distribution [8]. In this way, the validation criterion remains strongly correlated even in larger validation accuracy values, maintaining its utility in a greater range of hyperparameter configurations.\n\n## 5 Related Work\n\nPrior literature can be grouped according to three main strategies that attempt to improve a model's robustness to dataset bias [54]. (i) _Resampling_ strategies increase or decrease the frequency of biased\n\n\\begin{table}\n\\begin{tabular}{l c} \\hline \\hline \\multicolumn{2}{c}{Group-balanced Test Acc.} \\\\ \\hline ERM & \\(25.82_{\\pm 0.33}\\) \\\\ GroupDRO & \\(38.30\\) \\\\ \\hline uLA w. MoCoV2+ & \\(62.49_{\\pm 0.74}\\) \\\\ uLA w. BYOL & \\(59.73_{\\pm 2.03}\\) \\\\ uLA w. BarlowTwins & \\(50.08_{\\pm 1.23}\\) \\\\ \\hline \\hline \\end{tabular}\n\\end{table}\nTable 4: Ablation of SSL pretraining methods for the backbone model. Default \\(\\eta=1.0\\) and \\(\\tau=1.0\\) are used. We pretrain for \\(1000\\) epochs, and train the linear head of the bias proxy for \\(100\\) epochs. We report avg. group-balanced test accuracy (%) and std. dev. over 5 seeds on cCIFAR10 (1%).\n\nFigure 4: Validation vs test on cCIFAR10 (1%).\n\n\\begin{table}\n\\begin{tabular}{l c} \\hline \\hline Pretraining method & Pearson correlation \\\\ \\hline End-to-end supervised & \\(0.690\\) (\\(0.588,0.770\\)) \\\\ SSL + linear head & \\(0.819\\) (\\(0.721,0.885\\)) \\\\ \\hline \\hline \\end{tabular}\n\\end{table}\nTable 5: Effect of pretraining method for a bias proxy network on results of hyperparameter search using our validation criterion. Each training trial’s hyperparameters were sampled from the same prior. Pretraining with SSL results in trials whose best validation criterion correlates better with selected model’s group-balanced test accuracy on cCIFAR10 (1%). In parenthesis, we compute 95% confidence intervals.\n\nattributes in the input space [27; 55] or latent space [23; 74; 17]. (ii) _Loss reweighting_ methods balance class or feature importances during training [74; 21; 12; 65; 40]. Especially relevant to our work are the reweighting methods that improve robustness and generalization when training on biased datasets [61; 57; 17; 48]. (iii) _Post-hoc adaptation_ methods [54; 25; 19; 42; 41; 3; 70] correct the biases learned by already-trained models. Most relevant to our work is the logit adjustment technique proposed by Menon et al. [54] for long-tail learning, which leverages class frequencies to rebalance the model predictions after training or to train with a loss function which is aware of the class prior. We develop a logit adjustment technique for the problem of learning from biased data, which does not require previous knowledge about the dataset's biases.\n\nSeveral solutions have emerged for the group robustness problem, especially when bias attribute data is available. For example, GroupDRO [61] leverages explicit bias attributes to reduce worst-case group loss. Our method, however, reduces reliance on such data, recognizing that access to them can be impractical. Bias-unsupervised methods like Learning from Failure (LFF) [57] and Disentangled Feature Augmentation (DFA) [17] reweight the loss of the unbiased model using a co-trained biased model, removing the need for bias supervision. Conversely, Just-train-twice (JT) [48] reweights misclassified samples from an initial biased training, emphasizing worst-performing group data. Similarly, He et al. [33] exemplify the same intuition in that a second network is trained on examples that cannot be predicted already using the spurious structure. Liu et al. [50] proposed a bias-unsupervised logit adjustment technique (LC) also based on co-training a biased network. Utilizing domain knowledge, Clark et al. [18] describe a bias-supervised logit adjustment approach to debiasing Visual Question Answering (VQA) models by incorporating a bias proxy which is trained exclusively on question data. These methods only generalize to seen attribute groups during training and require bias knowledge during validation for optimal performance. We tackle these issues by employing SSL to pre-train a network, deriving a bias proxy for debiased model training and validation, as well as using it as initialization.\n\nIn order to derive bias-unsupervised solutions, literature has proposed to train reference models as proxies for the missing bias labels. Creager et al. [20] optimize a reference model for group assignments which maximally violate a relaxation of the Environment Invariance Criterion. Chen et al. [16] seek to establish conditional independence between the predictions of the proxy model and the target variable given the inferred groups. Our work follows more closely the approach of [48], in which a biased network is simply trained with ERM. As we demonstrate at Figure 6 of Appendix F, by utilizing a frozen backbone pretrained with SSL, our approach improves on the sensitivity to the number of training steps for the bias proxy.\n\nBardenhagen et al. [9] suggested a validation scheme dependent on early stopping of bias proxy network training. Our method, using SSL pretraining, avoids this by treating pretraining steps as tunable hyperparameters, maintaining performance of alternatives which were otherwise tuned with bias information. Chen et al. [16] perform experiments using a methodology dubbed as Training Environments Validation (TEV). Similar to us, TEV validates models based on inferred groups from training, however the methodology is unfortunately not well documented in the literature, making its reproducibility difficult. As we show in the ablation study of Figure 4, implementation details can make a large difference in the quality of the criterion. Finally, MaskTune[3] eliminates spurious shortcuts by masking input data during a secondary training phase. Despite its resilience in performance without using a bias-unsupervised o.o.d. model selection criterion, the need for a reliable validation strategy for group robustness remains.\n\n## 6 Conclusion\n\nWe explored group robust classification in synthetic and real tasks, proposing a generalization task with unseen attribute combinations. Current robust classification methods struggle in this setting, motivating our SSL-based logit adjustment approach. Importantly, we introduce **a methodology for training and validating robust models without group labels**. Empirical evaluations on five datasets show our method outperforms existing fully bias-unsupervised approaches and rivals those using bias annotations during validation. In terms of _limitations and broader impact_ of our contributions, as machine learning systems handle high-stakes applications, ensuring robustness to underrepresented samples is crucial. Our work reduces reliance on known data biases, but existing benchmarks differ from real-life scenarios with unknown biased attribute combinations. To bridge this gap, we proposed a synthetic benchmark and encourage further research on real data, revealing more system limitations.\n\n## References\n\n* Ahmed et al. [2021] F. Ahmed, Y. Bengio, H. van Seijen, and A. Courville. Systematic generalisation with group invariant predictions. In _International Conference on Learning Representations_, 2021.\n* Arjovsky et al. [2019] M. Arjovsky, L. Bottou, I. Gulrajani, and D. Lopez-Paz. Invariant risk minimization. _arXiv preprint arXiv:1907.02893_, 2019.\n* Asgari et al. [2022] S. Asgari, A. Khani, F. Khani, A. Gholami, L. Tran, A. Mahdavi Amiri, and G. Hamarneh. Masktune: Mitigating spurious correlations by forcing to explore. In _Advances in Neural Information Processing Systems_, 2022.\n* Asmussen and Glynn [2007] S. Asmussen and P. Glynn. _Stochastic Simulation: Algorithms and Analysis_. Stochastic Modelling and Applied Probability. Springer New York, 2007.\n* Assran et al. [2023] M. Assran, R. Balestriero, Q. Duval, F. Bordes, I. Misra, P. Bojanowski, P. Vincent, M. Rabbat, and N. Ballas. The hidden uniform cluster prior in self-supervised learning. In _International Conference on Learning Representations_, 2023.\n* Bahng et al. [2020] H. Bahng, S. Chun, S. Yun, J. Choo, and S. J. Oh. Learning de-biased representations with biased representations. In _International Conference on Machine Learning_, 2020.\n* Balestriero et al. [2023] R. Balestriero, M. Ibrahim, V. Sobal, A. Morcos, S. Shekhar, T. Goldstein, F. Bordes, A. Bardes, G. Mialon, Y. Tian, et al. A cookbook of self-supervised learning. _arXiv preprint arXiv:2304.12210_, 2023.\n* Bansal et al. [2020] Y. Bansal, G. Kaplun, and B. Barak. For self-supervised learning, rationality implies generalization, provably. _arXiv preprint arXiv:2010.08508_, 2020.\n* Bardenhagen et al. [2021] V. Bardenhagen, A. Tifrea, and F. Yang. Boosting worst-group accuracy without group annotations. In _NeurIPS 2021 Workshop on Distribution Shifts: Connecting Methods and Applications_, 2021.\n* Blodgett et al. [2016] S. L. Blodgett, L. Green, and B. O'Connor. Demographic dialectal variation in social media: A case study of african-american english. _arXiv preprint arXiv:1608.08868_, 2016.\n* Brodersen et al. [2010] K. H. Brodersen, C. S. Ong, K. E. Stephan, and J. M. Buhmann. The balanced accuracy and its posterior distribution. In _International Conference on Pattern Recognition_, 2010.\n* Cao et al. [2019] K. Cao, C. Wei, A. Gaidon, N. Arechiga, and T. Ma. Learning imbalanced datasets with label-distribution-aware margin loss. _Advances in Neural Information Processing Systems_, 2019.\n* Caton and Haas [2020] S. Caton and C. Haas. Fairness in machine learning: A survey. _arXiv preprint arXiv:2010.04053_, 2020.\n* Chen et al. [2020] T. Chen, S. Kornblith, M. Norouzi, and G. Hinton. A simple framework for contrastive learning of visual representations. In _International Conference on Machine Learning_, 2020.\n* Chen et al. [2020] X. Chen, H. Fan, R. Girshick, and K. He. Improved baselines with momentum contrastive learning. _arXiv preprint arXiv:2003.04297_, 2020.\n* Chen et al. [2022] Y. Chen, R. Xiong, Z.-M. Ma, and Y. Lan. When does group invariant learning survive spurious correlations? In _Advances in Neural Information Processing Systems_, 2022.\n* Chu et al. [2021] S. Chu, D. Kim, and B. Han. Learning debiased and disentangled representations for semantic segmentation. In _Advances in Neural Information Processing Systems_, 2021.\n* Clark et al. [2019] C. Clark, M. Yatskar, and L. Zettlemoyer. Don't take the easy way out: Ensemble based methods for avoiding known dataset biases. In _Proceedings of the 2019 Conference on Empirical Methods in Natural Language Processing and the 9th International Joint Conference on Natural Language Processing (EMNLP-IJCNLP)_. Association for Computational Linguistics, 2019.\n* Collell et al. [2016] G. Collell, D. Prelec, and K. R. Patil. Reviving threshold-moving: a simple plug-in bagging ensemble for binary and multiclass imbalanced data. _ArXiv_, abs/1606.08698, 2016.\n\n* [20] E. Creager, J.-H. Jacobsen, and R. Zemel. Environment inference for invariant learning. In _International Conference on Machine Learning_, 2021.\n* [21] Y. Cui, M. Jia, T.-Y. Lin, Y. Song, and S. Belongie. Class-balanced loss based on effective number of samples. In _Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition_, 2019.\n* [22] V. G. T. da Costa, E. Fini, M. Nabi, N. Sebe, and E. Ricci. solo-learn: A library of self-supervised methods for visual representation learning. _Journal of Machine Learning Research_, 2022.\n* [23] L. Darlow, S. Jastrzebski, and A. Storkey. Latent adversarial debiasing: Mitigating collider bias in deep neural networks. _arXiv preprint arXiv:2011.11486_, 2020.\n* [24] K. Eykholt, I. Evtimov, E. Fernandes, B. Li, A. Rahmati, C. Xiao, A. Prakash, T. Kohno, and D. Song. Robust physical-world attacks on deep learning visual classification. In _Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition_, 2018.\n* [25] T. Fawcett and F. Provost. Combining data mining and machine learning for effective user profiling. In _International Conference on Knowledge Discovery and Data Mining_, 1996.\n* [26] J. A. Fodor and Z. W. Pylyshyn. Connectionism and cognitive architecture: A critical analysis. _Cognition_, 28(1), 1988.\n* [27] R. Geirhos, P. Rubisch, C. Michaelis, M. Bethge, F. A. Wichmann, and W. Brendel. Imagenet-trained cnns are biased towards texture; increasing shape bias improves accuracy and robustness. In _International Conference on Learning Representations_, 2018.\n* [28] R. Geirhos, J.-H. Jacobsen, C. Michaelis, R. Zemel, W. Brendel, M. Bethge, and F. A. Wichmann. Shortcut learning in deep neural networks. _Nature Machine Intelligence_, 2020.\n* [29] M. W. Gondal, M. Wuthrich, D. Miladinovic, F. Locatello, M. Breidt, V. Volchkov, J. Akpo, O. Bachem, B. Scholkopf, and S. Bauer. On the transfer of inductive bias from simulation to the real world: a new disentanglement dataset. In _Advances in Neural Information Processing Systems_, 2019.\n* [30] J.-B. Grill, F. Strub, F. Altche, C. Tallec, P. Richemond, E. Buchatskaya, C. Doersch, B. Avila Pires, Z. Guo, M. Gheshlaghi Azar, et al. Bootstrap your own latent-a new approach to self-supervised learning. _Advances in Neural Information Processing Systems_, 2020.\n* [31] C. Guo, G. Pleiss, Y. Sun, and K. Q. Weinberger. On calibration of modern neural networks. In _International Conference on Machine Learning_, 2017.\n* [32] S. Gururangan, S. Swayamdipta, O. Levy, R. Schwartz, S. Bowman, and N. A. Smith. Annotation artifacts in natural language inference data. In _Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies_, 2018.\n* [33] H. He, S. Zha, and H. Wang. Unlearn dataset bias in natural language inference by fitting the residual. In _Proceedings of the 2nd Workshop on Deep Learning Approaches for Low-Resource NLP (DeepLo 2019)_. Association for Computational Linguistics, 2019.\n* [34] K. He, X. Zhang, S. Ren, and J. Sun. Deep residual learning for image recognition. In _Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition_, 2016.\n* [35] K. He, H. Fan, Y. Wu, S. Xie, and R. Girshick. Momentum contrast for unsupervised visual representation learning. In _Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition_, 2020.\n* [36] L. A. Hendricks, K. Burns, K. Saenko, T. Darrell, and A. Rohrbach. Women also snowboard: Overcoming bias in captioning models. In _European Conference on Computer Vision (ECCV)_, 2018.\n* [37] D. Hendrycks and T. Dietterich. Benchmarking neural network robustness to common corruptions and perturbations. In _International Conference on Learning Representations_, 2019.\n\n* [38] D. Hovy and A. Sogaard. Tagging performance correlates with author age. In _Association for Computational Linguistics_, 2015.\n* [39] P. Izmailov, P. Kirichenko, N. Gruver, and A. G. Wilson. On feature learning in the presence of spurious correlations. In _Advances in Neural Information Processing Systems_, 2022.\n* [40] M. A. Jamal, M. Brown, M.-H. Yang, L. Wang, and B. Gong. Rethinking class-balanced methods for long-tailed visual recognition from a domain adaptation perspective. In _Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition_, 2020.\n* [41] B. Kang, S. Xie, M. Rohrbach, Z. Yan, A. Gordo, J. Feng, and Y. Kalantidis. Decoupling representation and classifier for long-tailed recognition. _arXiv preprint arXiv:1910.09217_, 2019.\n* [42] B. Kim and J. Kim. Adjusting decision boundary for class imbalanced learning. _IEEE Access_, 2020.\n* [43] P. Kirichenko, P. Izmailov, and A. G. Wilson. Last layer re-training is sufficient for robustness to spurious correlations. In _International Conference on Learning Representations_, 2023.\n* [44] O. O. Koyejo, N. Natarajan, P. K. Ravikumar, and I. S. Dhillon. Consistent binary classification with generalized performance metrics. In _Advances in Neural Information Processing Systems_, 2014.\n* [45] A. Krizhevsky and G. Hinton. Learning multiple layers of features from tiny images. Technical report, University of Toronto, 2009.\n* [46] B. Lake and M. Baroni. Generalization without systematicity: On the compositional skills of sequence-to-sequence recurrent networks. In _International Conference on Machine Learning_, 2018.\n* [47] Y. LeCun, C. Cortes, and C. J. Burges. Mnist handwritten digit database. In _Proceedings of the IEEE_, 1998.\n* [48] E. Z. Liu, B. Haghgoo, A. S. Chen, A. Raghunathan, P. W. Koh, S. Sagawa, P. Liang, and C. Finn. Just train twice: Improving group robustness without training group information. In _International Conference on Machine Learning_, 2021.\n* [49] H. Liu, J. Z. HaoChen, A. Gaidon, and T. Ma. Self-supervised learning is more robust to dataset imbalance. In _NeurIPS 2021 Workshop on Distribution Shifts: Connecting Methods and Applications_, 2021.\n* [50] S. Liu, X. Zhang, N. Sekhar, Y. Wu, P. Singhal, and C. Fernandez-Granda. Avoiding spurious correlations via logit correction. In _International Conference on Learning Representations_, 2023.\n* [51] Z. Liu, P. Luo, X. Wang, and X. Tang. Deep learning face attributes in the wild. In _Proceedings of the IEEE international conference on computer vision_, 2015.\n* [52] I. Loshchilov and F. Hutter. Decoupled weight decay regularization. _arXiv preprint arXiv:1711.05101_, 2017.\n* [53] A. Menon, H. Narasimhan, S. Agarwal, and S. Chawla. On the statistical consistency of algorithms for binary classification under class imbalance. In _International Conference on Machine Learning_, 2013.\n* [54] A. K. Menon, S. Jayasumana, A. S. Rawat, H. Jain, A. Veit, and S. Kumar. Long-tail learning via logit adjustment. In _International Conference on Learning Representations_, 2021.\n* [55] M. Minderer, O. Bachem, N. Houlsby, and M. Tschannen. Automatic shortcut removal for self-supervised representation learning. In _International Conference on Machine Learning_, 2020.\n* [56] R. Muller, S. Kornblith, and G. E. Hinton. When does label smoothing help? _Advances in Neural Information Processing Systems_, 2019.\n\n* [57] J. Nam, H. Cha, S. Ahn, J. Lee, and J. Shin. Learning from failure: De-biasing classifier from biased classifier. _Advances in Neural Information Processing Systems_, 2020.\n* [58] A. v. d. Oord, Y. Li, and O. Vinyals. Representation learning with contrastive predictive coding. _arXiv preprint arXiv:1807.03748_, 2018.\n* [59] A. Paszke, S. Gross, F. Massa, A. Lerer, J. Bradbury, G. Chanan, T. Killeen, Z. Lin, N. Gimelshein, L. Antiga, et al. Pytorch: An imperative style, high-performance deep learning library. _Advances in Neural Information Processing Systems_, 2019.\n* [60] O. Russakovsky, J. Deng, H. Su, J. Krause, S. Satheesh, S. Ma, Z. Huang, A. Karpathy, A. Khosla, M. Bernstein, et al. Imagenet large scale visual recognition challenge. _International journal of computer vision_, 2015.\n* [61] S. Sagawa, P. W. Koh, T. B. Hashimoto, and P. Liang. Distributionally robust neural networks. In _International Conference on Learning Representations_, 2020.\n* [62] S. Sagawa, A. Raghunathan, P. W. Koh, and P. Liang. An investigation of why overparameterization exacerbates spurious correlations. In _International Conference on Machine Learning_, 2020.\n* [63] Y. Shi, I. Daunhawer, J. E. Vogt, P. Torr, and A. Sanyal. How robust is unsupervised representation learning to distribution shift? In _International Conference on Learning Representations_, 2023.\n* [64] R. Suter, D. Miladinovic, B. Scholkopf, and S. Bauer. Robustly disentangled causal mechanisms: Validating deep representations for interventional robustness. In _International Conference on Machine Learning_, 2019.\n* [65] J. Tan, C. Wang, B. Li, Q. Li, W. Ouyang, C. Yin, and J. Yan. Equalization loss for long-tailed object recognition. In _Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition_, 2020.\n* [66] R. Tatman. Gender and dialect bias in youtube's automatic captions. In _ACL workshop on ethics in natural language processing_, 2017.\n* [67] V. N. Vapnik. _Statistical Learning Theory_. Wiley-Interscience, 1998.\n* [68] C. Wah, S. Branson, P. Welinder, P. Perona, and S. Belongie. The caltech-ucsd birds-200-2011 dataset. Technical report, California Institute of Technology, 2011.\n* [69] H. Wang, Z. He, Z. C. Lipton, and E. P. Xing. Learning robust representations by projecting superficial statistics out. In _International Conference on Learning Representations_, 2018.\n* [70] J. Wei, H. Narasimhan, E. Amid, W.-S. Chu, Y. Liu, and A. Kumar. Distributionally robust post-hoc classifiers under prior shifts. In _International Conference on Learning Representations_, 2023.\n* [71] Y. Xue, K. Whitecross, and B. Mirzasoleiman. Investigating why contrastive learning benefits robustness against label noise. In _International Conference on Machine Learning_, 2022.\n* [72] Y. You, I. Gitman, and B. Ginsburg. Large batch training of convolutional networks. _arXiv preprint arXiv:1708.03888_, 2017.\n* [73] J. Zbontar, L. Jing, I. Misra, Y. LeCun, and S. Deny. Barlow twins: Self-supervised learning via redundancy reduction. In _International Conference on Machine Learning_, 2021.\n* [74] X. Zhang, Z. Fang, Y. Wen, Z. Li, and Y. Qiao. Range loss for deep face recognition with long-tailed training data. In _International Conference on Computer Vision_, 2017.\n* [75] Z. Zhang and M. Sabuncu. Generalized cross entropy loss for training deep neural networks with noisy labels. In _Advances in Neural Information Processing Systems_, 2018.\n\n* [76] C. Zhao, J. Mangat, S. Koujalgi, A. Squicciarini, and C. Caragea. Privacyalert: A dataset for image privacy prediction. In _Proceedings of the International AAAI Conference on Web and Social Media_, 2022.\n* [77] B. Zhou, A. Lapedriza, A. Khosla, A. Oliva, and A. Torralba. Places: A 10 million image database for scene recognition. _IEEE transactions on pattern analysis and machine intelligence_, 2017.\n\n## Appendix A Bias-Supervised Training with Logit Adjustment\n\nThe goal of this Appendix section is to develop a bias-supervised approach for maximizing the group-balanced accuracy (Equation (3)), which is going to serve as an initial point in the development of our entirely bias-unsupervised methodology in Section 3.\n\nUnder the problem formulation we introduced in Section 2, the metric that we would ultimately like to maximize given a scoring function \\(f_{Y|X}:\\mathcal{X}\\rightarrow\\mathbb{R}^{K}\\) is \\(\\operatorname{Acc}_{Y|X}(f_{Y|X};p_{\\text{test}})\\), the top-1 accuracy under the test distribution \\(p_{\\text{test}}\\). Simple algebraic manipulations reveal that this test accuracy corresponds to the group-balanced accuracy:\n\n\\[\\operatorname{Acc}_{Y|X}(f_{Y|X};p_{\\text{test}}) :=\\operatorname*{\\mathbb{E}}_{x,y\\sim p_{\\text{test}}}\\mathbb{1} \\left(y=\\operatorname*{argmax}_{y^{\\prime}}f_{Y|X}(x)_{y^{\\prime}}\\right) \\tag{8}\\] \\[=\\operatorname*{\\mathbb{E}}_{\\begin{subarray}{c}y,z\\sim p_{\\text {test}}\\\\ x\\sim p_{\\text{test}}(\\cdot\\,|y,z)\\end{subarray}}\\mathbb{1}\\left(y= \\operatorname*{argmax}_{y^{\\prime}}f_{Y|X}(x)_{y^{\\prime}}\\right)\\] \\[=\\frac{1}{KL}\\sum_{(y,z)\\in[K]\\times[L]}\\operatorname*{\\mathbb{E} }_{x\\sim p_{\\text{test}}(\\cdot\\,|y,z)}\\mathbb{1}\\left(y=\\operatorname*{argmax }_{y^{\\prime}}f_{Y|X}(x)_{y^{\\prime}}\\right).\\]\n\nA related quantity in the literature is the class-balanced top-1 accuracy, which is used in class-imbalanced classification problems [11, 53]. We can make the connection by imagining \\((y,z)\\) as a multi-label classification target in a multi-label, but class-imbalanced, problem. The multi-label class-balanced accuracy of a multi-label scoring function \\(f_{Y,Z|X}:\\mathcal{X}\\rightarrow\\mathbb{R}^{K}\\times\\mathbb{R}^{L}\\) would then be\n\n\\[\\operatorname{Acc}_{Y,Z|X}(f_{Y,Z|X};p_{\\text{test}})=\\frac{1}{KL}\\sum_{(y,z) \\in[K]\\times[L]}\\operatorname*{\\mathbb{E}}_{x\\sim p_{\\text{test}}(\\cdot\\,|y,z )}\\mathbb{1}\\left((y,z)=\\operatorname*{argmax}_{y^{\\prime},z^{\\prime}}f_{Y,Z|X }(x)_{y^{\\prime},z^{\\prime}}\\right). \\tag{9}\\]\n\nWe will set towards building a multi-label classifier which maximizes multi-label class-balanced accuracy and we will finally show that its part which only predicts \\(Y\\) from \\(X\\) can be shown to maximize a lower bound to the group-balanced accuracy.\n\nWe assume that \\(\\mathcal{D}\\) and \\(\\mathcal{D}_{\\text{valid}}\\) provide us with tuples of observations \\((x,y,z)\\) distributed according to \\(p_{\\text{data}}\\). In this case, results in the class-imbalance literature [53, 44, 19] state that the Bayes-optimal scoring function \\(f^{\\star}_{Y,Z|X}\\) for maximizing the class-balanced criterion (9) is given by\n\n\\[\\operatorname*{argmax}_{y,z}f^{\\star}_{Y,Z|X}(x)_{y,z}=\\operatorname*{argmax} _{y,z}p_{\\text{test}}(x\\,|\\,y,z). \\tag{10}\\]\n\nNote that under the invariance of mechanism assumption (2) we have that \\(p_{\\text{test}}(x\\,|\\,y,z)=p_{\\text{data}}(x\\,|\\,y,z)\\). This allows us to compute the mechanism conditional probability in \\(p_{\\text{data}}\\) distribution terms\n\n\\[p_{\\text{test}}(x\\,|\\,y,z)=\\frac{p_{\\text{data}}(y,z\\,|\\,x)}{p_{\\text{data}}( y,z)}p_{\\text{data}}(x) \\tag{11}\\]\n\nby utilizing the Bayes rule. This way we have a candidate strategy to maximize multi-label class-balanced accuracy in Equation (9): First, we will estimate \\(p_{\\text{data}}(y,z\\,|\\,x)\\) and \\(p_{\\text{data}}(y,z)\\) from data, and then we will divide the estimates. Notice that, in order to make a prediction about \\((y,z)\\) given \\(x\\), we do not need to model \\(p_{\\text{data}}(x)\\) as this quantity is constant in \\((y,z)\\), and thus does not influence the \\(\\operatorname*{argmax}\\).\n\n**Training process.** We can directly estimate \\(p_{\\text{data}}(y,z)\\) from available training data by computing the empirical frequencies of pairs \\((y,z)\\in[L]\\times[K]\\), or by a Bayesian estimate with a Dirichlet prior. On the other hand, we can train a parameterized discriminative model \\(p_{\\theta}\\) with maximum conditional likelihood estimation to estimate \\(p_{\\text{data}}(y,z|x)\\)\n\n\\[\\max_{\\theta}\\frac{1}{|\\mathcal{D}|}\\sum_{x,y,z\\in\\mathcal{D}}\\log p_{\\theta}( y,z\\,|\\,x) \\tag{12}\\]\n\nwhich is equivalent to the minimization of a KL divergence estimate between the \\(p_{\\text{data}}\\) and \\(p_{\\theta}\\) model conditionals.\n\nThe method described so far would have sufficed if the goal was to perform multi-label classification, however we need to devise a way to overcome the bias-supervision, since we only care about classifying \\(Y\\) without having access to \\(Z\\) annotations. For this reason, we need to proceed to modelling assumptions about \\(p_{\\theta}\\), which will allow us to develop the bias-unsupervised method described in Section 3. First, we recall that \\(p_{\\text{data}}\\) factorizes as\n\n\\[p_{\\text{data}}(y,z\\,|\\,x)=p_{\\text{data}}(y\\,|\\,x,z)p_{\\text{data}}(z\\,|\\,x). \\tag{13}\\]\n\nLet \\(h_{\\theta}:\\mathcal{X}\\to\\mathbb{R}^{K}\\) be a parameterized model (such as a neural network) tasked to predict unnormalized logits of \\(Y\\) given an observation \\(x\\). Then, we model \\(p_{\\text{data}}(y\\,|\\,x,z)\\) by\n\n\\[p_{\\theta}(y\\,|\\,x,z)\\propto\\exp\\big{(}h_{\\theta}(x)_{y}+\\log\\hat{p}_{\\text{ data}}(y\\,|\\,z)\\big{)}. \\tag{14}\\]\n\nUnpacking our modelling assumption Equation (14), we have the following: First, we are going to compute \\(\\hat{p}_{\\text{data}}(y\\,|\\,z)=\\frac{\\hat{p}_{\\text{data}}(y,z)}{\\hat{p}_{ \\text{data}}(z)}\\), by using our estimated \\(\\hat{p}_{\\text{data}}(y,z)\\) and by marginalizing out \\(y\\) for \\(\\hat{p}_{\\text{data}}(z)=\\sum_{y\\in[K]}\\hat{p}_{\\text{data}}(y,z)\\). Given a tuple of observables \\((x,y,z)\\) in a training batch, we use \\(\\log\\hat{p}_{\\text{data}}(y\\,|\\,z)\\) to adjust additively the outputs of our parametric discriminative model \\(h_{\\theta}\\) to the observed input \\(x\\). Notice that since \\(Y\\) is a categorical variable, \\(\\log\\hat{p}_{\\text{data}}(y\\,|\\,z)\\in(-\\infty,0]\\) for all \\(y\\in[K]\\). For this reason, we can intuitively interpret _logit adjustment_ as a soft masking operation for outputs of \\(h_{\\theta}\\) which are unlikely in the training data when we have observed \\(z\\). In this way, we account for the dependency of \\(Y\\) to \\(Z\\) which spuriously exists in the training distribution. By fitting the cross-entropy objective under logit adjustment, the network \\(h_{\\theta}\\) has to model the remaining relations for \\(Y\\,|\\,X\\) that are not spurious since those are already accounted for.\n\n**Prediction process.** Our modelling assumption enables the prediction of \\(y\\) without computing the maximizer over combinations of \\((y,z)\\in[K]\\times[L]\\). This is an important step towards developing a bias-unsupervised training procedure. This is because assumption 14 allows us to compute the maximizer for \\(y\\,|\\,x\\) and \\(z\\,|\\,x\\) separately. We show that under our modelling assumption, if we are only interested in predicting only the target variable \\(y\\) given an observation \\(x\\), we do not need to model \\(z\\,|\\,x\\) at all. To understand how this is possible, we take a close look at the total expression for optimal predictor. After fitting the model \\(p_{\\theta}\\) using Equation (12), we calculate the estimated scoring function for the multi-label class-imbalanced problem as in Equation (11)\n\n\\[\\hat{f}_{Y,Z|X}(x)=\\hat{p}_{\\text{test}}(x\\,|\\,y,z)=\\frac{\\hat{p} _{\\text{data}}(y,z\\,|\\,x)}{\\hat{p}_{\\text{data}}(y,z)}\\,\\hat{p}_{\\text{data}}(x) \\tag{15}\\] \\[=\\frac{p_{\\theta}(y\\,|\\,x,z)}{\\hat{p}_{\\text{data}}(y\\,|\\,z)}\\, \\frac{\\hat{p}_{\\text{data}}(z\\,|\\,x)}{\\hat{p}_{\\text{data}}(z)}\\,\\hat{p}_{ \\text{data}}(x)\\] \\[=\\frac{\\exp\\big{(}h_{\\theta}(x)_{y}+\\log\\hat{p}_{\\text{data}}(y \\,|\\,z)\\big{)}}{Z_{\\theta}(z,x)\\,\\hat{p}_{\\text{data}}(y\\,|\\,z)}\\,\\frac{\\hat{p }_{\\text{data}}(z\\,|\\,x)}{\\hat{p}_{\\text{data}}(z)}\\,\\hat{p}_{\\text{data}}(x)\\] \\[=\\exp\\big{(}h_{\\theta}(x)_{y}\\big{)}\\,\\frac{\\hat{p}_{\\text{data}}( z\\,|\\,x)\\hat{p}_{\\text{data}}(x)}{Z_{\\theta}(z,x)\\hat{p}_{\\text{data}}(z)},\\]\n\nwhere \\(Z_{\\theta}(z,x)\\coloneqq\\sum_{y\\in[K]}\\exp\\big{(}h_{\\theta}(x)_{y}+\\log\\hat{p }_{\\text{data}}(y\\,|\\,z)\\big{)}\\) is the partition function of \\(p_{\\theta}(y|x,z)\\). Notice that the expression simplifies into a multiplication of two terms; the first, \\(\\exp\\big{(}h_{\\theta}(x)_{y}\\big{)}\\), solely depends on \\((x,y)\\), while the second fraction only depends on \\((x,z)\\). Then, predicting according to Equation (10) amounts to computing\n\n\\[\\max_{y,z}\\hat{p}_{\\text{test}}(x\\,|\\,y,z)=\\max_{y}\\exp\\big{(}h_{\\theta}(x)_{ y}\\big{)}\\,\\max_{z}\\frac{\\hat{p}_{\\text{data}}(z|x)\\hat{p}_{\\text{data}}(x)}{Z_{ \\theta}(z,x)\\hat{p}_{\\text{data}}(z)}. \\tag{16}\\]\n\nEssentially, information about \\(Z\\) is only need during training time and not during prediction of \\(Y\\) given \\(X\\). In practice, during prediction we adapt the model by removing the logit adjustment term in order to acquire the unbiased network for the task. Consequently, the obtained scoring function for debiased \\(Y\\) predictions is just the trained neural network\n\n\\[\\hat{f}_{Y|X}(x;\\theta)\\coloneqq h_{\\theta}(x). \\tag{17}\\]\n\nNotice that our design decisions are well-justified, in the sense that, under our class of models (eq. 14), we have \\(\\operatorname{Acc}_{Y,Z|X}(\\hat{f}_{Y,Z|X};p_{\\text{test}})\\leq\\operatorname{ Acc}_{Y|X}(\\hat{f}_{Y|X};p_{\\text{test}})\\). This means that by maximizing the multi-label class-balanced accuracy we indirectly maximize the top-1 accuracy of the candidate scoring function \\(\\hat{f}_{Y|X}\\) under the test distribution \\(p_{\\text{test}}\\), which is the group-balanced accuracy as we have demonstrated.\n\n**Proposition A.1** (sLA optimizes the group-balanced accuracy).: _Under the assumption that the hypothesis class of \\(p_{\\theta}(y|x,z)\\) (eq. 4) contains \\(p_{\\text{data}}(y|x,z)\\), the minimizer network \\(h_{\\theta^{*}}\\) of the cross-entropy loss maximizes a lower bound to the group-balanced accuracy \\(\\operatorname{Acc}_{Y|X}(h_{\\theta^{*}};p_{\\text{test}})\\)._\n\nProof.: We will start by the quantity\n\n\\[\\operatorname{Acc}_{Y,Z|X}\\bigl{(}f;p_{\\text{test}}(\\cdot|y,z)\\bigr{)}\\coloneqq \\mathbb{P}_{x|y,z}^{\\text{test}}\\bigl{(}(y,z)=\\operatorname*{argmax}_{y^{\\prime },z^{\\prime}}f(x)_{y^{\\prime},z^{\\prime}}\\bigr{)} \\tag{18}\\]\n\nthe classification accuracy conditioned on the pair \\((y,z)\\). Recall that under our modelling assumption Equation (14), the estimated Bayes-optimal scoring function for the multi-label class-imbalanced classification problem is\n\n\\[\\hat{f}_{Y,Z|X}(x)=\\hat{p}_{\\text{test}}(x\\,|\\,y,z)=\\exp\\big{(}h_{\\theta^{*}}( x)_{y}\\big{)}C(z,x), \\tag{19}\\]\n\nwhere \\(C(z,x)\\coloneqq\\frac{\\hat{p}_{\\text{data}}(z|x)\\hat{p}_{\\text{data}}(x)}{\\hat {Z}_{\\theta}(z,x)\\hat{p}_{\\text{data}}(z)}\\) is a constant in \\(y\\). We then substitute to get\n\n\\[\\operatorname{Acc}_{Y,Z|X}\\bigl{(}\\hat{f}_{Y,Z|X};p_{\\text{test}}(\\cdot|y,z) \\bigr{)}=\\mathbb{P}_{x|y,z}^{\\text{test}}\\Bigl{(}(y,z)=\\operatorname*{argmax}_ {y^{\\prime},z^{\\prime}}\\exp\\big{(}h_{\\theta^{*}}(x)_{y^{\\prime}}\\big{)}C(z^{ \\prime},x)\\Bigr{)}. \\tag{20}\\]\n\nHowever \\(\\max_{z^{\\prime}}\\) gets pushed to the innermost term since \\(\\exp\\big{(}h_{\\theta^{*}}(x)_{y^{\\prime}}\\big{)}\\) is independent of \\(z^{\\prime}\\), as well as \\(\\max_{z^{\\prime}}C(z^{\\prime},x)\\) is independent of \\(y^{\\prime}\\). This means that we can take the \\(\\operatorname{argmax}\\) of individual expression independently\n\n\\[\\operatorname{Acc}_{Y,Z|X}\\bigl{(}\\hat{f}_{Y,Z|X};p_{\\text{test}} (\\cdot|y,z)\\bigr{)}= \\tag{21}\\] \\[\\mathbb{P}_{x|y,z}^{\\text{(test)}}\\bigl{(}\\{y=\\operatorname*{ argmax}_{y^{\\prime}}\\exp\\big{(}h_{\\theta^{*}}(x)_{y^{\\prime}}\\big{)}\\}\\cap\\{z= \\operatorname*{argmax}_{z^{\\prime}}C_{2}(z^{\\prime},x)\\}\\bigr{)},\\]\n\nand since \\(\\mathbb{P}(A\\cap B)\\leq\\mathbb{P}(A)\\) and \\(\\exp\\) is a strictly increasing function, we get\n\n\\[\\operatorname{Acc}_{Y,Z|X}\\bigl{(}\\hat{f}_{Y,Z|X};p_{\\text{test}}(\\cdot|y,z) \\bigr{)}\\leq\\mathbb{P}_{x|y,z}^{\\text{(test)}}\\bigl{(}y=\\operatorname*{argmax} _{y^{\\prime}}h_{\\theta^{*}}(x)_{y^{\\prime}}\\bigr{)}. \\tag{22}\\]\n\nWe recognise the scoring function \\(\\hat{f}_{Y|X}(x)\\coloneqq h_{\\theta^{*}}(x)\\) on the right hand side of the expression. Finally, by summing up all inequalities for all \\((y,z)\\in[K]\\times[L]\\), and dividing by \\(KL\\), we get\n\n\\[\\frac{1}{KL}\\sum_{y,z}\\operatorname{Acc}_{Y,Z|X}\\bigl{(}\\hat{f}_ {Y,Z|X};p_{\\text{test}}(\\cdot|y,z)\\bigr{)}\\leq\\frac{1}{KL}\\sum_{y,z}\\mathbb{ P}_{x|y,z}^{\\text{(test)}}\\bigl{(}y=\\operatorname*{argmax}_{y^{\\prime}}\\hat{f}_{Y|X} (x)_{y^{\\prime}}\\bigr{)} \\tag{23}\\] \\[\\operatorname{Acc}_{Y,Z|X}\\bigl{(}\\hat{f}_{Y,Z|X};p_{\\text{test}} \\bigr{)}\\leq\\operatorname{Acc}_{Y|X}(\\hat{f}_{Y|X};p_{\\text{test}}). \\tag{24}\\]\n\nThis shows that the scoring function \\(\\hat{f}_{Y|X}(x)\\coloneqq h_{\\theta^{*}}(x)\\) maximizes a lower bound to the group-balanced accuracy. \n\n## Appendix B Self-supervised Learning Algorithms\n\n\\begin{table}\n\\begin{tabular}{l c c c c} \\hline \\hline  & \\multicolumn{2}{c}{Target Attribute} & \\multicolumn{2}{c}{Bias Attribute} \\\\ \\cline{2-5}  & \\(p_{\\text{data}}\\) & \\(p_{\\text{test}}\\) & \\(p_{\\text{data}}\\) & \\(p_{\\text{test}}\\) \\\\ \\hline cMNIST (\\%) & \\(98.86\\) & \\(27.01\\) & \\(99.88\\) & \\(96.04\\) \\\\ cCIFAR10 (\\(10\\)) & \\(98.93\\) & \\(36.53\\) & \\(99.60\\) & \\(82.05\\) \\\\ SMP19D (\\(C=3\\)) & \\(98.19\\) & \\(66.08\\) & \\(99.91\\) & \\(87.22\\) \\\\ CelebA\\({}^{*}\\) & \\(94.01\\) & \\(75.44\\) & \\(95.74\\) & \\(90.60\\) \\\\ \\hline \\hline \\end{tabular}\n\\end{table}\nTable 6: Downstream online linear classification against the _target_ and _bias_ attributes of each task considered (see Appendix D). We present top-1 (average) accuracy (%) of the corresponding classifiers evaluated at last epoch of MoCoV2+ training, under samples from the i.i.d. validation set (\\(p_{\\text{data}}\\)) and the o.o.d. test set (\\(p_{\\text{test}}\\)). \\({}^{*}\\)For CelebA, since the test split is not controlled to be group-balanced, we simulate the accuracy under \\(p_{\\text{test}}\\) by utilizing the existing bias annotations to estimate a group-balanced test accuracy.\n\nIn our work, we are using a contrastive learning algorithm [58, 14, 35, 15] to pretrain a base model for the bias proxy, as well as to initialize the debiased model. In contrastive learning, an instance discrimination pretext task is defined from unlabeled data. In particular, pairs of data points (positives), which are derived by independently augmenting the same observation (views), should have representations that are closer in distance than representations of other samples in the dataset (negatives). To achieve that we optimize a form of the InfoNCE loss [58]\n\n\\[\\mathcal{L}(q,k^{+},\\{k^{-}_{i}\\}_{i=1}^{N})=-\\log\\frac{\\exp(q^{\\top}k^{+}/ \\tau)}{\\exp(q^{\\top}k^{+}/\\tau)+\\sum_{i=1}^{N}\\exp(q^{\\top}k^{-}_{i}/\\tau)}, \\tag{25}\\]\n\nwhere \\((q,k^{+})\\) is the positive pair and \\(\\{k^{-}_{i}\\}_{i=1}^{N}\\) are the negatives. Typically, representations are projected to the unit hypersphere for training stability. In the MoCoV2+ framework [15], gradients are backpropagated only through the query representation (\\(q\\)) to the encoding network, while the key representations (\\(k^{+}\\) and \\(k^{-}\\)) are extracted by a network derived from the exponential moving average of its parameters. Finally, positive key representations of past batches are kept in a queue in memory to serve as negative key representations for the subsequent batches.\n\n## Appendix C uLA: Algorithm\n\nIn Algorithm 1, we provide a high-level description of the uLA methodology for training and validation of group-robust models without any bias annotations. In addition, a PyTorch [59] implementation is available at the following repository: [https://github.com/tsirif/uLA](https://github.com/tsirif/uLA).\n\n## Appendix D Datasets\n\n### Systematic MPI3D\n\nsMPI3D is our contributed dataset which we use to study combinatorial systematicity in classifiers. By this term, we designate the ability of classifiers to generalize to samples generated from novel combinations of generative attributes, which under \\(p_{\\text{data}}\\) have \\(0\\) probability. However, all the constituent values of the individual generative attributes in the novel combinations have been observed under some combination in the training data. The _real_ split of MPI3D [29] consists of photographs of a robotic arm that has a colored rigid object attached to its end effector. The images are captured in a way that controls their generative attributes, such as the shape of the rigid object or the position of the robotic arm.\n\nWe use this dataset to create development, and deployment splits to test for systematic generalization. In particular, we consider the shape of the rigid object to be the target variable \\(Y_{\\text{shape}}\\), while its color to be the bias \\(Z_{\\text{color}}\\). There are \\(6\\) possible shapes for the objects and \\(6\\) possible colors, totaling \\(36\\) combinations of attributes. Figure 1(d) illustrates pairs of attributes used to sample an example systematic training split. A training set is created by assigning \\(C\\) number of colors per shape so that: first, \\(p_{\\text{data}}(z_{\\text{color}}\\mid y_{\\text{shape}})=\\frac{1}{C}\\) if \\((y_{\\text{shape}},z_{\\text{color}})\\) exists in the elected pairs of attributes for training; otherwise \\(0\\). Second, we make sure that marginally all colors and shapes are represented uniformly, which means that \\(p_{\\text{data}}(y)=\\frac{1}{6}\\) and \\(p_{\\text{data}}(z)=\\frac{1}{6}\\). During testing, we desire to generalize to samples generated from \\(p_{\\text{test}}\\) which distributes pairs \\((y_{\\text{shape}},z_{\\text{color}})\\) uniformly, covering combinations that are entirely missing from the training set. In our benchmarks, we are testing methods for \\(C\\in\\{2,3,4,5\\}\\), against the same 5 sets of systematic splits which we generate independently per \\(C\\).\n\n**Systematic split generation.** As we described above, sMPI3D is generated using the _real_ \"split\" of the MPI3D dataset [29]. The dataset is distributed as a tensor of shape \\((6,6,2,3,3,40,40,64,64,3)\\), with the last 3 axis corresponding to pixel values of a \\(64\\times 64\\) RGB image and the rest to coordinates for generative attribute values. The first two coordinates correspond to the color and shape attributes for the rigid object depicted in the image. The procedure for generating splits of sMPI3D is stochastic, meaning that it depends on a random seed set by the user. The reason for this is that we would like to benchmark methods against the possibility that some systematic splits are consistently easier to generalize than others. To generate a train-validation-test split of sMPI3D, we first generate the pattern of included-excluded combinations for the development dataset (training and validation datasets), like the one presented in Figure 1(d). The pattern depends on the argument \\(C\\), the number of color values observed per shape value. Then, we randomly permute data points across the the first two axis of the data tensor, which correspond to the color and shape attributes. Afterwards, we split the data tensor according to whether a certain position \\((\\operatorname{color},\\operatorname{shape})\\) belongs to the development dataset or not. We subsample the deployment dataset uniformly across i.i.d. combinations for \\(180k\\) and \\(18k\\) mutually exclusive data points for the training and validation sets respectively, for all \\(C\\in\\{2,3,4,5\\}\\). Finally, the remaining data points are combined and \\(54k\\) images are sampled uniformly from all combinations to create the unbiased test set. We provide code for generating systematic splits on MPI3D in the supplementary material.\n\n```\nData:Training split: \\(\\mathcal{D}\\coloneqq\\{x_{m},y_{m}\\}_{m=1}^{M}\\)\n1Data:Validation split: \\(\\mathcal{D}_{\\operatorname{valid}}\\coloneqq\\{x_{n},y_{n}\\}_{n=1}^{N}\\)\n2Data:Hyperparameters:SSL checkpoint \\(T_{\\operatorname{sid}}\\), linear probe training steps \\(T_{\\operatorname{sid}}\\), logit adjustment strength \\(\\eta\\) and calibration \\(\\tau\\), and optimization hyperparameters Result:Debiased model \\(h_{\\theta^{*}}\\) and its validation_score\n3\n4 /* Load pretrained SSL encoder */\n5 \\(f_{\\operatorname{base}}\\leftarrow\\operatorname{load}(T_{\\operatorname{ssl}})\\)\n6 /* Train Linear Head */\n7 Define a classifier \\(h_{\\operatorname{bias}}=g_{\\phi}\\circ f_{\\operatorname{bias}}\\) from the composition of pretrained \\(f_{\\operatorname{base}}\\) and linear classifier \\(g_{\\phi}\\) with parameters \\(\\phi\\)\n8 Predict biased predictions using \\(\\hat{y}(x;h_{\\operatorname{bias}})=\\operatorname{argmax}_{y}h_{\\operatorname{ bias}}(x)_{y}\\)\n9for\\(t=1\\)to\\(T_{\\operatorname{sid}}\\)do\n10 Sample batch \\(B\\) from \\(\\mathcal{D}\\)\n11 Update \\(\\phi\\) so that linear model on top of \\(h_{\\operatorname{bias}}(x)\\) minimizes average cross-entropy on \\(B\\)\n12\n13 end for\n14\n15 end for /* Finetune with logit adjustment */\n16 Define calibrated model of biased predictions \\(p_{\\operatorname{bias}}(y_{\\operatorname{bias}}|x)\\propto\\exp\\left(h_{ \\operatorname{bias}}(x)_{y_{\\operatorname{bias}}}/\\tau\\right)\\)\n17 Compute confusion matrix \\(\\hat{p}_{\\operatorname{data}}(y,y_{\\operatorname{bias}})\\) // See Equation (5)\n18 Define \\(h_{\\theta}\\) to be the debiased model\n19 Initialize base model in \\(h_{\\theta}\\) from pretrained \\(f_{\\operatorname{base}}\\)\n20 Predict debiased predictions using \\(\\hat{y}(x;h_{\\theta})=\\operatorname{argmax}_{y}h_{\\theta}(x)_{y}\\)\n21\n22 checkpoints \\(\\leftarrow\\) [ ]\n23for\\(t=1\\)to maximum number of iterationsdo\n24 Sample batch \\(B\\) from \\(\\mathcal{D}\\)\n25 Update \\(\\theta\\) so that model \\(p_{\\theta}(y|x)\\propto\\exp\\left(h_{\\theta}(x)_{y}+\\eta\\log\\hat{p}_{\\operatorname {data}}\\big{(}y|\\hat{y}(x;h_{\\operatorname{bias}})\\big{)}\\right)\\) minimizes average cross-entropy on \\(B\\)\n26 validation_score \\(\\leftarrow\\) compute_validation_criterion(\\(\\mathcal{D}_{\\operatorname{valid}},h_{\\operatorname{bias}},h_{\\theta}\\)) // see Equation (7)\n27 checkpoints \\(\\leftarrow\\) checkpoints \\(+\\left[(\\texttt{validation\\_score},h_{\\theta})\\right]\\)\n28\n29 end for\n30 Final debiased model, \\(h_{\\theta^{*}}\\leftarrow\\max_{\\texttt{validation\\_score}\\texttt{ checkpoints}}\\)\n```\n\n**Algorithm 1**ULA: Logit adjustment without bias labels during training or model selection.\n\n### Non-systematic Benchmarks\n\n**Colored MNIST.** (cMNIST) is an RGB version of MNIST dataset [47], in which the digit target variable \\(Y_{\\operatorname{digit}}\\) is paired with a color bias variable \\(Z_{\\operatorname{color}}\\) to draw an image of that digit using a specific color. In this work, we study the dataset which follows the data generation procedure from Nam et al. [57]. Each of the digits \\(k\\in\\{0,...,9\\}\\) is paired with a distinct color out of a choice of ten, with probability \\(p_{\\operatorname{data}}(Z_{\\operatorname{color}}=k\\,|\\,Y_{\\operatorname{digit }}=k)=1-\\beta\\) of the available training samples for that digit \\(k\\). The rest of the probability mass is split uniformly across the remaining color options, \\(p_{\\operatorname{data}}(Z_{\\operatorname{color}}=l\\,|\\,Y_{\\operatorname{digit }}=k)=\\frac{\\beta}{9}\\) for \\(l\\neq k\\). Essentially, \\(\\beta\\) controls for the percentage of bias-conflicting samples in the training set. The goal is to train a classifier that performs well under the test set in which digits are paired with colors with random chance \\(1/10^{2}\\). We obtain four different tasks by varying \\(\\beta\\in\\{0.5\\%,1\\%,2\\%,5\\%\\}\\).\n\n**Corrupted CIFAR10.** (CCIFAR10) is a modification of the CIFAR10 dataset [45] by Hendrycks and Dietterich [37], in which images are affected by a type of texture noise which correlates with the original target category. Similarly to cMNIST, there are 10 different types of texture noise each of which is predominantly paired to a unique CIFAR10 label. The training set is created under \\(p_{\\text{data}}(Z_{\\text{noise}}=k\\,|\\,Y_{\\text{cifar}}=k)=1-\\beta\\) with the rest of probability spread uniformly to the bias-conflicting options of texture noise. Our goal again is to perform well under the test set in which pairs of texture noise and labels are distributed uniformly. Four development datasets are used in the experiments with \\(\\beta\\in\\{0.5\\%,1\\%,2\\%,5\\%\\}\\).\n\n**Waterbirds.** Our study employs the Waterbirds dataset, devised by Sagawa et al. [61]. This dataset is derived from the CUB dataset's bird images [68], superimposed on backgrounds from the Places dataset [77]. The dataset classifies seabirds and waterfowl as waterbirds, and all other species as landbirds. Ocean and natural lake backgrounds from Places are labeled as 'water background', while bamboo and broadleaf forest backgrounds are termed 'land background'. Four groups emerge: land background with waterbird and landbird, and water background with waterbird and landbird. The first two are minority groups due to fewer examples, while the latter two constitute majority groups. The dataset maintains the original training, validation, and test splits from Sagawa et al. [61]. In training, 95% of waterbirds and landbirds are respectively paired with water and land backgrounds, while we care for the worst-group test accuracy.\n\n**CelebA.** We examine a fair classification task using the CelebA dataset [51] of celebrity facial attributes. Same to Sagawa et al. [61], we consider the _BlondHair_ attribute to be the target variable. Then, a spurious correlation can naturally be observed with the binarized gender attribute _Male_. Four groups emerge for which the minorities are males with blond hair, followed by females with non-blond hair. We use the standard splits defined in the literature, and we report the worst-group test accuracy.\n\n## Appendix E Experiment Details\n\n**Architecture, Optimization, Augmentation and Validation.** For cMNIST, we train a 3-hidden layer MLP with 100 hidden neurons for each layer, while we use a ResNet18 [34] for cCIFAR10 and sMPI3D, and a ResNet50 for Waterbirds and CelebA. For all datasets except Waterbirds, we pretrain the base model with the MoCoV2+ [15] process, while for Waterbirds we use an already pretrained ResNet50 on Imagenet [60]. Training of the linear probe for the bias network and finetuning for the logit adjusted debiased network happen with AdamW [52] optimizer using default momentum hyperparameters \\(\\beta=(0.9,0.999)\\), except for Waterbirds where we use SGD with \\(0.9\\) momentum. The learning rate for all dataset, excluding Waterbirds, is scheduled with a cosine decaying rule. Batch size is set to \\(256\\) for cMNIST, cCIFAR10, sMPI3D and CelebA, and \\(64\\) for Waterbirds. We use minimal data augmentation for training the linear probe, as well as the debiased logit adjusted model, following the baselines we compare against for fair comparison. In more details, we use random resized crops (RRC) and random horizontal flipping for all datasets, except cMNIST where no augmentation is used. Finally, training happens for a maximum of \\(100\\)\n\n\\begin{table}\n\\begin{tabular}{l c c c c} \\hline \\hline  & Learning Rate & Batch Size & Weight Decay & Temperature \\\\ \\hline cMNIST & \\(1.0\\) & \\(256\\) & \\(1\\)e-\\(3\\) & \\(0.1\\) \\\\ cCIFAR10 & \\(0.9\\) & \\(256\\) & \\(1\\)e-\\(4\\) & \\(0.1\\) \\\\ sMPI3D & \\(0.9\\) & \\(256\\) & \\(1\\)e-\\(4\\) & \\(0.1\\) \\\\ CelebA & \\(0.3\\) & \\(128\\) & \\(3\\)e-\\(5\\) & \\(0.1\\) \\\\ \\hline \\hline \\multicolumn{5}{c}{SSL Augmentations} \\\\ \\hline cMNIST & \\multicolumn{3}{c}{RRC + Gaussian Blur} \\\\ cCIFAR10 & RRC + Color Jitter + Gray Scale + Gaussian Blur + Horizontal Flip \\\\ sMPI3D & RRC + Color Jitter + Gray Scale + Gaussian Blur + Horizontal Flip \\\\ CelebA & \\multicolumn{3}{c}{RRC + Color Jitter + Solarization + Horizontal Flip} \\\\ \\hline \\hline \\end{tabular}\n\\end{table}\nTable 7: Hyperparameters used during SSL pretraining.\n\nepochs for all datasets (except Waterbirds which we train for \\(200\\)), and the best model across the training duration is selected for each trial, by evaluating our proposed validation score at the end of each epoch and monitoring for the maximum value. For cMNIST, cCIFAR10 and sMPI3D, we use our bias-unsupervised group-balanced validation score (see Equation (7)), whereas for Waterbirds and CelebA the corresponding worst-group version.\n\nSelf-supervised Pretraining.For all cases, we adapt self-supervised pretraining recipes from _solo-learn_ software library [22]. In particular, we optimize the MoCoV2+ loss with a LARS optimizer [72] using \\(\\eta_{\\text{LARS}}=0.002\\). Furthermore, an MLP projector with a single hidden layer is used to project features to a hypersphere in a \\(256\\)-dimensional space. Cosine scheduling is used for the learning rate and for the exponential moving average coefficient of the momentum encoder. We train for a maximum of \\(100\\) epochs for the cMNIST, sMPI3D and CelebA tasks, and for \\(500\\) epochs for the cCIFAR10 datasets. We save periodically checkpoints of base models across training, as we are going to decide which one to use as a basis for the bias proxy and for the debiased model as a hyperparameter of uLA. For each dataset, we perform a small random search over contrastive temperature and data augmentation by maximizing the i.i.d. validation accuracy of an online linear classifier on-top of extracted representations. In general, we fix the contrastive temperature to \\(0.1\\) and use default Imagenet augmentations for sMPI3D and cCIFAR10, while we search for effective variants of those for cMNIST and CelebA. Details about the hyperparameter combinations used are displayed at Table 7.\n\nIn all cases, we verify that the augmentations at Table 7**do not cause representations to be invariant to variations of the bias attribute**. We take extra care in order to delineate that, especially for synthetic tasks, improvements are observed due to our uLA methodology, and not due to the use of data augmentations which nullify the bias attribute; something which, in real-life tasks such as CelebA, might be impossible to do with hand-crafted transformation. We classify against the bias attribute using a linear classifier on extracted representations, and we find that, for the hyperparameters used, bias information is indeed retained (see Table 6).\n\nHyperparameter Search for uLA.For each dataset and task, we search over the respective spaces defined at Table 8. In particular, we sample uniformly \\(64\\) independent configurations (\\(128\\) for Waterbirds) and we choose the one with the best bias-unsupervised validation score (Equation (7)). To make the space a bit smaller, we consider the same weight decay for the training of the bias proxy and the debiased logit adjusted network, as well as we consider the same number of training epochs\n\n\\begin{table}\n\\begin{tabular}{l c c c} \\hline \\hline  & Learning Rate & Weight Decay & \\(\\eta\\) \\\\ \\hline cMNIST & \\{5e-4, 1e-3, 2e-3, 5e-3\\} & \\{0.0,1e-4, 1e-3, 1e-2, 1e-1\\} & \\{1.0,1.25, 1.5\\} \\\\ cCIFAR10 & \\{1e-5, 5e-5, 1e-4, 5e-4, 1e-3\\} & \\{0.0,1e-4, 1e-3, 1e-2, 1e-1\\} & \\{1.0,1.25, 1.5, 1.75, 2.0\\} \\\\ sMPI3D & \\{1e-5, 5e-5, 1e-4, 5e-4, 1e-3\\} & \\{0.0,1e-4, 1e-3, 1e-2, 1e-1\\} & 1.0 \\\\ Waterbirds & \\{1e-4, 5e-4, 1e-3, 5e-3\\} & \\{0.0, 1e-4, 1e-2\\} & \\{1.0,1.5, 2.0, 2.5, 3.0\\} \\\\ CelebA & \\{1e-5, 5e-5, 1e-4, 5e-4\\} & \\{0.0, 1e-5, 1e-4, 1e-3, 1e-2\\} & \\{1.0,1.25, 1.5\\} \\\\ \\hline \\multicolumn{3}{c}{\\(\\tau\\)} & \\multicolumn{3}{c}{\\(T_{\\text{sal}}/T_{\\text{stop}}\\)} \\\\ \\hline cMNIST & \\{0.1,1.0,2.0\\} & \\{10,20,50,100\\} epochs \\\\ cCIFAR10 & \\{0.5,1.0,2.0\\} & \\{100,200,300,400,500\\} epochs \\\\ sMPI3D & 1.0 & \\{10,20,50,100\\} epochs \\\\ Waterbirds & \\{0.1,0.5,1.0,2.0\\} & \\{50,100,200,500,1000,200,5000\\} steps \\\\ CelebA & \\{0.5,1.0,2.0\\} & \\{10,20,50,100\\} epochs \\\\ \\hline \\hline \\end{tabular}\n\\end{table}\nTable 8: Search space used to tune hyperparameters for each of the datasets considered\n\n\\begin{table}\n\\begin{tabular}{l c c c c c} \\hline \\hline  & Learning Rate & Weight Decay & \\(\\eta\\) & \\(\\tau\\) & \\(T_{\\text{sal}}\\) (epochs) & \\(T_{\\text{stop}}\\) \\\\ \\hline cMNIST (1\\%) & 2e-3 & 1e-3 & \\(1.5\\) & \\(2.0\\) & \\(100\\) & \\(100\\) epochs \\\\ cCIFAR10 (1\\%) & 5e-5 & 1e-4 & \\(1.0\\) & \\(1.0\\) & \\(500\\) & \\(500\\) epochs \\\\ sMPI3D (\\(C=3\\)) & 5e-5 & 1e-4 & \\(1.0\\) & \\(1.0\\) & \\(100\\) & \\(100\\) epochs \\\\ Waterbirds & 1e-3 & \\(0.0\\) & \\(3.0\\) & \\(0.1\\) & - & \\(50\\) steps \\\\ CelebA & 1e-4 & \\(0.0\\) & \\(1.5\\) & \\(0.5\\) & \\(10\\) & \\(10\\) epochs \\\\ \\hline \\hline \\end{tabular}\n\\end{table}\nTable 9: Best hyperparameters selected by our proposed bias-unsupervised validation score.\n\nfor SSL-pretrained base network and the linear head for the bias proxy (which we train online in our implementation). For Waterbirds, the linear head is trained on top of a pretrained ResNet50 on Imagenet, which is taken from a PyTorch [59] model repository. The pretrained model's parameters remain frozen during the training of the bias proxy, so in this case there is no number of pretraining steps to select. The hyperparameters used for the benchmarking experiments, that we presented in Section 4, can be found at Table 9.", "response": "## Review\n\n### Summary\n\nThis paper introduces a novel bias-unsupervised method for mitigating spurious correlations in classifiers without requiring group annotations during training or validation. The authors propose a Self-Supervised Learning (SSL)-based approach that utilizes a pre-trained model to inform the model selection and training process, specifically through a logit adjustment technique. The method is empirically validated against various benchmarks, including the newly introduced sMPI3D dataset, which highlights its effectiveness in systematic generalization tasks. Overall, while the method shows promise in addressing limitations of previous techniques, further clarity in some theoretical aspects and connections to existing methods would enhance its contributions.\n\n### Strengths\n\n- The proposed method, uLA, effectively addresses spurious correlations without group annotations.\n- Clear and well-structured presentation, making it easy to follow.\n- The introduction of the sMPI3D dataset is a significant contribution, providing a realistic testing ground for systematic generalization.\n- Empirical results demonstrate the method's efficacy across various benchmarks.\n\n### Weaknesses\n\n- Lack of clarity on how the proposed method achieves improvements in systematic generalization.\n- The paper does not sufficiently discuss important related works in the field, which could provide context for its contributions.\n- Certain sections, such as the explanation of the bias variable extraction and dataset construction, require more exposition.\n- Concerns about the efficiency and robustness of the uLA method, particularly regarding the choice of SSL algorithms and hyperparameter sensitivity.\n\n### Questions\n\n- What theoretical or intuitive justifications support the claim that the proposed method benefits systematic generalization?\n- Could the authors clarify the choice of SSL algorithm and its impact on the method's performance?\n- What empirical analyses support the reliability of the estimation method used in the proposed approach?\n\n### Soundness\n\n**Score:** 3\n\n**Description:** Good. The paper presents a solid theoretical foundation and empirical results, but some claims need clearer justification.\n\n### Presentation\n\n**Score:** 3\n\n**Description:** Good. The paper is generally well-written, though certain areas could benefit from improved clarity and exposition.\n\n### Contribution\n\n**Score:** 4\n\n**Description:** Excellent. The introduction of a new dataset and a novel method significantly advance the field of bias-unsupervised learning.\n\n### Rating\n\n**Score:** 6\n\n**Description:** Weak Accept: The paper is technically solid and has moderate-to-high impact potential, but it requires some clarifications and additional discussions.\n\n### Paper Decision\n\n**Decision:** Accept (poster)\n\n**Reasons:** The paper presents a novel approach to a significant problem in machine learning, showing empirical promise and introducing new methodologies and datasets. While some aspects require clarification, the overall contribution to the field is valuable enough to warrant acceptance.\n"}
{"query": "You are a highly experienced, conscientious, and fair academic reviewer, please help me review this paper. The review should be organized into nine sections: \n1. Summary: A summary of the paper in 100-150 words.\n2. Strengths/Weaknesses/Questions: The Strengths/Weaknesses/Questions of paper, which should be listed in bullet points, with each point supported by specific examples from the article where possible.\n3. Soundness/Contribution/Presentation: Rate the paper's Soundness/Contribution/Presentation, and match this score to the corresponding description from the list below and provide the result. The possible scores and their descriptions are: \n    1 poor\n    2 fair\n    3 good\n    4 excellent\n4. Rating: Give this paper an appropriate rating, match this rating to the corresponding description from the list below and provide the result. The possible Ratings and their descriptions are: \n    1 strong reject\n    2 reject, significant issues present\n    3 reject, not good enough\n    4 possibly reject, but has redeeming facets\n    5 marginally below the acceptance threshold\n    6 marginally above the acceptance threshold\n    7 accept, but needs minor improvements \n    8 accept, good paper\n    9 strong accept, excellent work\n    10 strong accept, should be highlighted at the conference   \n5. Paper Decision: It must include the Decision itself(Accept or Reject) and the reasons for this decision, based on the criteria of originality, methodological soundness, significance of results, and clarity and logic of presentation.\n\nHere is the template for a review format, you must follow this format to output your review result:\n**Summary:**\nSummary content\n\n**Strengths:**\n- Strength 1\n- Strength 2\n- ...\n\n**Weaknesses:**\n- Weakness 1\n- Weakness 2\n- ...\n\n**Questions:**\n- Question 1\n- Question 2\n- ...\n\n**Soundness:**\nSoundness result\n\n**Presentation:**\nPresentation result\n\n**Contribution:**\nContribution result\n\n**Rating:**\nRating result\n\n**Paper Decision:**\n- Decision: Accept/Reject\n- Reasons: reasons content\n\n\nPlease ensure your feedback is objective and constructive. The paper is as follows:\n\n# Double Auctions with Two-sided Bandit Feedback\n\n Soumya Basu\n\nGoogle Mountain View\n\nbasusoumya@google.com &Abishek Sankararaman\n\nAWS\n\nabishek.90@gmail.com\n\n###### Abstract\n\nDouble Auction enables decentralized transfer of goods between multiple buyers and sellers, thus underpinning functioning of many online marketplaces. Buyers and sellers compete in these markets through bidding, but do not often know their own valuation a-priori. As the allocation and pricing happens through bids, the profitability of participants, hence sustainability of such markets, depends crucially on learning respective valuations through repeated interactions. We initiate the study of Double Auction markets under bandit feedback on both buyers' and sellers' side. We show with confidence bound based bidding, and 'Average Pricing' there is an efficient price discovery among the participants. In particular, the regret on combined valuation of the buyers and the sellers - a.k.a. the social regret - is \\(O(\\log(T)/\\Delta)\\) in \\(T\\) rounds, where \\(\\Delta\\) is the minimum price gap. Moreover, the buyers and sellers exchanging goods attain \\(O(\\sqrt{T})\\) regret, individually. The buyers and sellers who do not benefit from exchange in turn only experience \\(O(\\log T/\\Delta)\\) regret individually in \\(T\\) rounds. We augment our upper bound by showing that \\(\\omega(\\sqrt{T})\\) individual regret, and \\(\\omega(\\log T)\\) social regret is unattainable in certain Double Auction markets. Our paper is the first to provide decentralized learning algorithms in a two-sided market where _both sides have uncertain preference_ that need to be learned.\n\n0\nFootnote 0: Work done when AS was affiliated with UC Berkeley. All opinions are that of the authors and do not necessarily represent that of the employers\n\n## 1 Introduction\n\nOnline marketplaces, such as eBay, Craigslist, Task Rabbit, Doordash, Uber, enables allocation of resources between supply and demand side agents at a scale through market mechanisms, and dynamic pricing. In many of these markets, the valuation of the resources are often personalized across agents (both supply and demand side), and remain apriori unknown. The agents learn their own respective valuations through repeated interactions while competing in the marketplace. In turn, the learning influences the outcomes of the market mechanisms. In a recent line of research, this interplay between learning and competition in markets has been studied in multiple systems, such as bipartite matching markets [29, 30, 39, 7], centralized basic auctions [26, 22]. These works follow the 'protocol model' [2], where multiple agents follow a similar protocol/algorithm, while each agent executes her protocol using only her own observations/world-view up to the point of execution.\n\nIn this paper, we initiate the study of the decentralized Double Auction market where multiple sellers and buyers, each with their own valuation, trades an indistinguishable good. In each round, the sellers and the buyers present bids for the goods.1 The auctioneer is then tasked with creating an allocation, and pricing for the goods. All sellers with bids smaller than the price set by the auctioneer sell at that price, whereas all the buyers with higher bids buy at that price. Each buyer and seller, is oblivious to all the prices including her own. Only a buyer, or a seller participating in the market observesher own valuation of the good (with added noise). Notably, our work tackles two-sided uncertainty, whereas the previous works mainly focused on one-sided uncertainty in the market.\n\nDouble auction is used in e-commerce [43] - including business-to-business and peer-to-peer markets, bandwidth allocation [24; 23], power allocation [31]. We focus on the 'Average Mechanism' for double auction. Average mechanism guarantees that the auctioneer and the auction participants incur no losses in the trade. It also ensures that each commodity is given to the participant that values it the highest, thus maximizing social welfare. Additionally, average mechanism can be implemented through simple transactions. These properties make average mechanism a suitable choice in large social markets, such as energy markets [32], environmental markets [35], cloud-computing markets [38; 32], bidding on wireless spectrum [18]. Our objective is to design a bandit average mechanism for double auction markets when the participants are a-prioriu unaware of their exact valuation.\n\nUnder average mechanism (detailed in Section 2.2) first an allocation is found, by maximizing \\(K\\) such that the \\(K\\) highest bidding buyers all bid higher than the \\(K\\) lowest bidding sellers. The price is set as the average of the \\(K\\)-th lowest bid among the buyers, and the \\(K\\) highest bid among the sellers for the chosen \\(K\\). We have two-sided uncertainty in the market, as both buyers and sellers do not know their own valuation. The uncertainty in bids manifests in two ways. Firstly, each buyer needs to compete with others by bidding high enough to get allotted so that she can discover her own price. Similarly, the sellers compete by bidding lower for price discovery. The competition-driven increase in buyers' bids, and decrease in the sellers' bids may decrease the utility that a buyer or seller generates. Secondly, as the valuation needs to be estimated, the price set in each round as a function of these estimated valuations (communicated to the auctioneer in the form of bids) remains noisy. This noise in price also decreases the utility. However, when price discovery is slow the noise in price increases. Therefore, the main challenge in decentralized double auction with learning is to strike a balance between the competition-driven increase/decrease of bids, and controlling the noise in price.\n\n### Main Contributions\n\nOur main contributions in this paper are as follows.\n\n1. Our paper is the first to provide decentralized learning algorithms in a two-sided market where _both sides have uncertain preference_ that need to be learned. Unlike in the setting with one sided uncertainity only, we identify that with _two-sided uncertainty_ in double auction markets _optimism_ in the face of uncertainty in learning (OFUL) from both sides causes _information flow bottleneck_ and thus not a good strategy. We introduce the notion of _domination of information flow_ - that increases the chance of trade and price discovery. The sellers bid the lower confidence bound (LCB), and buyers bid the upper confidence bound (UCB) of their respective valuation. By using UCB bids the buyers, and using the LCB bids the sellers decrease their reward and facilitate price discovery. Formally, with the above bids under average mechanism\n\n* We show that the social welfare regret, i.e. the regret in the combined valuation of all the sellers and the buyers in the market is \\(O(\\log(T)/\\Delta)\\) regret in \\(T\\) rounds with a minimum reward gap \\(\\Delta\\). We also show a \\(\\Omega(\\log(T)/\\Delta)\\) lower bound on social-welfare regret and thus our upper bound is order-optimal.\n* For individual regret, we show that each of the the sellers and the buyers that do not participate under the true valuations incur \\(O(\\log(T)/\\Delta)\\) regret, while the optimal participating buyers and sellers incur a \\(O(\\sqrt{T\\log(T)})\\) regret. Our upper bound holds for heterogeneous confidence widths, making it robust against the choices of the individual agents.\n\n2. We complement the upper bounds by showing price discovery itself is \\(\\Omega(\\sqrt{T})\\) hard in the minimax sense. Specifically, we consider a relaxed system where _(i)_ the price of the good is known to all, and _(ii)_ an infinite pool of resource exists, and hence any buyer willing to pay the price gets to buy, and any seller willing to sell at the price does so. We show under this setup, for any buyer or seller, there exists a system where that agent must incur a regret of \\(\\Omega(\\sqrt{T})\\). Similarly, we establish a \\(\\Omega(\\log(T))\\) lower bound for the social-welfare regret by showing that the centralized system can be reduced to a combinatorial semi-bandit and using the results of [14].\n\nSystem Model\n\nThe market consists of \\(N\\geq 1\\) buyers and \\(M\\geq 1\\) sellers, trading a _single type_ of item which are indistinguishable across sellers. This set of \\(M+N\\) market participants, repeatedly participate in the market for \\(T\\) rounds. Each buyer \\(i\\in[N]\\) has valuation \\(B_{i}\\geq 0\\), for the item and each seller \\(j\\in[M]\\) has valuation \\(S_{i}\\geq 0\\). No participant knows of their valuation apriori and learn it while repeatedly participating in the market over \\(T\\) rounds.\n\n### Interaction Protocol\n\nThe buyers and sellers interact through an auctioneer who implements a bilateral trade mechanism at each round. At each round \\(t\\geq 1\\), every buyer \\(i\\in[N]\\) submits bids \\(b_{i}(t)\\) and seller \\(j\\in[M]\\) submits asking price 2\\(s_{j}(t)\\) simultaneously. Based on the bids and asking prices in round \\(t\\), the auctioneer outputs _(i)_ subsets \\(\\mathcal{P}_{b}(t)\\subseteq[N]\\) and \\(\\mathcal{P}_{s}(t)\\subseteq[M]\\) of participating buyers and sellers with equal cardinality \\(K(t)\\leq\\min(M,N)\\), and _(ii)_ the trading price \\(p(t)\\) for the participating buyers and sellers in this round. Subsequently, every buyer \\(i\\in[N]\\) is _(i)_ either part of the trade at time \\(t\\), in which case she gets utility \\(r_{i}^{(B)}(t):=B_{i}+\\nu_{b,i}(t)-p(t)\\), or _(ii)_ is not part of the trade at time \\(t\\) and receives \\(0\\) utility along with a signal that she did not participate. Similarly, each seller \\(j\\in[M]\\) is either part of the trade and receives utility \\(r_{j}^{(S)}(t):=p(t)-S_{j}-\\nu_{s,j}(t)\\), or is informed she is not part of the trade and receives \\(0\\) utility. The random variables \\(\\nu_{b,i}(t)\\) and \\(\\nu_{s,j}(t)\\) for all \\(i\\in[N]\\), \\(j\\in[M]\\) and \\(t\\in[T]\\) are i.i.d., \\(0\\) mean, \\(1\\) sub-Gaussian random variables.3\n\nFootnote 2: Throughout, we refer to sellers ‘bids’ as their asking price\n\nFootnote 3: We study the system with \\(1\\)-sub-Gaussian to avoid clutter. Extension to general \\(\\sigma\\)-sub-Gaussian is trivial when \\(\\sigma\\) or an upper bound to it is known.\n\n### Average price mechanism\n\nThroughout the paper, we assume the auctioneer implements the average price mechanism in every round \\(t\\). Under this mechanism, at each round \\(t\\), the auctioneer orders the bids by the 'natural order', i.e., sorts the buyers bids in descending order and the seller's bids in ascending order. Denote by the sorted bids from the buyer and seller as \\(b_{i_{1}}(t)\\geq\\cdots b_{i_{N}}(t)\\) and the sorted sellers bids by \\(s_{j_{1}}(t)\\leq\\cdots s_{j_{M}}(t)\\). Denote by the index \\(K(t)\\) to be the largest index such that \\(b_{i_{K(t)}}(t)\\geq s_{j_{K(t)}}(t)\\). In words, \\(K(t)\\) is the 'break-even index' such that all buyers \\(i_{1},\\cdots,i_{K(t)}\\) have placed bids offering to buy at a price strictly larger than the asking price submitted by sellers \\(j_{1},\\cdots,j_{K(t)}\\). The auctioneer then selects the participating buyers \\(\\mathcal{P}_{b}(t)=\\{i_{1},\\cdots,i_{K(t)}\\}\\), and participating sellers \\(\\mathcal{P}_{s}(t)=\\{j_{1},\\cdots,j_{K(t)}\\}\\). The price is set to \\(p(t):=\\frac{b_{i_{K(t)}}+s_{j_{K(t)}}}{2}\\), and thus the name of the mechanism is deemed as the average mechanism.\n\n### Regret definition\n\nFor the given bilateral trade mechanism, and true valuations \\((B_{i})_{i\\in[N]}\\) and \\((S_{j})_{j\\in[M]}\\), denote by \\(K^{*}\\leq\\min(M,N)\\) be the number of matches and by \\(p^{*}\\) to be the price under the average mechanism when all the buyers and sellers bid their true valuations. Let \\(\\mathcal{P}_{b}^{*}\\) be to be set of the _optimal participating buyers_, and \\(\\mathcal{P}_{s}^{*}\\) to be set of the _optimal participating sellers_. For any buyer \\(i\\in[N]\\), we denote by \\((B_{i}-p^{*})\\) to be the true utility of the buyer. Similarly, for any seller \\(j\\in[N]\\), we denote by \\((p^{*}-S_{j})\\) to be the true utility of seller \\(j\\). From the description of the average mechanism, in the system with true valuations, all participating agents have non-negative true utilities.\n\nRecall from the protocol description in Section 2.1 that at any time \\(t\\), if buyer \\(i\\in[N]\\) participates, then she receives a mean utility of \\((B_{i}-p(t))\\). For a participating seller \\(j\\in[M]\\) her mean utility of \\((p(t)-S_{j})\\) in round \\(t\\). If in any round \\(t\\), if a buyer \\(i\\in[N]\\) or a seller \\(j\\in[M]\\) does not participate, then she receives a deterministic utility \\(0\\). The expected individual regret of a buyer \\(i\\), namely \\(R_{b,i}(T)\\)\n\nFigure 1: Average Mechanism with 6 Buyers and 5 Sellers\n\nand a seller \\(j\\), namely \\(R_{s,j}(T)\\), are defined as\n\n\\[R_{b,i}(T)=T(B_{i}-p^{*})\\mathbbm{1}(i\\leq K^{*})-\\mathbb{E}\\Big{[} \\sum_{t:i\\in\\mathcal{P}_{b}(t)}(B_{i}-p(t))\\Big{]},\\] \\[R_{s,j}(T)=T(p^{*}-S_{j})\\mathbbm{1}(j\\leq K^{*})-\\mathbb{E}\\Big{[} \\sum_{t:j\\in\\mathcal{P}_{s}(t)}(p(t)-S_{j})\\Big{]}.\\]\n\nAuctioneer has no regret as average mechanism is _budget balanced_, i.e. auctioneer does not gain or lose any utility during the process.\n\nWe also define the social welfare regret similar to gain from trade regret in [10]. The social welfare is defined as the total valuation of the goods after the transfer of goods from seller to buyer in each round. The expected (w.r.t reward noise) total valuation after transfer is thus defined as \\(\\Big{(}\\sum_{i\\in\\mathcal{P}_{b}(t)}B_{i}+\\sum_{j\\in[M]\\setminus\\mathcal{P}_{s }(t)}S_{j}\\Big{)}\\), while the the expected total valuation under oracle average mechanism is \\(\\Big{(}\\sum_{i\\in\\mathcal{P}_{b}^{*}}B_{i}+\\sum_{j\\in[M]\\setminus\\mathcal{P}_{ s}^{*}}S_{j}\\Big{)}\\). Therefore, the expected social welfare regret is defined as\n\n\\[R_{SW}(T)=T\\big{(}\\sum_{i\\in\\mathcal{P}_{b}^{*}}B_{i}+\\sum_{j\\in[M]\\setminus \\mathcal{P}_{s}^{*}}S_{j}\\big{)}-\\mathbb{E}\\Big{[}\\sum_{t=1}^{T}\\big{(}\\sum_{i \\in\\mathcal{P}_{b}(t)}B_{i}+\\sum_{j\\in[M]\\setminus\\mathcal{P}_{s}(t)}S_{j} \\Big{)}\\Big{]}. \\tag{1}\\]\n\n## 3 Decentralized Bidding for Domination of Information Flow\n\nWe consider the decentralized system where each market participant bids based on their own observation, without any additional communication. The core idea is balancing _domination of information flow_ and _overunder bidding_, i.e. ensuring the number of allocation is not less than \\(K^{*}\\) in each round with high probability, and the bids converge to each agent's true valuation.\n\nEach seller \\(j\\in[M]\\), with \\(n_{s,j}(t)\\) participation upto round \\(t\\), at time \\(t+1\\) bids the lower confidence bound (scaled by \\(\\alpha_{s,j}\\)), LCB(\\(\\alpha_{s,j}\\)) in short, of its own valuation of the item. Each buyer \\(i\\in[N]\\), with \\(n_{s,j}(t)\\) participation upto round \\(t\\), at time \\(t+1\\), bids the upper confidence bound (scaled by \\(\\alpha_{b,i}\\)), UCB(\\(\\alpha_{b,i}\\)) in short, of its own valuation of the item. The bids are specified in Equation 2.\n\n\\[s_{j}(t+1)=\\hat{s}_{j}(t)-\\sqrt{\\frac{\\alpha_{s,j}\\log(t)}{n_{s,j}(t)}},\\quad b _{j}(t+1)=\\hat{b}_{i}(t)+\\sqrt{\\frac{\\alpha_{b,i}\\log(t)}{n_{b,i}(t)}} \\tag{2}\\]\n\nHere, \\(\\hat{s}_{j}(t)=\\frac{1}{n_{s,j}(t)}\\sum_{t^{\\prime}\\leq t:j\\in\\mathcal{P}_{s}( t^{\\prime})}Y_{s,j}(t^{\\prime})\\), and \\(\\hat{b}_{i}(t)=\\frac{1}{n_{b,i}(t)}\\sum_{t^{\\prime}\\leq t:i\\in\\mathcal{P}_{b}( t^{\\prime})}Y_{b,i}(t^{\\prime})\\) are the observed empirical valuation of the item upto time \\(t\\) by seller \\(j\\), and buyer \\(i\\), respectively.\n\nOur buyers and sellers follow the _protocol model_ (which is ubiquitous in bandit learning for markets) and agree on UCB and LCB based bids, respectively. They are _heterogeneous_ as they may use different \\(\\alpha_{b,i}\\) and \\(\\alpha_{s,j}\\) scaling parameters. The only restriction (as seen in Section 4) is \\(\\min\\{\\alpha_{b,i},\\alpha_{s,j}\\}\\geq 4\\) which they agree on as part of the protocol.\n\n**Key Insights:** We now contrast our algorithm design from standard multi-armed-bandit (MAB) problems. In a typical MAB problems, including other multi-agent settings, algorithm is designed based on optimism in the face of uncertainty in learning (OFUL) principle [1]. The UCB-type indices under OFUL arises as optimistic estimate of the rewards of arms/actions. However, such optimism used from both sides, i.e. both buyer and seller using UCB indices, may lead to a standstill. The bids of the buyers with constant probability can remain below the sellers' bids. Instead, we emphasize information flow through trade. In our algorithm buyers' UCB bids and sellers' LCB bids ensure that the system increases the chance for buyer and seller to participate in each round as compared to using their true valuations. There is _domination of information flow_, and consequently they discover their own valuation in the market.\n\nHowever, too aggressive over or under bidding can disrupt the price setting process. In particular, if even one non-participating buyer is bidding high enough (due to UCB bids) to exceed a non-participating seller's price, firstly she participates and accrues regret. More importantly, the participating sets deviates, resulting in a deviation of the price of the good from \\(p^{*}\\). Thus resulting in regret for all participating agents as well. Similar problems arise if one or more non-participatingseller predicts lower. On the other hand, too low aggression is also harmful as the price discovery may not happen resulting in deviation of participating set, deviation of price, and high regret. In the next section, we show the aggression remains within desired range, i.e. the regret of the agents remain low, even with heterogeneous UCBs and LCBs.\n\n## 4 Regret Upper Bound\n\nIn this section, we derive the regret upper bound for all the buyers and sellers in the system. Without loss of generality, let us assume that the buyers are sorted in decreasing order of valuation \\(B_{1}>B_{2}>...>B_{N}\\). The sellers are sorted in increasing order of valuation \\(S_{1}<S_{2}<...<S_{N}\\). The buyers \\(i=1,\\ldots,K^{*}\\) are the optimal participating buyers, and, similarly, the sellers \\(j=1,\\ldots,K^{*}\\) are the optimal participating sellers. Let us define \\(\\alpha_{\\min}:=\\min\\{\\alpha_{b,i},\\alpha_{s,j}\\}\\), and \\(\\alpha_{\\max}:=\\max\\{\\alpha_{b,i},\\alpha_{s,j}\\}\\). We define the minimum distance of an agent's true valuation from the true price \\(p^{*}\\) as \\(\\Delta=\\min_{i,\\in[N],j\\in[M]}\\{|p^{*}-S_{j}|,|B_{i}-p^{*}|\\}\\). As an item is sold only if the matched buyer bids strictly greater than the matched seller's bid we have \\(\\Delta>0\\).\n\nOur first main result, Theorem 1, proves a \\(O(\\log(T)/\\Delta)\\) upper bound for the social welfare regret.\n\n**Theorem 1**.: _The expected social welfare regret of the Average mechanism with buyers bidding UCB(\\(\\mathbf{\\alpha}_{b}\\)), and sellers bidding LCB(\\(\\mathbf{\\alpha}_{s}\\)) of their estimated valuation, for \\(\\alpha_{\\min}>2\\), is bounded as:_\n\n\\[R_{SW}(T)\\leq\\sum_{i\\leq K^{*}}\\sum_{i^{\\prime}>K^{*}}\\frac{( \\sqrt{\\alpha_{\\max}}+2)^{2}}{(B_{i}-B_{i^{\\prime}})}\\log(T)+\\sum_{j\\leq K^{*}} \\sum_{j^{\\prime}>K^{*}}\\frac{(\\sqrt{\\alpha_{\\max}}+2)^{2}}{(S_{j^{\\prime}}-S_{ j})}\\log(T)\\\\ +\\sum_{j^{\\prime}>K^{*}}\\sum_{i^{\\prime}>K^{*}}\\frac{(\\sqrt{ \\alpha_{\\max}}+2)^{2}}{(S_{j^{\\prime}}-B_{i^{\\prime}})}\\log(T)+MNb_{max}\\zeta( \\alpha_{\\min}/2),\\]\n\n_where \\(\\zeta(x)\\) is the Riemann-zeta function which is finite for \\(x>1\\)._\n\nThe following theorem provides the individual regret bounds for all sellers and buyers in \\(T\\) rounds.\n\n**Theorem 2**.: _The expected regret of the Average mechanism with buyers bidding UCB(\\(\\mathbf{\\alpha}_{b}\\)), and sellers bidding LCB(\\(\\mathbf{\\alpha}_{s}\\)) of their estimated valuation, for \\(\\alpha_{\\min}>2\\), is bounded as:_\n\n* _for a participating buyer_ \\(i\\in[K^{*}]\\) _as_ \\(R_{b,i}(T)\\leq(2+\\sqrt{\\alpha_{\\max}})\\sqrt{T\\log(T)}+C_{b^{\\prime},i}\\log(T)\\)_,_\n* _for a participating seller_ \\(j\\in[K^{*}]\\) _as_ \\(R_{s,j}(T)\\leq(2+\\sqrt{\\alpha_{\\max}})\\sqrt{T\\log(T)}+C_{s^{\\prime},j}\\log(T)\\)_,_\n* _for a non-participating buyer_ \\(i\\geq(K^{*}+1)\\) _as_ \\(R_{b,i}(T)\\leq\\frac{\\sqrt{(M-K^{*}+1)(2+\\sqrt{\\alpha_{\\max}})^{2}}}{(B_{K^{*} }-B_{i})}\\log(T)\\)_,_\n* _for a non-participating seller_ \\(j\\geq(K^{*}+1)\\) _as_ \\(R_{s,j}(T)\\leq\\frac{\\sqrt{(N-K^{*}+1)(2+\\sqrt{\\alpha_{\\max}})^{2}}}{(S_{j}-S_{ K^{*}})}\\log(T)\\)_._\n\n_Here \\(C_{b^{\\prime},i}\\) and \\(C_{s^{\\prime},j}\\) are \\(O\\left(\\frac{(M-K^{*}+1)(N-K^{*}+1)}{\\Delta}\\right)\\) constants (see Theorem 16 in Appendix A)._\n\nWe can summarize our main results (with lower bounds taken from Section 5) as\n\nSeveral comments on our main results are in order.\n\n**Social and Individual Regret:** The social regret as well as the individual regret of the optimal non-participating buyers and sellers under the average mechanism with confidence bound bids grow as \\(O(\\log(T))\\). However, the individual regret of the optimal participating agents grow as \\(O(\\sqrt{T\\log(T)})\\). The social regret, is determined by how many times a participant buyers fails to participate, and non-participants end up participating. Also, the non-partic\n\n\\begin{table}\n\\begin{tabular}{|l|c|c|c|c|} \\hline \\multirow{2}{*}{**Regret**} & \\multirow{2}{*}{**Social**} & \\multirow{2}{*}{**Participant**} & \\multicolumn{2}{c|}{**Non-participant**} \\\\ \\cline{3-4}  & & & & **Buyer** & **Seller** \\\\ \\hline\n**Upper** & \\(O(\\frac{MN}{\\Delta}\\log(T))\\) & \\(O((\\sqrt{T}+\\frac{MN}{\\Delta})\\log(T))\\) & \\(O(\\frac{\\sqrt{M}}{\\Delta}\\log(T))\\) & \\(O(\\frac{\\sqrt{N}}{\\Delta}\\log(T))\\) \\\\ \\hline\n**Lower** & \\(\\Omega(\\frac{M+N}{\\Delta}\\log(T))\\) & \\(\\Omega(\\sqrt{T})^{*}\\) & 0 & 0 \\\\ \\hline \\end{tabular}\n\\end{table}\nTable 1: Regret bounds derived in this paper. The participant regret lower bound is instance independent indicated by (\\(*\\)). Rest are instance dependent. Lower bound of 0 indicates absence of non-trivial bound.\n\nregret through participation. The effect of price setting is not present in both the cases, and counting the number of bad events lead to \\(O(\\log(T))\\) regret. For the participant agents the error in price-setting dominates their respective individual regret of \\(O(\\sqrt{T\\log(T)})\\).\n\n**Individually rational for \\(\\mathbf{\\Delta}=\\mathbf{\\omega}\\big{(}\\sqrt{\\log(T)/T}\\big{)}\\):** We observe that the non-participating buyers and sellers, are only having individual regret \\(O(\\log(T)/\\Delta)\\). And the participating buyers and sellers are having individual regret \\(O(\\sqrt{T\\log(T)}\\). This is reassuring as this does not discourage buyers and sellers from participation for a large range of system a-priori. Indeed, for the last participating buyer and seller the utility is \\((B_{K}^{*}-S_{K}^{*})/2\\), which is close to \\(\\Delta\\). Hence, as long as \\(\\Delta T=\\omega(\\max\\{\\log(T)/\\Delta,\\sqrt{T\\log(T)}\\})\\) or \\(\\Delta=\\omega(\\sqrt{\\log(T)/T})\\) a non-participating buyer or seller prefers entering the market then discovering her price and getting out, as compared to not participating in the beginning. Also, participating buyer or seller is guaranteed return through participation.\n\n**Incentives and Deviations:** We now discuss the incentives of individual users closely following the notion of symmetric equilibrium in double auction [42]. In this setting, a _myopic agent_ who knows her true valuation, and greedily maximizes her reward in each round assuming all the non-strategic agents use confidence-based bidding. For average mechanism only the price-setting agents (i.e. the \\(K^{*}\\)-th buyer and the \\(K^{*}\\)-th seller) have incentive to deviate from their true valuation to increase their single-round reward. The _non-price-setting agents are truthful_. When the \\(K^{*}\\)-th buyer deviates then each participating buyer has an average per round surplus of \\((B_{K^{*}}-\\max(S_{K^{*}},B_{K^{*}+1}))/2\\), and each participating seller has the same deficit. On the contrary, when the \\(K^{*}\\)-th seller deviates each participating seller has average per round surplus of \\((S_{K^{*}}-\\min(B_{K^{*}},S_{K^{*}+1}))/2\\) surplus, and each participating buyer has the same average deficit in each round. See Appendix C for more discussions.\n\n**Scaling of Regret with \\(N\\), \\(M\\), \\(K^{*}\\), \\(\\mathbf{\\Delta}\\):**\n\n_Social regret:_ The social regret scales as \\(O((MN-(K^{*})^{2})\\log(T)/\\Delta)\\). When participant buyers are replaced by non-participant buyers the valuation decreases which leads to \\(O(K^{*}(N-K^{*})\\log(T)/\\Delta)\\) regret. Similarly, participant seller getting replaced by non-participant seller introduces \\(O(K^{*}(M-K^{*})\\log(T)/\\Delta)\\) social regret. Finally, as goods move from non-participant buyers to non-participant sellers we obtain \\(O((M-K^{*})(N-K^{*})\\log(T)/\\Delta)\\) regret.\n\n_Non-participant regret:_ The individual regret scales as \\(O(\\sqrt{M}\\log(T)/\\Delta)\\) for a non-participant buyer, whereas it scales as \\(O(\\sqrt{N}\\log(T)/\\Delta)\\) for a non-participant seller. For any non-participant buyer once it has \\(O(\\log(T)/\\Delta^{2})\\) samples it no longer falls in top \\(K^{*}\\) buyer. However, it can keep participating until each non-participant seller collects enough samples, i.e. \\(O(\\log(T)/\\Delta^{2})\\) samples. Finally, regret is shown to be \\(O(\\sqrt{\\#\\text{participant}\\log(T)})\\) which leads to the \\(O(\\sqrt{M}\\log(T)/\\Delta)\\).\n\n_Participants:_ The leading \\(O(\\sqrt{T\\log(T)})\\) term in the regret for each optimal participating buyer and seller do not scale with the size of the system. This leading term depends mainly on the random fluctuation of the bid of the lowest bidding participating buyer and the highest bidding participating seller. The \\(O\\left(\\frac{MN}{\\Delta}\\log(T)\\right)\\) regret for the participating buyer and seller comes because each time a non-participating buyer or seller ends up participating the price deviates.\n\n_Special Case \\(K^{*}\\approx N\\approx M\\):_ We see that when the system the number of participants is very high, i.e. \\((N-K^{*}),(M-K^{*})=O(1)\\), then the \\(O(\\log(T)/\\Delta)\\) component, hence the regret, per buyer/seller does not scale with the system size. Furthermore, the social regret also scales as \\(O((MN-(K^{*})^{2})\\log(T)/\\Delta)\\approx O(\\log(T)/\\Delta)\\). This indicates that there is rapid learning, as all the participants are discovering her own price almost every round.\n\n### Proof Outline of Regret Upper Bounds\n\nWe now present an outline to the proof of Theorem 2. The full proof is given in Appendix A. A salient challenge in our proof comes from _two-sided uncertainty_. In Double auction, the outcomes of the buyers' and sellers' sides are inherently coupled. Hence error in buyers' side propagates to the sellers' side, and vice versa, under two-sided uncertainty. For example, if the buyers are bidding lower than their true valuation then the optimal non-participating sellers, even with perfect knowledge of their own valuation, ends up participating. We show that optimal participants (both buyers and sellers) get decoupled, whereas in optimal non-participants regret leads to new information in both buyer and seller side. This breaks the two-sided uncertainty obstacle.\n\n**Monotonicity and Information Flow:** Our proof leverages _monotonicity_ of the average mechanism, which means if the bid of each buyer is equal or higher, and simultaneously bid of each seller is equal or lower, then the number of participants increases (Proposition 7). Additionally, UCB ensures each buyer w.h.p. bids higher than its true value, and LCB ensures each seller w.h.p. bids lower than its true value. Therefore, we have _domination of the information flow_: \\(K(t)\\geq K^{*}\\) w.h.p. for all \\(t\\geq 1\\).\n\n**Social Regret Decomposition:** For any sample path with \\(K(t)\\geq K^{*}\\) the social regret, namely \\(r_{SW}(T)\\), under average mechanism can be bounded as\n\n\\[r_{SW}(T)\\leq\\sum_{i\\leq K^{*}<i^{\\prime}}(B_{i}-B_{i^{\\prime}}) \\sum_{t=1}^{T}\\mathbbm{1}(i\\sharp\\mathcal{P}_{b}(t),i^{\\prime}\\in\\mathcal{P}_ {b}(t))\\] \\[+\\sum_{j\\leq K^{*}<j^{\\prime}}(S_{j^{\\prime}}-S_{j})\\sum_{t=1}^{T} \\mathbbm{1}(j\\sharp\\mathcal{P}_{s}(t),j^{\\prime}\\in\\mathcal{P}_{s}(t))+\\sum_{ i^{\\prime},j^{\\prime}>K^{*}}(S_{j^{\\prime}}-B_{i^{\\prime}})\\sum_{t=1}^{T} \\mathbbm{1}(j^{\\prime}\\in\\mathcal{P}_{s}(t),i^{\\prime}\\in\\mathcal{P}_{b}(t)).\\]\n\nThe first term corresponds to a non-participant buyer replacing a participant buyer. The second term is the same for sellers, whereas the final term corresponds to two non-participant buyer and seller getting matched.\n\n**Individual Regret Decomposition:** We now turn to the individual regrets. Regret of a non-participant buyer \\(i\\), can be bounded by\n\n\\[\\sum_{t:i\\in\\mathcal{P}_{b}(t)}(p(t)-B_{i})\\leq\\sum_{t:i\\in \\mathcal{P}_{b}(t)}(b_{i}(t)-B_{i})\\lessapprox\\sum_{t:i\\in\\mathcal{P}_{b}(t)} \\sqrt{\\frac{\\alpha_{b,i}\\log(t)}{n_{b,i}(t)}}\\lessapprox\\sqrt{\\alpha_{b,i}n_{b, i}(T)\\log(T)}.\\]\n\nWe have \\(p(t)\\) lesser than \\(b_{i}(t)\\) because \\(i\\)-th buyer participates in round \\(t\\). By UCB property w.h.p. \\(b_{i}(t)\\) is at most \\(\\sqrt{\\frac{\\alpha_{b,i}\\log(t)}{n_{b,i}(t)}}\\) away from \\(B_{i}\\). A similar argument shows for a non-participating seller \\(j\\) the regret is roughly \\(\\sqrt{\\alpha_{s,j}n_{s,j}(T)\\log(T)}\\).\n\nFor a participating buyer \\(i\\), we have the regret bounded as\n\n\\[\\sum_{t:i\\notin\\mathcal{P}_{b}(t)}(B_{i}-p^{*})+\\sum_{t:i\\in \\mathcal{P}_{b}(t)}(p(t)-p^{*})=(T-n_{b,i}(T))(B_{i}-p^{*})+\\sum_{t:i\\in \\mathcal{P}_{b}(t)}(p(t)-p^{*}).\\]\n\nSimilarly, a participating seller \\(j\\) has regret bound \\((T-n_{s,j}(T))(p^{*}-S_{j})+\\sum_{t:j\\in\\mathcal{P}_{s}(t)}(p^{*}-p(t))\\).\n\n**Decoupling Participants:** In Lemma 10 in Appendix A, we show that learning is decoupled for the optimal participating buyers, and optimal participating sellers. It lower bounds the number of participation for optimal participating buyers and sellers.\n\n**Lemma 3** (Informal statement of Lemma 10).: _For \\(\\alpha_{\\min}>4\\), w.h.p., for every \\(i,j\\in[K^{*}]\\), and \\(i^{\\prime},j^{\\prime}>K^{*}\\)_\n\n\\[(T-n_{b,i}(T))\\lessapprox\\sum_{i^{\\prime\\prime}\\geq K^{*}+1}\\frac{ \\alpha_{b,i^{\\prime\\prime}}\\log(T)}{(B_{i}-B_{i^{\\prime\\prime}})^{2}},(T-n_{s,j }(T))\\lessapprox\\sum_{j^{\\prime\\prime}\\geq K^{*}+1}\\frac{\\alpha_{s,i^{\\prime \\prime}}\\log(T)}{(S_{j^{\\prime\\prime}}-S_{j})^{2}},\\] \\[\\sum_{t=1}^{T}\\mathbbm{1}(i\\sharp\\mathcal{P}_{b}(t),i^{\\prime}\\in \\mathcal{P}_{b}(t))\\lessapprox\\frac{\\alpha_{b,i^{\\prime}}\\log(T)}{(B_{i}-B_{i^ {\\prime}})^{2}},\\text{ and}\\sum_{t=1}^{T}\\mathbbm{1}(j\\sharp\\mathcal{P}_{s}(t),j^ {\\prime}\\in\\mathcal{P}_{s}(t))\\lessapprox\\frac{\\alpha_{b,i^{\\prime}}\\log(T)}{(S _{j^{\\prime}}-S_{j})^{2}}.\\]\n\nIt argues after a optimal non-participant \\(i^{\\prime\\prime}\\) gets \\(O(\\frac{\\log(T)}{(B_{i}-B_{i^{\\prime\\prime}})^{2}})\\) samples it can not participate while \\(i\\) does not participate as the bid of \\(i\\) is higher than the bid of \\(i^{\\prime\\prime}\\) with high probability. Similarly seller side result follow.\n\n**Non-Participant regret leads to two-sided Learning:** Unlike optimal participating agents, the effect of uncertainties on the optimal non-participating buyers and sellers cannot be decoupled directly. With at least one non-participating buyer with large estimation error in her valuation present, the non-participating sellers can keep participating even with perfect knowledge. However, this does not ensure directly that the estimation error of this non-participating buyer decreases. Next, in Lemma 12 in Appendix A, we upper bound the number of times a non-participant can participate. Informally we have\n\n**Lemma 4** (Informal).: _For \\(\\alpha_{\\min}>4\\), with high probability for any \\(i,j\\geq(K^{*}+1)\\),_\n\n\\[n_{b,i}(T)\\lessapprox\\frac{\\alpha_{b,i}\\log(T)}{(B_{K^{*}}-B_{i})^{2}}+\\sum_{j^{ \\prime}\\geq(K^{*}+1)}\\frac{\\alpha_{s,i^{\\prime}}\\log(T)}{(S_{j^{\\prime}}-B_{i}) ^{2}},n_{s,j}(T)\\lessapprox\\frac{\\alpha_{s,i}\\log(T)}{(S_{j}-S_{K^{*}})^{2}}+ \\sum_{i^{\\prime}\\geq(K^{*}+1)}\\frac{\\alpha_{b,i^{\\prime}}\\log(T)}{(S_{j}-B_{i^ {\\prime}})^{2}},\\]\n\n_and \\(\\sum_{t=1}^{T}\\mathbbm{1}(j^{\\prime}\\in\\mathcal{P}_{s}(t),i^{\\prime}\\in \\mathcal{P}_{b}(t))\\lessapprox\\frac{\\alpha_{\\max}\\log(T)}{(S_{j^{\\prime}}-B_{i ^{\\prime}})^{2}}\\)._\n\nA non-participant buyer \\(i\\) after obtaining \\(O(\\frac{\\log(T)}{(B_{K^{*}}-B_{i})^{2}})\\) samples does not belong to top \\(K^{*}\\) w.h.p. Hence, she participates with non-negligible probability only if a seller \\(j^{\\prime}\\geq(K^{*}+1)\\) participates. However, this implies that with each new match of buyer \\(i\\), additionally at least one non-participating seller is matched, decreasing both their uncertainties. For this buyer \\(i\\), we argue such spurious participation happens a total of \\(\\sum_{j^{\\prime}\\geq(K^{*}+1)}O(\\frac{\\log(T)}{(S_{j^{\\prime}}-B_{i})^{2}})\\) times. After that all the non-participating sellers \\(j\\geq(K^{*}+1)\\) will have enough samples so that their LCB bids will separate from the \\(i\\)-th buyer's UCB bid. Reversing sellers' and buyers' roles does the rest.\n\n**Bounding Price Difference:** The final part of the proof establishes bound on the cumulative difference of price from the true price \\(p^{*}\\) (see, Lemma 15 in Appendix A).\n\n**Lemma 5**.: _(Informal statement of Lemma 15) For \\(\\alpha_{\\min}>4\\), w.h.p. \\(\\sum_{t=1}^{T}|p(t)-p^{*}|\\lessapprox C\\log(T)+\\sqrt{\\alpha_{max}T\\log(T)},\\) where \\(C=O\\big{(}\\frac{(M-K^{*})(N-K^{*})}{\\Delta}\\big{)}\\)._\n\nLet us focus on the first upper bound, i.e. of the cumulative value of \\((p(t)-p^{*})\\). The proof breaks down the difference into two terms, \\(2(p(t)-p^{*})=(\\min_{i\\in\\mathcal{P}_{b}(t)}b_{i}(t)-B_{K^{*}})+(S_{K^{*}}- \\max_{j\\in\\mathcal{P}_{s}(t)}s_{j}(t))\\). For rounds when the buyer \\(K^{*}\\) is present (which happens all but \\(O\\big{(}\\frac{(N-K^{*})\\log(T)}{(B_{i}-p^{*})^{2}}\\big{)}\\) rounds) we can replace \\(\\min_{i\\in\\mathcal{P}_{b}(t)}b_{i}(t)\\) with \\(b_{K^{*}}(t)\\). Finally noticing that \\(\\sum_{t}(b_{K^{*}}(t)-B_{K^{*}})\\lessapprox\\sqrt{\\alpha_{max}}\\sqrt{T\\log(T)}\\) takes care of the first term. For the second term, we need to study the process \\(\\max_{j\\in\\mathcal{P}_{s}(t)}s_{j}(t)\\). First we bound the number of times sellers \\(1\\) to \\((K^{*}-1)\\) crosses the seller \\(K^{*}\\). Next we eliminate all the rounds where at least one seller \\(j\\geq(K^{*}+1)\\) are participating. Such an elimination comes at a cost of \\(O\\big{(}\\frac{(M-K^{*})}{\\Delta}\\big{)}\\) For any seller \\(j\\geq(K^{*}+1)\\) this happens \\(O\\big{(}\\frac{(N-K^{*})\\log(T)}{\\Delta_{j}^{2}}\\big{)}\\) times for some appropriate \\(\\Delta_{j}\\), and gives \\(\\Delta_{j}\\) regret in each round. This final step gives us the dominating \\(O\\big{(}\\frac{(M-K^{*})(N-K^{*})}{\\Delta}\\big{)}\\) term. The bound for the cumulative value of \\((p^{*}-p(t))\\) follows analogously.\n\n## 5 Lower Bounds\n\n### \\(\\Omega(\\sqrt{T})\\) minimax lower bound on individual regret\n\nWe show a minimax regret lower bound of \\(\\Omega(\\sqrt{T})\\) in Lemma 21 by considering a simpler system that decouples learning and competition. In this system, the seller is assumed to _(i)_ know her exact valuation, and _(ii)_ always ask her true valuation as the selling price, i.e., is truthful in her asking price in all the rounds. Furthermore, the pricing at every round is fixed to the average \\(p_{t}=\\frac{B_{t}+S_{t}}{2}\\) in the event that \\(B_{t}\\geq S\\). We show in Corollary 19 through a coupling argument that any algorithm for the classical two armed bandit problem can be converted to an algorithm for this special case. Then we use well known lower bounds for the bandit problem to give the \\(\\Omega(\\sqrt{T})\\) lower bound in Lemma 21. All technical details are in Appendix B.\n\n### \\(\\Omega(\\log(T))\\) instance dependent lower bound on Social Welfare Regret\n\nThe key observation is that social-welfare regret in Equation (1) is _independent_ of the pricing mechanism and only depends on the participating buyers \\(\\mathcal{P}_{b}(t)\\) and sellers \\(\\mathcal{P}_{s}(t)\\) at each time \\(t\\). We will establish a lower bound on a centralized decision maker (DM), who at each time, observes all the rewards obtained by all agents thus far, and decides \\(\\mathcal{P}_{b}(t)\\) and \\(\\mathcal{P}_{s}(t)\\) for each \\(t\\). In Appendix B.7, we show that the actions of the DM can be coupled to that of a combinatorial semi-bandit model [14], where the base set of arms are the set of all buyers and sellers, the valid subset of arms are those subsets having an equal number of buyers and sellers and the mean reward of any valid subset\\(\\mathcal{A}\\subseteq 2^{\\mathcal{D}}\\) is the difference between the sum of all valuations of buyers in \\(\\mathcal{A}\\) and of sellers in \\(\\mathcal{A}\\). In Appendix B we exploit this connection to semi-bandits to give a \\(\\Omega(\\log(T))\\) regret lower bound for the centralized DM. Thus our upper bound of \\(O(\\log(T))\\) social welfare regret under the decentralized setting is order optimal since even a centralized system must incur \\(\\Omega(\\log(T))\\) regret.\n\n## 6 Simulation Study\n\nWe perform synthetic studies to augment our theoretical guarantees. For a fixed system of \\(N\\) buyers, \\(M\\) sellers, \\(K^{*}\\) participants, and \\(\\Delta\\) gap, the rewards are Bernoulli, with means themselves chosen uniformly at random. We vary the confidence width of the buyers, \\(\\alpha_{b}\\), and seller, \\(\\alpha_{s}\\), in \\([\\alpha_{1},\\alpha_{2}]\\). Next we simulate the performance of the UCB(\\(\\alpha_{b}\\)) and LCB(\\(\\alpha_{s}\\)) over \\(100\\) independent sample paths with \\(T=50k\\). We report the mean, 25% and 75% value of the trajectories. We plot the cumulative regret of the buyers, \\(R_{b,i}(t)\\), and the sellers, \\(R_{s,j}(t)\\), the number of matches in the system \\(K(t)\\), and the price difference \\((p(t)-p^{*})\\). In Figure 2, we have a \\(8\\times 8\\) system with \\(K^{*}=5\\). We see that \\(K(t)\\) converges to \\(5\\), where as \\((p(t)-p^{*})\\) converges to \\(0\\). The social regret grows as \\(log(T)\\). The participant and non-participant individual regret of this instance is presented in the appendix in Figure 3. We defer the simulation of other systems used to study behavior of heterogeneous \\(\\alpha\\), varying gaps, and different system sizes to Appendix D.\n\n## 7 Related Work\n\n**Classical mechanism design in double auctions:** There is a large body of work on mechanism design for double auctions, following Myerson et al. [36]. The average-mechanism, which is the subject of focus in this paper achieves all the above desiderata except for being incentive compatible. The VCG mechanism was developed in a series of works [40], [12] and [20] achieves all desiderata except being budget balanced. This mechanism requires the auctioneer to subsidize the trade. More sophisticated trade mechanisms known as the McAfee mechanism [33], trade reduction, and the probabilistic mechanism [5] all trade-off some of the desiderata for others. However, the key assumption in all of these lines of work was that all participants know their own valuations, and do not need to learn through repeated interactions.\n\n**Bandit learning in matching markets:** In recent times, online learning for the two-sided matching markets have been extensively studied in [29], [30], [39], [7], [16]. This line of work studies two sided markets when one of the side does not know of their preferences apriori and learn it through interactions. However, unlike the price-discovery aspect of the present paper, the space of preferences that each participant has to learn is discrete and finite, while the valuations that agents need to learn form a continuum. This model was improved upon by [9] that added notions of price and transfer. The paper of [25] studied a contextual version of the two-sided markets where the agents preferences depend on the reveled context through an unknown function that is learnt through interactions.\n\n**Learning in auctions:** Online learning in simple auctions has a rich history - [6], [13], [34], [8], [15], [41], [22], [17] to name a few, each of which study a separate angle towards learning from repeated samples in auctions. However, unlike the our setting, one side of the market knows of their true valuations apriori. The work of [26] is the closest to ours, where the participants apriori do not know their true valuations. However, they consider the VCG mechanism in the centralized setting, i.e., all participants can observe the utilities of all participants in every round.\n\n**Learning in bi-lateral trade:** One of the first studies on learning in bilateral trade is [10]. The work of [10] considers the single buyer and seller model under a weaker ordinal feedback model, while in the present work we consider the multi-agent model under a stronger bandit feedback model. The feedback model in [10] is more restrictive since the gain from the trade cannot be estimated by the agents based on ordinal feedback while in our model, the gain from the trade can be estimated by each agent. However, our work considers the impact of multi-agent competition on regret minimization, which is not studied in [10]. In each round a buyer and a seller draw her own valuation i.i.d. from their respective distributions. Then an arbiter sets the price, with trade happening if the price is between the seller's and buyer's price. They show with full-information follow-the-leader type algorithm achieves \\(O(\\sqrt{T})\\) regret, where as with realistic (bandit-like) feedback by learning the two distributions approximately \\(O(T^{2/3})\\) regret can be achieved.\n\n**Protocol Model:** The 'protocol model' alludes to multiple agents following a similar protocol/algorithm with each agent executing protocol with private information (e.g. Platform-as-a-Service (PAAS) [21]). In the works on learning and markets mentioned here, although the decentralized systems can be modeled as games, the protocol model is studied as a tractable way to capture the essence of the problem [4; 3]. The technical basis for this assumption is that, in the limit when the number of participants are large, and the impact of any single participant is small, a protocol based algorithm is the limit in a precise sense of any equilibrium solution to the multi-agent game [19].\n\n## 8 Conclusion and Future Work\n\nWe study the Double auction with Average mechanism where the buyers and sellers need to know their own valuation from her own feedback. Using confidence based bounds - UCB for the buyers and LCB for the sellers, we show that it is possible to obtain \\(O(\\sqrt{T\\log(T)})\\) individual regret for the true participant buyers and sellers in \\(T\\) rounds. Whereas, the true non-participant buyers and sellers obtain a \\(O(\\log(T)/\\Delta)\\) individual regret where \\(\\Delta\\) is the smallest gap. The social regret of the proposed algorithm also admits a \\(O(\\log(T)/\\Delta)\\) bound. We show that there are simpler systems where each buyer and seller must obtain a \\(\\Omega(\\sqrt{T})\\) individual regret in the minimax sense. Moreover, in our setting we show even a centralized controller obtains \\(\\Omega(\\log(T))\\) social regret. Obtaining a minimax matching a \\(O(\\sqrt{T})\\) regret remains open. Another important future avenue is, developing a framework and bidding strategy with provable 'good' regret for general Double auction mechanisms. Alleviating drawbacks of protocol model, such as collusion among participants, and platform disintermediation are important directions to explore.\n\n## References\n\n* [1] Yasin Abbasi-Yadkori, David Pal, and Csaba Szepesvari. Improved algorithms for linear stochastic bandits. _Advances in neural information processing systems_, 24, 2011.\n* [2] Dana Angluin, James Aspnes, David Eisenstat, and Eric Ruppert. The computational power of population protocols. _Distributed Computing_, 20(4):279-304, 2007.\n* [3] Nick Arnosti, Ramesh Johari, and Yash Kanoria. Managing congestion in decentralized matching markets. In _Proceedings of the fifteenth ACM conference on Economics and computation_, pages 451-451, 2014.\n* [4] Itai Ashlagi, Mark Braverman, Yash Kanoria, and Peng Shi. Communication requirements and informative signaling in matching markets. In _EC_, page 263, 2017.\n* [5] Moshe Babaioff and Noam Nisan. Concurrent auctions across the supply chain. _Journal of Artificial Intelligence Research_, 21:595-629, 2004.\n* [6] Maria-Florina Balcan, Avrim Blum, and Yishay Mansour. Single price mechanisms for revenue maximization in unlimited supply combinatorial auctions. Technical report, Technical Report CMU-CS-07-111, Carnegie Mellon University, 2007. 3, 2007.\n* [7] Soumya Basu, Karthik Abinav Sankararaman, and Abishek Sankararaman. Beyond \\(\\log^{2}(t)\\) regret for decentralized bandits in matching markets. In _International Conference on Machine Learning_, pages 705-715. PMLR, 2021.\n\n* [8] Avrim Blum, Vijay Kumar, Atri Rudra, and Felix Wu. Online learning in online auctions. _Theoretical Computer Science_, 324(2-3):137-146, 2004.\n* [9] Sarah H Cen and Devavrat Shah. Regret, stability & fairness in matching markets with bandit learners. In _International Conference on Artificial Intelligence and Statistics_, pages 8938-8968. PMLR, 2022.\n* [10] Nicolo Cesa-Bianchi, Tommaso R Cesari, Roberto Colomboni, Federico Fusco, and Stefano Leonardi. A regret analysis of bilateral trade. In _Proceedings of the 22nd ACM Conference on Economics and Computation_, pages 289-309, 2021.\n* [11] Nicolo Cesa-Bianchi and Gabor Lugosi. Combinatorial bandits. _Journal of Computer and System Sciences_, 78(5):1404-1422, 2012.\n* [12] Edward H Clarke. Multipart pricing of public goods. _Public choice_, pages 17-33, 1971.\n* [13] Richard Cole and Tim Roughgarden. The sample complexity of revenue maximization. In _Proceedings of the forty-sixth annual ACM symposium on Theory of computing_, pages 243-252, 2014.\n* [14] Richard Combes, Mohammad Sadegh Talebi Mazraeh Shahi, Alexandre Proutiere, et al. Combinatorial bandits revisited. _Advances in neural information processing systems_, 28, 2015.\n* [15] Lorenzo Croissant, Marc Abeille, and Clement Calauzenes. Real-time optimisation for online learning in auctions. In _International Conference on Machine Learning_, pages 2217-2226. PMLR, 2020.\n* [16] Xiaowu Dai and Michael Jordan. Learning in multi-stage decentralized matching markets. _Advances in Neural Information Processing Systems_, 34, 2021.\n* [17] Constantinos Daskalakis and Vasilis Syrgkanis. Learning in auctions: Regret is hard, envy is easy. _Games and Economic Behavior_, 2022.\n* [18] Xiaojun Feng, Yanjiao Chen, Jin Zhang, Qian Zhang, and Bo Li. Tahes: A truthful double auction mechanism for heterogeneous spectrums. _IEEE Transactions on Wireless Communications_, 11(11):4038-4047, 2012.\n* [19] Markus Fischer. On the connection between symmetric n-player games and mean field games. _The Annals of Applied Probability_, 27:757-810, 2017.\n* [20] Theodore Groves. Incentives in teams. _Econometrica: Journal of the Econometric Society_, pages 617-631, 1973.\n* [21] Xin Guo, Anran Hu, Renyuan Xu, and Junzi Zhang. Learning mean-field games. _Advances in Neural Information Processing Systems_, 32, 2019.\n* [22] Yanjun Han, Zhengyuan Zhou, and Tsachy Weissman. Optimal no-regret learning in repeated first-price auctions. _arXiv preprint arXiv:2003.09795_, 2020.\n* [23] George Iosifidis, Lin Gao, Jianwei Huang, and Leandros Tassiulas. A double-auction mechanism for mobile data-offloading markets. _IEEE/ACM Transactions On Networking_, 23(5):1634-1647, 2014.\n* [24] George Iosifidis and Iordanis Koutsopoulos. Double auction mechanisms for resource allocation in autonomous networks. _IEEE Journal on Selected Areas in Communications_, 28(1):95-102, 2009.\n* [25] Meena Jagadeesan, Alexander Wei, Yixin Wang, Michael Jordan, and Jacob Steinhardt. Learning equilibria in matching markets from bandit feedback. _Advances in Neural Information Processing Systems_, 34, 2021.\n* [26] Kirthevasan Kandasamy, Joseph E Gonzalez, Michael I Jordan, and Ion Stoica. Mechanism design with bandit feedback. _arXiv preprint arXiv:2004.08924_, 2020.\n\n* [27] Elon Kohlberg, Shmuel Zamir, et al. Repeated games of incomplete information: The symmetric case. _Annals of Statistics_, 2(5):1040-1041, 1974.\n* [28] Tor Lattimore and Csaba Szepesvari. _Bandit algorithms_. Cambridge University Press, 2020.\n* [29] Lydia T Liu, Horia Mania, and Michael Jordan. Competing bandits in matching markets. In _International Conference on Artificial Intelligence and Statistics_, pages 1618-1628. PMLR, 2020.\n* [30] Lydia T Liu, Feng Ruan, Horia Mania, and Michael I Jordan. Bandit learning in decentralized matching markets. _Journal of Machine Learning Research_, 22(211):1-34, 2021.\n* [31] Bodhisattwa P Majumder, M Nazif Faqiry, Sanjoy Das, and Anil Pahwa. An efficient iterative double auction for energy trading in microgrids. In _2014 IEEE Symposium on Computational Intelligence Applications in Smart Grid (CIASG)_, pages 1-7. IEEE, 2014.\n* [32] Sweta Malik, Subhasis Thakur, Maeve Duffy, and John G Breslin. Double auction mechanisms for peer-to-peer energy trading: A comparative analysis. In _2022 IEEE 7th International Energy Conference (ENERGYCON)_, pages 1-6. IEEE, 2022.\n* [33] R Preston McAfee. A dominant strategy double auction. _Journal of economic Theory_, 56(2):434-450, 1992.\n* [34] Mehryar Mohri and Andres Munoz Medina. Learning theory and algorithms for revenue optimization in second price auctions with reserve. In _International conference on machine learning_, pages 262-270. PMLR, 2014.\n* [35] R Andrew Muller, Stuart Mestelman, John Spraggon, and Rob Godby. Can double auctions control monopoly and monopsony power in emissions trading markets? _Journal of environmental economics and management_, 44(1):70-92, 2002.\n* [36] Roger B Myerson and Mark A Satterthwaite. Efficient mechanisms for bilateral trading. _Journal of economic theory_, 29(2):265-281, 1983.\n* [37] Jean-Pierre Ponssard and Shmuel Zamir. Zero-sum sequential games with incomplete information. _International Journal of Game Theory_, 2(1):99-107, 1973.\n* [38] SM Reza Dibaj, Ali Miri, and SeyedAkbar Mostafavi. A cloud priority-based dynamic online double auction mechanism (pb-dodam). _Journal of Cloud Computing_, 9:1-26, 2020.\n* [39] Abishek Sankararaman, Soumya Basu, and Karthik Abinav Sankararaman. Dominate or delete: Decentralized competing bandits in serial dictatorship. In _International Conference on Artificial Intelligence and Statistics_, pages 1252-1260. PMLR, 2021.\n* [40] William Vickrey. Counterspeculation, auctions, and competitive sealed tenders. _The Journal of finance_, 16(1):8-37, 1961.\n* [41] Jonathan Weed, Vianney Perchet, and Philippe Rigollet. Online learning in repeated auctions. In _Conference on Learning Theory_, pages 1562-1583. PMLR, 2016.\n* [42] Robert Wilson. Incentive efficiency of double auctions. _Econometrica: Journal of the Econometric Society_, pages 1101-1115, 1985.\n* [43] Peter R Wurman, William E Walsh, and Michael P Wellman. Flexible double auctions for electronic commerce: Theory and implementation. _Decision Support Systems_, 24(1):17-27, 1998.", "response": "## Review\n\n### Summary\n\nThis paper investigates a two-sided double auction setting where buyers and sellers face uncertainty over their valuations, which are learned through interactions. The proposed algorithm, based on upper confidence bounds for buyers and lower confidence bounds for sellers, aims to minimize regret both socially and individually. The authors derive regret bounds showing that the social regret is upper-bounded by O(log T/Δ), while individual regret is at most O(T). This work is significant as it is the first to study repeated double auctions with unknown valuations, integrating insights from bandit learning theory and providing theoretical and simulation results to support the findings.\n\n### Strengths\n\n- Novelty of the setting, being the first to tackle learning in double auctions with unknown valuations.\n- Interesting connection between double auctions and bandit learning.\n- Well-written and organized with clear explanations.\n- The proposed algorithm is easy to implement and guarantees good regret bounds.\n- Tight regret results for social welfare and individual utility.\n- Good discussion on incentives and deviations complemented by simulation studies.\n\n### Weaknesses\n\n- The algorithmic approach is straightforward and does not introduce substantial new ideas.\n- Coordination between buyers and sellers on the bidding algorithm could encourage collusion.\n- The proof techniques may not easily generalize to more complex settings.\n- Minor lack of clarity regarding the definition of social welfare used in the paper.\n\n### Questions\n\n- Which many-to-many double auction bandit settings do you view as the right motivation for this work?\n- Please clarify the use of both instance-dependent and instance-independent regret bounds.\n- How to understand the regret bounds in cases of ties, and what happens if Δ=0?\n- Can the proof outline be made more concise to include results of simulation experiments?\n\n### Soundness\n\n**Score:** 3\n\n**Description:** 3 = good; the theoretical foundations are solid, though there are some concerns about the generalizability of the proof techniques.\n\n### Presentation\n\n**Score:** 3\n\n**Description:** 3 = good; the paper is well-organized and clear, but minor improvements in clarity and structure could enhance comprehension.\n\n### Contribution\n\n**Score:** 3\n\n**Description:** 3 = good; while the contribution is significant in the context of auction theory and online learning, it lacks groundbreaking new ideas.\n\n### Rating\n\n**Score:** 7\n\n**Description:** 7 = accept, but needs minor improvements; the paper is technically solid with a high impact on its area, yet it requires some clarifications and enhancements.\n\n### Paper Decision\n\n**Decision:** Accept (poster)\n\n**Reasons:** The paper presents a novel approach to a relevant problem in economic theory and online learning, with solid theoretical foundations and interesting results. While there are some minor weaknesses and questions regarding clarity and the approach, the overall contribution is significant, justifying an acceptance decision.\n"}
{"query": "You are a highly experienced, conscientious, and fair academic reviewer, please help me review this paper. The review should be organized into nine sections: \n1. Summary: A summary of the paper in 100-150 words.\n2. Strengths/Weaknesses/Questions: The Strengths/Weaknesses/Questions of paper, which should be listed in bullet points, with each point supported by specific examples from the article where possible.\n3. Soundness/Contribution/Presentation: Rate the paper's Soundness/Contribution/Presentation, and match this score to the corresponding description from the list below and provide the result. The possible scores and their descriptions are: \n    1 poor\n    2 fair\n    3 good\n    4 excellent\n4. Rating: Give this paper an appropriate rating, match this rating to the corresponding description from the list below and provide the result. The possible Ratings and their descriptions are: \n    1 strong reject\n    2 reject, significant issues present\n    3 reject, not good enough\n    4 possibly reject, but has redeeming facets\n    5 marginally below the acceptance threshold\n    6 marginally above the acceptance threshold\n    7 accept, but needs minor improvements \n    8 accept, good paper\n    9 strong accept, excellent work\n    10 strong accept, should be highlighted at the conference   \n5. Paper Decision: It must include the Decision itself(Accept or Reject) and the reasons for this decision, based on the criteria of originality, methodological soundness, significance of results, and clarity and logic of presentation.\n\nHere is the template for a review format, you must follow this format to output your review result:\n**Summary:**\nSummary content\n\n**Strengths:**\n- Strength 1\n- Strength 2\n- ...\n\n**Weaknesses:**\n- Weakness 1\n- Weakness 2\n- ...\n\n**Questions:**\n- Question 1\n- Question 2\n- ...\n\n**Soundness:**\nSoundness result\n\n**Presentation:**\nPresentation result\n\n**Contribution:**\nContribution result\n\n**Rating:**\nRating result\n\n**Paper Decision:**\n- Decision: Accept/Reject\n- Reasons: reasons content\n\n\nPlease ensure your feedback is objective and constructive. The paper is as follows:\n\n# DFRD: Data-Free Robustness Distillation for Heterogeneous Federated Learning\n\nKangyang Luo\\({}^{1}\\), Shuai Wang\\({}^{1}\\), Yexuan Fu\\({}^{1}\\), Xiang Li\\({}^{1}\\)1, Yunshi Lan\\({}^{1}\\), Ming Gao\\({}^{1,2}\\)\n\nSchool of Data Science & Engineering\\({}^{1}\\)\n\nKLATASDS-MOE in School of Statistics\\({}^{2}\\)\n\nEast China Normal University\n\nShanghai, China\n\n{52205901003, 51215903058,51215903042}@stu.ecnu.edu.cn\n\n{xiangli, yslan, mgao}@dase.ecnu.edu.cn\n\nFootnote 1: corresponding author\n\n###### Abstract\n\nFederated Learning (FL) is a privacy-constrained decentralized machine learning paradigm in which clients enable collaborative training without compromising private data. However, how to learn a robust global model in the data-heterogeneous and model-heterogeneous FL scenarios is challenging. To address it, we resort to data-free knowledge distillation to propose a new FL method (namely DFRD). DFRD equips a conditional generator on the server to approximate the training space of the local models uploaded by clients, and systematically investigates its training in terms of _fidelity_, _transferability_ and _diversity_. To overcome the catastrophic forgetting of the global model caused by the distribution shifts of the generator across communication rounds, we maintain an exponential moving average copy of the generator on the server. Additionally, we propose dynamic weighting and label sampling to accurately extract knowledge from local models. Finally, our extensive experiments on various image classification tasks illustrate that DFRD achieves significant performance gains compared to SOTA baselines. Our code is here: [https://anonymous.4open.science/r/DFRD-0C83/](https://anonymous.4open.science/r/DFRD-0C83/).\n\n## 1 Introduction\n\nWith the surge of data, deep learning algorithms have made significant progress in both established and emerging fields [1, 2, 3]. However, in many real-world applications (e.g., mobile devices [4], IoT [5], and autonomous driving [6], etc.), data is generally dispersed across different clients (i.e., data silos). Owing to the high cost of data collection and strict privacy protection regulations, the centralized training that integrates data together is prohibited [7, 8]. Driven by this reality, Federated Learning (FL) [9, 10] has gained considerable attention in industry and academia as a promising distributed learning paradigm that allows multiple clients to participate in the collaborative training of a global model without access to their private data, thereby ensuring the basic privacy.\n\nDespite its remarkable success, the inevitable hurdle that plagues FL research is the vast heterogeneity among real-world clients [11]. Specifically, the distribution of data among clients may be non-IID (identical and independently distributed), resulting in **data heterogeneity**[42, 12, 13, 8, 41]. It has been confirmed that the vanilla FL method FedAvg [14] suffers from _client drift_ in this case, which leads to severe performance degradation. To ameliorate this issue, a plethora of modifications [15, 16, 17, 34, 39] for FedAvg focus on regularizing the objectives of the local models to align the global optimization objective. All of the above methods follow the widely accepted assumption of model homogeneity, where local models have to share the same architecture as the global model. Nevertheless,when deploying FL systems, different clients may have distinct hardware and computing resources, and can only train the model architecture matching their own resource budgets [22, 24], resulting in **model heterogeneity**. In this case, to enable the FL systems with model homogeneity, on the one hand, clients with low resource budgets, which may be critical for enhancing the FL systems, will be discarded at the expense of training bias [43, 8, 23]. On the other hand, keeping a low complexity for the global model to accommodate all clients leads to performance drop due to the limited model capacity [30]. Therefore, the primary challenge of model-heterogeneous FL is how to conduct model aggregation of heterogeneous architectures among clients to enhance the inclusiveness of federated systems. To solve this challenge, existing efforts fall broadly into two categories: knowledge distillation (KD)-based methods [40, 45, 25, 26, 27] and partial training (PT)-based methods [30, 28, 29, 31], yet each of them has its own limitations. Concretely, KD-based methods require additional public data to align the logits outputs between the global model (student) and local models (teachers). But the desired public data is not always available in practice and the performance may decrease dramatically if the apparent disparity in distributions exists between public data and clients' private data [33]. PT-based methods send width-based sub-models to clients, which are extracted by the server from a larger global model according to each client's resource budget, and then aggregate these trained sub-models to update the global model. PT-based methods can be considered as an extension of FedAvg to model-heterogeneous scenarios, which means they are implementation-friendly and computationally efficient, but they also suffer from the same adverse effects from data heterogeneity as FedAvg or even more severely. In a word, how to learn a robust global model in FL with both data and model heterogeneity is a highly meaningful and urgent problem.\n\nTo this end, we systematically investigate the training of a robust global model in FL with both data and model heterogeneity with the aid of data-free knowledge distillation (DFKD) [35, 36, 53, 59]. See the related work in Appendix A for more DFKD methods. Note that the strategy of integrating DFKD to FL is not unique to us. Recently, FedFTG [48] leverages DFKD to fine-tune the global model in model-homogeneous FL to overcome data heterogeneity, and DENSE [49] aggregates knowledge from heterogeneous local models based on DFKD to train a global model for one-shot FL. They all equip the server, which possesses powerful hardware and computing resources, with a generator to approximate the training space of the local models (teachers), and train the generator and the global model (student) in an adversarial manner. However, the local models uploaded per communication round are not only architecturally heterogeneous but also trained on non-IID distributed private data in the situation of both data and model heterogeneity. _In this case, the generator tends to deviate from the real data distribution. Also, its output distribution may undergo large shifts (i.e., distribution shifts) across communication rounds, causing the global model to catastrophically forget useful knowledge learned in previous rounds and suffer from performance degradation._ To confront the mentioned issues, we propose a novel **D**ata-**F**ree **R**obust **D**istillation FL method called DFRD, which utilizes a conditional generator to generate synthetic data and thoroughly studies how to effectively and accurately simulate the local models' training space in terms of _fidelity_, _transferability_ and\n\nFigure 1: The full pipeline for DFRD combined with a PT-based method. DFRD works on the server and contains two phases, _training generator_ and _robust model distillation_, where \\(\\mathcal{L}_{tran},\\mathcal{L}_{div},\\mathcal{L}_{fid}\\) and \\(\\mathcal{L}_{kl},\\widetilde{\\mathcal{L}}_{kl}\\) are the loss objectives of the conditional generator and the global model, respectively.\n\n_diversity_[48; 49]. To mitigate catastrophic forgetting of the global model, an exponential moving average (EMA) copy of the conditional generator is maintained on the server to store previous knowledge learned from the local models. The EMA generator, along with the current generator, provides training data for updates of the global model. Also, we propose dynamic weighting and label sampling to aggregate the logits outputs of the local models and sample labels respectively, thereby properly exploring the knowledge of the local models. We revisit FedFTG and DENSE, and argue that DFRD as a fine-tuning method (similar to FedFTG) can significantly enhance the global model. So, we readily associate the PT-based methods in model-heterogeneous FL with the ability to rapidly provide a preliminary global model, which will be fine-tuned by DFRD. We illustrate the schematic for DFRD as a fine-tuning method based on a PT-based method in Fig. 1. Although FedFTG and DENSE can also be applied to fine-tune the global model from the PT-based methods after simple extensions, we empirically find that they do not perform as well, and the performance of the global model is even inferior to that of local models tailored to clients' resource budgets.\n\nOur main contributions of this work are summarized as follows. First, we propose a new FL method termed DFRD that enables a robust global model in both data and model heterogeneity settings with the help of DFKD. Second, we systematically study the training of the conditional generator w.r.t. _fidelity_, _transferability_ and _diversity_ to ensure the generation of high-quality synthetic data. Additionally, we maintain an EMA generator on the server to overcome the global model's catastrophic forgetting caused by the distribution shifts of the generator. Third, we propose dynamic weighting and label sampling to accurately extract the knowledge of local models. At last, our extensive experiments on six real-world image classification datasets verify the superiority of DFRD.\n\n## 2 Notations and Preliminaries\n\n**Notations.** In this paper, we focus on the centralized setup that consists of a central server and \\(N\\) clients owning private labeled datasets \\(\\{(\\mathbf{X}_{i},\\mathbf{Y}_{i})\\}_{i=1}^{N}\\), where \\(\\mathbf{X}_{i}=\\{\\mathbf{x}_{i}^{b}\\}_{b=1}^{n_{i}}\\) follows the data distribution \\(\\mathcal{D}_{i}\\) over feature space \\(\\mathcal{X}_{i}\\), i.e., \\(\\mathbf{x}_{i}^{b}\\sim\\mathcal{D}_{i}\\), and \\(\\mathbf{Y}_{i}=\\{y_{i}^{b}\\}_{b=1}^{n_{i}}\\subset[C]:=\\{1,\\cdots,C\\}\\) denotes the ground-truth labels of \\(\\mathbf{X}_{i}\\). And \\(C\\) refers to the total number of labels. Remarkably, the heterogeneity for FL in our focus includes both data heterogeneity and model heterogeneity. **For the former**, we consider the same feature space, yet the data distribution may be different among clients, that is, label distribution skewness in clients (i.e., \\(\\mathcal{X}_{i}=\\mathcal{X}_{j}\\) and \\(\\mathcal{D}_{i}\\neq\\mathcal{D}_{j},\\forall i\\neq j,i,j\\in[N]\\)). **For the latter**, each client \\(i\\) holds an on-demand local model \\(f_{i}\\) parameterized by \\(\\mathbf{\\theta}_{i}\\). Due to the difference in resource budgets, the model capacity of each client may vary, i.e., \\(|\\mathbf{\\theta}_{i}|\\neq|\\mathbf{\\theta}_{j}|,\\exists i\\neq j,i,j\\in[N]\\). In PT-based methods, we define a confined width capability \\(R_{i}\\in(0,1]\\) according to the resource budget of client \\(i\\), which is the proportion of nodes extracted from each layer in the global model \\(f\\). Note that \\(f\\) is parameterized by \\(\\mathbf{\\theta}\\), and \\(|\\mathbf{a}|\\) denotes the number of elements in vector \\(\\mathbf{a}\\).\n\n**PT-based method** is a solution for model-heterogeneous FL, which strives to extract a matching width-based slimmed-down sub-model from the global model as a local model according to each client's budget. As with FedAvg, it requires the server to periodically communicate with the clients. In each round, there are two phases: local training and server aggregation. In local training, each client trains the sub-model received from the server utilizing the local optimizer. In server aggregation, the server collects the heterogeneous sub-models and aggregates them by straightforward selective averaging to update the global model, as follows [28; 29; 30; 31]:\n\n\\[\\mathbf{\\theta}_{[l,k]}^{t}=\\frac{1}{\\sum_{j\\in\\mathcal{S}_{t}}p_{j}}\\sum_{i\\in \\mathcal{S}_{t}}p_{i}\\mathbf{\\theta}_{i,[l,k]}^{t}, \\tag{1}\\]\n\nwhere \\(\\mathcal{S}_{t}\\) is a subset sampled from \\([N]\\) and \\(p_{i}\\) is the weight of client \\(i\\), which generally indicates the size of data held by client \\(i\\). At round \\(t\\), \\(\\mathbf{\\theta}_{[l,k]}^{t}\\) denotes the \\(k^{th}\\) parameter of layer \\(l\\) of the global model and \\(\\mathbf{\\theta}_{i,[l,k]}^{t}\\) denotes the parameter \\(\\mathbf{\\theta}_{[l,k]}^{t}\\) updated by client \\(i\\). We can clearly see that Eq. (1) independently calculates the average of each parameter for the global model according to how many clients update that parameter in round \\(t\\). Instead, the parameter remains unchanged if no clients update it. Notably, if \\(|\\mathbf{\\theta}_{i}^{t}|=|\\mathbf{\\theta}^{t}|\\) for any \\(i\\in[N]\\), PT-based method becomes FedAvg. The key to PT-based method is to select \\(\\mathbf{\\theta}_{i}^{t}\\) from the global model \\(\\mathbf{\\theta}^{t}\\) when given \\(R_{i}\\). And existing sub-model extraction schemes fall into three categories: _static_[28; 29], _random_[30] and _rolling_[31].\n\nProposed Method\n\nIn this section, we detail the proposed method DFRD. We mainly work on considering DFRD as a fine-tuning method to enhance the PT-based methods, thus enabling a robust global model in FL with both data and model heterogeneity. Fig. 1 visualizes the training procedure of DFRD combined with a PT-based method, consisting of four stages on the server side: _training generator_, _robust model distillation_, _sub-model extraction_ and _model aggregation_. Note that _sub-model extraction_ and _model aggregation_ are consistent with that in the PT-based methods, so we detail the other two stages. Moreover, we present pseudocode for DFRD in Appendix C.\n\n### Training Generator\n\nAt this stage, we aim to train a well-behaved generator to capture the training space of local models uploaded from active clients. Specifically, we consider a conditional generator \\(G(\\cdot)\\) parameterized by \\(\\mathbf{w}\\). It takes as input a random noise \\(\\mathbf{z}\\in\\mathbb{R}^{d}\\) sampled from standard normal distribution \\(\\mathcal{N}(\\mathbf{0},\\mathbf{I})\\), and a random label \\(y\\in[C]\\) sampled from label distribution \\(p(y)\\), i.e., the probability of sampling \\(y\\), thus generating the synthetic data \\(\\mathbf{s}=G(\\mathbf{h}=o(\\mathbf{z},y),\\mathbf{w})\\). Note that \\(o(\\mathbf{z},y)\\) represents the merge operator of \\(\\mathbf{z}\\) and \\(y\\). To the best of our knowledge, synthetic data generated by a well-trained generator should satisfy several key characteristics: _fidelity_, _transferability_, and _diversity_[48, 49]. Therefore, in this section, we construct the loss objective from the referred aspects to ensure the quality and utility of \\(G(\\cdot)\\).\n\n**Fidelity.** To commence, we study the fidelity of the synthetic data. Specifically, we expect \\(G(\\cdot)\\) to simulate the training space of the local models to generate the synthetic dataset with a similar distribution to the original dataset. To put it differently, we want the synthetic data \\(\\mathbf{s}\\) to approximate the training data with label \\(y\\) without access to clients' training data. To achieve it, we form the fidelity loss \\(\\mathcal{L}_{fid}\\) at logits level:\n\n\\[\\mathcal{L}_{fid}=CE(\\sum_{i\\in\\mathcal{S}_{t}}\\tau_{i,y}f_{i}(\\mathbf{s},\\mathbf{\\theta }_{i}),y), \\tag{2}\\]\n\nwhere \\(CE\\) denotes the cross-entropy function, \\(f_{i}(\\mathbf{s},\\mathbf{\\theta}_{i})\\) is the logits output of the local model from client \\(i\\) when \\(\\mathbf{s}\\) is given, \\(\\tau_{i,y}\\) dominates the weight of logits from different clients \\(\\{i|i\\in\\mathcal{S}_{t}\\}\\) when \\(y\\) is given. And \\(\\mathcal{L}_{fid}\\) is the cross-entropy loss between the weighted average logits \\(\\sum_{i\\in\\mathcal{S}_{t}}\\tau_{i,y}f_{i}(\\mathbf{s},\\mathbf{\\theta}_{i})\\) and the label \\(y\\). By minimizing \\(\\mathcal{L}_{fid}\\), \\(\\mathbf{s}\\) is enforced to be classified into label \\(y\\) with a high probability, thus facilitating the fidelity of \\(\\mathbf{s}\\).\n\nIn reality, the conditional generator \\(G(\\cdot)\\) easily generates synthetic data with low classification errors (i.e. \\(\\mathcal{L}_{fid}\\) close to \\(0\\)) as the training proceeds. This may cause the synthetic data to fall into a space far from the decision boundary of the ensemble model (i.e., \\(\\sum_{i\\in\\mathcal{S}_{t}}\\tau_{i,y}f_{i}(\\cdot,\\mathbf{\\theta}_{i})\\)) if only \\(\\mathcal{L}_{fid}\\) is optimized, as shown in the synthetic data represented by red circles in Fig. 2 (a). Note that \\(d_{S}\\) and \\(d_{T}\\) denote the decision boundaries of the global model (student) and the ensemble model (teacher), respectively. An obvious observation is that the red circles are correctly classified on the same side of the two decision boundaries (i.e., \\(d_{S}\\) and \\(d_{T}\\)), making it difficult to transfer teacher's knowledge to student. We next explore how to augment the transferability of the synthetic data to ameliorate this pitfall.\n\n**Transferability** is intended to guide \\(G(\\cdot)\\) in generating synthetic data that moves the decision boundary of the global model towards that of the ensemble model, such as synthetic data with black circles in Fig. 2 (b). However, during the training of \\(d_{S}\\) to approach \\(d_{T}\\), we find that \\(G(\\cdot)\\) can easily generate two other types of synthetic data, the yellow and purple circles in Fig. 2 (c). Both of them are misclassified by the ensemble model (\\(d_{T}\\)), while the yellow circles are correctly classified and\n\nFigure 2: The visualization of synthetic data and decision boundaries of global model (student) and ensemble model (teacher). _Left panel_: synthetic data (red circles) are far away from the decision boundary \\(d_{T}\\). _Middle panel_: synthetic data (black circles) near the decision boundaries \\(d_{T}\\). _Right panel_: synthetic data (yellow and purple circles) cross over the decision boundary \\(d_{T}\\).\n\nthe purple circles are misclassified by the global model (\\(d_{S}\\)). For the conditional generator \\(G(\\cdot)\\) that takes label information as one of the inputs, yellow and purple circles can mislead the generator, thereby leading to \\(d_{S}\\) approximating \\(d_{T}\\) with a large deviation, as shown in Fig. 2 (c). Based on the above observation, we reckon that the synthetic data \\(\\mathbf{s}=G(\\mathbf{h}=o(\\mathbf{z},y),\\mathbf{w})\\) is useful if it is classified as \\(y\\) by the ensemble model but classified not as \\(y\\) by the global model. To realize it, we maximize the logits discrepancy between the global model and the ensemble model on synthetic data with black circles by leveraging Kullback-Leibler divergence loss, which takes the form:\n\n\\[\\mathcal{L}_{tran}=-\\varepsilon\\cdot KL(\\sum_{i\\in\\mathcal{S}_{t}}\\tau_{i,y}f_ {i}(\\mathbf{s},\\mathbf{\\theta}_{i}),f(\\mathbf{s},\\mathbf{\\theta})), \\tag{3}\\]\n\nwhere \\(KL\\) is Kullback-Leibler divergence function and \\(f(\\mathbf{s},\\mathbf{\\theta})\\) denotes the logits output of the global model on \\(\\mathbf{s}\\) with label \\(y\\). Note that \\(\\varepsilon=1\\) if \\(\\arg\\max f(\\mathbf{s},\\mathbf{\\theta})\\neq y\\) and \\(\\arg\\max\\sum_{i\\in\\mathcal{S}_{t}}\\tau_{i,y}f_{i}(\\mathbf{s},\\mathbf{\\theta}_{i})=y\\) hold, otherwise \\(\\varepsilon=0\\). (\\(\\diamondsuit\\))\n\nWe would like to point out that the existing works [48] and [49] are in line with our research perspective on the transferability of generator, which aims to generate more synthetic data with black circles. However, they do not carefully frame learning objective for enhancing the transferability of generator. Concretely, [48] does not consider the type of synthetic data, i.e., \\(\\varepsilon=1\\) always holds, thus inducing the generation of synthetic data with yellow and purple circles. (\\(\\triangle\\)) [49] focuses on synthetic data satisfying \\(\\arg\\max f(\\mathbf{s},\\mathbf{\\theta})\\neq\\arg\\max\\sum_{i\\in\\mathcal{S}_{t}}\\tau_{i,y} f_{i}(\\mathbf{s},\\mathbf{\\theta}_{i})\\), but enables the generation of synthetic data with purple circles yet. (\\(\\bigtriangledown\\)) 2\n\nFootnote 2: Note that \\(\\triangle\\), \\(\\bigtriangledown\\) and \\(\\diamondsuit\\) denote the strategies for the transferability of generator in [48], in [49] and in this paper, respectively.\n\n**Diversity.** Although we enable \\(G(\\cdot)\\) to generate synthetic data that falls around the real data by optimizing \\(\\mathcal{L}_{fid}\\) and \\(\\mathcal{L}_{tran}\\), the diversity of synthetic data is insufficient. Due to the fact that the generator may get stuck in _local equilibria_ as the training proceeds, model collapse occurs [57, 58]. In this case, the generator may produce similar data points for each class with little diversity. Also, the synthetic data points may not differ significantly among classes. This causes the empirical distribution estimated by \\(G(\\cdot)\\) to cover only a small manifold in the real data space, and thus only partial knowledge of the ensemble model is extracted. To alleviate this issue, we introduce a diversity loss \\(\\mathcal{L}_{div}\\) with label information to increase the diversity of synthetic data as follows:\n\n\\[\\mathcal{L}_{div}=\\exp{(-\\sum_{j,k\\in[B]}\\|\\mathbf{s}_{j}-\\mathbf{s}_{k}\\|_{2}*\\|\\mathbf{h }_{j}-\\mathbf{h}_{k}\\|_{2}/B^{2})}, \\tag{4}\\]\n\nwhere \\(B\\) denotes the batch size and \\(\\mathbf{s}_{j/k}=G(\\mathbf{h}_{j/k}=o(\\mathbf{z}_{j/k},y_{j/k}),\\mathbf{w})\\). Intuitively, \\(\\mathcal{L}_{div}\\) takes \\(\\|\\mathbf{h}_{j}-\\mathbf{h}_{k}\\|_{2}\\) as a weight, and then multiplies it by the corresponding \\(\\|\\mathbf{s}_{j}-\\mathbf{s}_{k}\\|_{2}\\) in each batch \\(B\\), thus imposing a larger weight on the synthetic data points pair (\\(\\mathbf{s}_{j}\\) and \\(\\mathbf{s}_{k}\\)) at the more distant input pair (\\(\\mathbf{h}_{j}\\) and \\(\\mathbf{h}_{k}\\)). Notably, we merge the random noise \\(\\mathbf{z}\\) with label \\(y\\) as the input of \\(G(\\cdot)\\) to overcome spurious solutions [53]. Further, we propose a multiplicative merge operator, i.e., \\(o(\\mathbf{z},y)=\\mathbf{z}\\times\\mathcal{E}(y)\\), where \\(\\mathcal{E}\\) is a trainable embedding and \\(\\times\\) means vector element-wise product. We find that our merge operator enables synthetic data with more diversity compared to others, possibly because the label information is effectively absorbed into the stochasticity of \\(\\mathbf{z}\\) by multiplying them when updating \\(\\mathcal{E}\\). See Section 4.3 for more details and empirical justification.\n\nCombining \\(\\mathcal{L}_{fid}\\), \\(\\mathcal{L}_{tran}\\) and \\(\\mathcal{L}_{div}\\), the overall objective of the generator can be formalized as follows:\n\n\\[\\mathcal{L}_{gen}=\\mathcal{L}_{fid}+\\beta_{tran}\\cdot\\mathcal{L}_{tran}+\\beta _{div}\\cdot\\mathcal{L}_{div}, \\tag{5}\\]\n\nwhere \\(\\beta_{tran}\\) and \\(\\beta_{div}\\) are tunable hyper-parameters. Of note, the synthetic data generated by a well-trained generator should be visually distinct from the real data for privacy protection, while it can capture the common knowledge of the local models to ensure similarity to the real data distribution for utility. More privacy protection is discussed in Appendices A and A.\n\n### Robust Model Distillation\n\nNow we update the global model. Normally, the global model attempts to learn as much as possible logits outputs of the ensemble model on the synthetic data generated by the generator based on knowledge distillation [46, 48, 40, 49]. The updated global model and the ensemble model are then served to train \\(G(\\cdot)\\) with the goal of generating synthetic data that maximizes the mismatch between them in terms of logits outputs (see _transferability_ discussed in the previous section). This adversarial game enables the generator to rapidly explore the training space of the local models to help knowledge transfer from them to the global model. However, it also leads to dramatic shifts in the output distribution of \\(G(\\cdot)\\) across communication rounds under heterogeneous FL scenario (i.e., distribution shifts), causing the global model to catastrophically forget useful knowledge gained in previous rounds. To tackle the deficiency, we propose to equip the server with a generator \\(\\widetilde{G}(\\cdot)\\) parameterized by \\(\\widetilde{\\mathbf{w}}\\) that is an exponential moving average (EMA) copy of \\(G(\\cdot)\\). Its parameters at the \\(t^{th}\\) communication round are computed by\n\n\\[\\widetilde{\\mathbf{w}}^{t}=\\lambda\\cdot\\widetilde{\\mathbf{w}}^{t-1}+(1-\\lambda)\\cdot \\mathbf{w}^{t}, \\tag{6}\\]\n\nwhere \\(\\lambda\\in(0,1)\\) is the momentum. We can easily see that the parameters of \\(\\widetilde{G}(\\cdot)\\) vary very little compared to those of \\(G(\\cdot)\\) over communication rounds, if \\(\\lambda\\) is close to \\(1\\). We further utilize synthetic data from \\(\\widetilde{G}(\\cdot)\\) as additional training data for the global model outside of \\(G(\\cdot)\\), mitigating the huge exploratory distribution shift induced by the large update of \\(G(\\cdot)\\) and achieving stable updates of the global model. Particularly, we compute the Kullback-Leibler divergence between logits of the ensemble model and the global model on the synthetic data points \\(\\mathbf{s}=G(\\mathbf{h}=o(\\mathbf{z},y),\\mathbf{w})\\) and \\(\\widetilde{\\mathbf{s}}=\\widetilde{G}(\\widetilde{\\mathbf{h}}=o(\\widetilde{\\mathbf{z}}, \\widetilde{y}),\\widetilde{\\mathbf{w}})\\) respectively, which is formulated as follows:\n\n\\[\\mathcal{L}_{md}=\\mathcal{L}_{kl}+\\alpha\\widetilde{\\mathcal{L}}_{kl}=KL(f(\\bm {s},\\mathbf{\\theta}),\\sum_{i\\in\\mathcal{S}_{t}}\\tau_{i,y}f_{i}(\\mathbf{s},\\mathbf{\\theta} _{i}))+\\alpha\\cdot KL(f(\\widetilde{\\mathbf{s}},\\mathbf{\\theta}),\\sum_{i\\in\\mathcal{S}_ {t}}\\tau_{i,\\widetilde{y}}f_{i}(\\widetilde{\\mathbf{s}},\\mathbf{\\theta}_{i})), \\tag{7}\\]\n\nwhere \\(\\alpha\\) is a tunable hyper-parameter for balancing different loss items.\n\n**Dynamic Weighting and Label Sampling.** So far, how to determine \\(\\tau_{i,y}\\) and \\(p(y)\\) is unclear. The appropriate \\(\\tau_{i,y}\\) and \\(p(y)\\) are essential for effective extraction of knowledge from local models. For clarity, we propose dynamic weighting and label sampling, i.e., \\(\\tau_{i,y}=n_{i,t}^{y}/n_{\\mathcal{S}_{t},t}^{y}\\) and \\(p(y)=n_{\\mathcal{S}_{t},t}^{y}/\\sum_{y\\in[C]}n_{\\mathcal{S}_{t},t}^{y}\\), where \\(n_{\\mathcal{S}_{t},t}^{y}=\\sum_{j\\in[\\mathcal{S}_{t}]}n_{j,t}^{y}\\) and \\(n_{i,t}^{y}\\) denotes the number of data with label \\(y\\) involved in training on client \\(i\\) at round \\(t\\). Due to space limitations, see Appendix F for their detail study and experimental justification.\n\n## 4 Experiments\n\n### Experimental Settings\n\n**Datasets.** In this paper, we evaluate different methods with six real-world image classification task-related datasets, namely Fashion-MNIST [69] (FMNIST in short), SVHN [70], CIFAR-10, CIFAR-100 [71], Tiny-imageNet3 and Food101 [73]. We detail the six datasets in Appendix B. To simulate data heterogeneity across clients, as in previous works [34; 37; 38], we use Dirichlet process \\(Dir(\\omega)\\) to partition the training set for each dataset, thereby allocating local training data for each client. It is worth noting that \\(\\omega\\) is the concentration parameter and smaller \\(\\omega\\) corresponds to stronger data heterogeneity.\n\nFootnote 3: [http://cs231n.stanford.edu/tiny-imagenet-200.zip](http://cs231n.stanford.edu/tiny-imagenet-200.zip)\n\n**Baselines.** We compare DFRD to FedFTG [48] and DENSE [49], which are the most relevant methods to our work. To verify the superiority of DFRD, on the one hand, DFRD, FedFTG and DENSE are directly adopted on the server to transfer the knowledge of the local models to a randomly initialized global model. We call them collectively **data-free methods**. On the other hand, they are utilized as **fine-tuning methods** to improve the global model's performance after computing weighted average per communication round. In this case, in each communication round, the preliminary global model is obtained using FedAvg [14] in FL with homogeneous models, whereas in FL with heterogeneous models, the PT-based methods, including HeteroFL [29], Federated Dropout [30] (FedDp for short) and FedRolex [31], are employed to get the preliminary global model. 4\n\nFootnote 4: As an example, DFRD+FedRelox indicates that DFRD is used as a fine-tuning method to improve the performance of FedRelox, and others are similar. See Tables 1 and 2 for details.\n\n**Configurations.** Unless otherwise stated, all experiments are performed on a centralized network with \\(N=10\\) active clients. We set \\(\\omega\\in\\{0.01,0.1,1.0\\}\\) to mimic different data heterogeneity scenarios. To simulate model-heterogeneous scenarios, we formulate exponentially distributed budgets for a given \\(N\\): \\(R_{i}=[\\frac{1}{2}]^{\\min\\{\\sigma_{i}[\\frac{\\omega}{N}]\\}}(i\\in[N])\\), where \\(\\sigma\\) and \\(\\rho\\) are both positive integers. We fix \\(\\sigma=4\\)and consider \\(\\rho\\in\\{5,10,40\\}\\). See Appendix D for more details. Unless otherwise specified, we set \\(\\beta_{tran}\\) and \\(\\beta_{div}\\) both to \\(1\\) in _training generator_, while in _robust model distillation_, we set \\(\\lambda=0.5\\) and \\(\\alpha=0.5\\). And all baselines leverage the same setting as ours. Due to space limitations, see Appendix E for the full experimental setup.\n\n**Evaluation Metrics.** We evaluate the performance of different FL methods by local and global test accuracy. To be specific, for local test accuracy (_L.acc_ for short), we randomly and evenly distribute the test set to each client and harness the test set on each client to verify the performance of local models. In terms of global test accuracy (_G.acc_ for short), we employ the global model on the server to evaluate the global performance of different FL methods via utilizing the original test set. Note that _L.acc_ is reported in **round brackets**. To ensure reliability, we report the average for each experiment over \\(3\\) different random seeds.\n\n### Results Comparison\n\n**Impacts of varying \\(\\omega\\).** We study the performance of different methods at different levels of data heterogeneity on FMNIST, SVHN, CIFAR-10 and CIFAR-100, as shown in Table 1. One can see that the performance of all methods degrades severely as \\(\\omega\\) decreases, with DFRD being the only method that is robust while consistently leading other baselines with an overwhelming margin w.r.t. _G.acc_. Also, Fig. 3 (a)-(b) show that the learning efficiency of DFRD consistently beats other baselines (see Fig. 8-9 in Appendix H for complete curves). Notably, DFRD, FedFTG and DENSE as fine-tuning methods uniformly surpass FedAvg w.r.t. _G.acc_ and _L.acc_. However, their global test accuracies suffer from dramatic deterioration or even substantially worse than that of FedAvg when they act as data-free methods. We conjecture that FedAvg aggregates the knowledge of local models more effectively than data-free methods. Also, when DFRD is used to fine-tune FedAvg, it can significantly enhance the global model, yet improve the performance of local models to a less extent.\n\n**Impacts of different \\(\\rho\\).** We explore the impacts of different model heterogeneity distributions on different methods with SVHN, CIFAR-10, Tiny-ImageNet and FOOD101. A higher \\(\\rho\\) means more clients with \\(\\frac{1}{16}\\)-width capacity w.r.t. the global model. From Table 2, we can clearly see that the performance of all methods improves uniformly with decreasing \\(\\rho\\), where DFRD consistently and overwhelmingly dominates other baselines in terms of \\(G.acc\\). Specifically, DFRD improves \\(G.acc\\) by an average of \\(11.07\\%\\) and \\(7.54\\%\\) on SVHN and CIFAR-10 respectively, compared to PT-based methods (including HeteroFL, FedDP and FedRolex). Meanwhile, DFRD uniformly and significantly outstrips FedFTG and DENSE w.r.t. \\(G.acc\\). The selected learning curve shown in Fig. 3 (c) also verifies the above statement (see Fig. 10-12 in Appendix H for more results). The above empirical results show that DFRD not only is robust to varying \\(\\rho\\), but also has significantly intensified effects on the global model for different PT-based methods. However, the results on Tiny-ImageNet and FOOD101 indicate that PT-based methods suffer from inferior test accuracy. Although DFRD improves their test accuracy, the improvement is slight. Notably, DFRD improves negligibly over PT-based methods when all clients exemplify \\(\\frac{1}{16}\\)-width capability. We thus argue that weak clients performing complex image classification tasks learn little useful local knowledge, resulting in the inability to provide effective information for the global model.\n\n### Ablation Study\n\nIn this section, we carefully demonstrate the efficacy and indispensability of core modules and key parameters in our method on SVHN, CIFAR-10 and CIFAR-100. Thereafter, we resort to FedRolex+DFRD to yield all results. For SVHN (CIFAR-10, CIFAR-100), we set \\(\\omega=0.1\\)\\((0.1,1.0)\\)\n\n\\begin{table}\n\\begin{tabular}{c c c c|c c c|c c c|c c} \\hline \\hline \\multicolumn{2}{c|}{} & \\multicolumn{2}{c|}{PRDST} & \\multicolumn{2}{c|}{SVHN} & \\multicolumn{2}{c|}{CIFAR-10} & \\multicolumn{2}{c}{CIFAR-100} \\\\ \\hline \\multirow{2}{*}{Alex.} & \\(\\omega\\)\\(\\in\\)\\(0.1\\) & \\(\\omega\\)\\(\\in\\)\\(0.1\\) & \\(\\omega\\)\\(\\in\\)\\(0.01\\) & \\(\\omega\\)\\(\\in\\)\\(0.1\\) & \\(\\omega\\)\\(\\in\\)\\(0.01\\) & \\(\\omega\\)\\(\\in\\)\\(0.01\\) & \\(\\omega\\)\\(\\in\\)\\(0.01\\) & \\(\\omega\\)\\(\\in\\)\\(0.01\\) & \\(\\omega\\)\\(\\in\\)\\(0.01\\) & \\(\\omega\\)\\(\\in\\)\\(0.01\\) \\\\ \\cline{2-14}  & \\(\\omega\\)\\(\\in\\)\\(0.1\\) & \\(\\omega\\)\\(\\in\\)\\(0.1\\) & \\(\\omega\\)\\(\\in\\)\\(0.01\\) & \\(\\omega\\)\\(\\in\\)\\(0.1\\) & \\(\\omega\\)\\(\\in\\)\\(0.01\\) & \\(\\omega\\)\\(\\in\\)\\(0.01\\) & \\(\\omega\\)\\(\\in\\)\\(0.01\\) & \\(\\omega\\)\\(\\in\\)\\(0.01\\) & \\(\\omega\\)\\(\\in\\)\\(0.01\\) & \\(\\omega\\)\\(\\in\\)\\(0.01\\) & \\(\\omega\\)\\(\\in\\)\\(0.01\\) \\\\ \\hline \\multirow{2}{*}{ResNet} & 10.99\\(\\pm\\)1.99 & 13.99\\(\\pm\\)1.99 & 13.99\\(\\pm\\)1.99 & 13.99\\(\\pm\\)1.99 & 14.19\\(\\pm\\)1.99 & 13.99\\(\\pm\\)1.99 & 13.99\\(\\pm\\)1.99 & 13.99\\(\\pm\\)1.99 & 13.99\\(\\pm\\)1.99 & 13.99\\(\\pm\\)1.99 & 13.99\\(\\pm\\)1.99 & 13.99\\(\\pm\\)1.99 & 13.99\\(\\pm\\)1.99 \\\\ \\cline{2-14}  & \\(\\omega\\)\\(\\in\\)\\(0.1\\) & \\(\\omega\\)\\(\\in\\)\\(0.1\\) & \\(\\omega\\)\\(\\in\\)\\(0.1\\) & \\(\\omega\\)\\(\\in\\)\\(0.1\\) & \\(\\omega\\)\\(\\in\\)\\(0.1\\) & \\(\\omega\\)\\(\\in\\)\\(0.1\\) & \\(\\omega\\)\\(\\in\\)\\(0.1\\) & \\(\\omega\\)\\(\\in\\)\\(0.1\\) & \\(\\omega\\)\\(\\in\\)\\(0.1\\) & \\(\\omega\\)\\(\\in\\)\\(0.1\\) & \\(\\omega\\)\\(\\in\\)\\(0.1\\) & \\(\\omega\\)\\(\\in\\)\\(0.1\\) \\\\ \\hline \\multirow{2}{*}{Neural} & 10.69\\(\\pm\\)1.99 & 10.59\\(\\pm\\)1.99 & 10.59\\(\\pm\\)1.99 & 10.59\\(\\pm\\)1.99 & 10.59\\(\\pm\\)1.99 & 10.59\\(\\pm\\)1.99 & 10.59\\(\\pm\\)1.99 & 10.59\\(\\pm\\)1.99 & 10.59\\(\\pm\\)1.99 & 11.59\\(\\pm\\)1.99 & 11.59\\(\\pm\\)1.99 \\\\ \\cline{2-14}  & **0.770\\(\\pm\\)1.99** & 10.59\\(\\pm\\)1.99 & **10.59\\(\\pm\\)1.99** & **10.59\\(\\pm\\)1.99** & **10.59\\(\\pm\\)1.99** & **10.59\\(\\pm\\)1.99** & **10.59\\(\\pm\\)1.99** & **10.59\\(\\pm\\)1.99** & **10.59\\(\\pm\\)1.99** & **10.59\\(\\pm\\)1.99** & **10.59\\(\\pm\\)1.99** \\\\ \\hline \\multirow{2}{*}{Deep} & 10.79\\(\\pm\\)1.99 & 10.59\\(\\pm\\)1.99 & 10.59\\(\\pm\\)1.99 & 10.59\\(\\pm\\)1.99 & 10.59\\(\\pm\\)1.99 & 10.59\\(\\pm\\)1.99 & 10.59\\(\\pm\\)1.99 & 10.59\\(\\pm\\)1.99 & 10.59\\(\\pm\\)1.99 & 10.59\\(\\pm\\)1.99 & 11.59\\(\\pm\\)1.99 \\\\ \\cline{2-14}  & 0.677\\(\\pm\\)1.99 & 10.59\\(\\pm\\)1.99 & 10.59\\(\\pm\\)1.99 & 10.59\\(\\pm\\)1.99 & 10.59\\(\\pm\\)1.99 & 10.59\\(\\pm\\)1.99 & 10.59\\(\\pm\\)1.99 & 10.59\\(\\pm\\)1.99 & 10.59\\(\\pm\\)1.99 & 11.59\\(\\pm\\)1.99 \\\\ \\hline \\multirow{2}{*}{Neural} & 10.19\\(\\pm\\)1.99 & 10.59\\(\\pm\\)1.99 & 10.59\\(\\pm\\)1.99 & 10.59\\(\\pm\\)1.99 & 10.59\\(\\pm\\)1.99 & 10.59\\(\\pm\\)1.99 & 10.59\\(\\pm\\)1.99 & 10.59\\(\\pm\\)1.99 & 10.59\\(\\pm\\)1.99 & 10.59\\(\\pm\\)1.99 & 10.59\\(\\pm\\)1.99 \\\\ \\cline{2-14}  & **0.667\\(\\pm\\)1.99** & 10.59\\(\\pm\\)1.99 & 10.59\\(\\pm\\)1.99 & 10.59\\(\\pm\\)1.99 & 10.59\\(\\pm\\)1.99 & 10.59\\(\\pm\\)1.99 & 10.59\\(\\pm\\)1.99 & 10.59\\(\\pm\\)1.99 & 10.59\\(\\pm\\)1.99 & 10.59\\(\\pm\\)1.99 \\\\ \\hline \\multirow{2}{*}{Neural} & 10.19\\(\\pm\\)1.99 & 10.59\\(\\pm\\)1.99 & 10.59\\(\\pm\\)1.99 & 10.59\\(\\pm\\)1.99 & 10.59\\(\\pm\\)1.99 & 10.59\\(\\pm\\)1.99 & 10.59\\(\\pm\\)1.99 & 10.59\\(\\pm\\)1.99 & 10.59\\(\\pm\\)1.99 & 10.59\\(\\pm\\)1.99 & 10.59\\(\\pm\\)1.99 \\\\ \\cline{2-14}  & **0.667\\(\\pm\\)1.99** & 10.59\\(\\pm\\)1.99 & 10.59\\(\\pm\\)1.99 & 10.59\\(\\pm\\)1.99 & 10.59\\(\\pm\\)1.99 & 10.59\\(\\pm\\)1.99 & 10.59\\(\\pm\\)1.99 & 10.59\\(\\pm\\)1.99 & 10.59\\(\\pm\\)1.99 & 10.59\\(\\pm\\)1.99 & 10.59\\(\\pm\\)1.99 \\\\ \\hline \\multirow{2}{*}{Neural} & 10.19\\(\\pm\\)1.99 & 10.59\\(\\pm\\)1.\n\nand \\(\\rho=10\\)\\((10,5)\\). Note that we also study the performance of DFRD with different numbers of clients and stragglers, see Appendix G for details.\n\n**Impacts of different transferability constraints.** It can be observed from Table 3 that our proposed transferability constraint \\(\\lozenge\\) uniformly beats \\(\\triangle\\) and \\(\\bigtriangledown\\) w.r.t. global test performance over three datasets. This means that \\(\\lozenge\\) can guide the generator to generate more effective synthetic data, thereby improving the test performance of the global model. Also, we conjecture that generating more synthetic data like those with the black circles (see Fig. 2) may positively impact on the performance of local models, since \\(\\lozenge\\) also slightly and consistently trumps other competitors w.r.t. \\(L.acc\\).\n\n**Impacts of varying merge operators.** To look into the utility of the merge operator, we consider multiple merge operators, including \\(mul\\), \\(add\\), \\(cat\\), \\(ncat\\) and \\(none\\). Among them, \\(mul\\) is \\(o(\\mathbf{z},y)=\\mathbf{z}\\times\\mathcal{E}(y)\\), \\(add\\) is \\(o(\\mathbf{z},y)=\\mathbf{z}+\\mathcal{E}(y)\\), \\(cat\\) is \\(o(\\mathbf{z},y)=[\\mathbf{z},\\mathcal{E}(y)]\\), \\(ncat\\) is \\(o(\\mathbf{z},y)=[\\mathbf{z},y]\\) and \\(none\\) is \\(o(\\mathbf{z},y)=\\mathbf{z}\\). From Table 4, we can observe that our proposed \\(mul\\) significantly outperforms other competitors in terms of \\(G.acc\\). This suggests that \\(mul\\) can better exploit the diversity constraint to make the generator generate more diverse synthetic data, thus effectively fine-tuning the global model. Also, the visualization of the synthetic data in Appendix I validates this statement.\n\n\\begin{table}\n\\begin{tabular}{c c|c|c|c} \\hline \\hline M. O. & SVHN & CIFAR-10 & CIFAR-100 \\\\ \\hline \\(mul\\)\\(mul\\)\\(mul\\)\\(\\bm\\bm\\)\\(\\bm\\)\\(\\bm\\)\\(\\bm\\)\\(\\bm\\)\\(\\bm\\bm\\) & **25.57\\(\\bm\\bm\\)\\(\\bm\\)\\(\\bm\\bm\\)**\\(\\bm\\bm\\)**\\(\\bm\\bm\\)**\\(\\bm\\bm\\)**\\(\\bm\\bm\\)**\\(\\bm\\)**\\(\\bm\\bm\\)**\\(\\bm\\bm\\)**\\(\\bm\\bm\\)**\\(\\bm\\bm\\)**\\(\\bm\\bm\\)**\\(\\bm\\bm\\)**\\(\\bm\\bm\\)**\\(\\bm\\bm\\)**\\(\\bm\\bm\\)**\\(\\bm\\bm\\)**\\(\\bm\\bm\\)**\\(\\bm\\bm\\)**\\(\\bm\\bm\\)**\\(\\bm\\bm\\)**\\(\\bm\\bm\\)**\\(\\bm\\bm\\)**\\(\\bm\\bm\\)**\\(\\bm\\bm\\)**\\(\\bm\\bm\\)**\\(\\bm\\bm\\)**\\(\\bm\\bm\\)**\\(\\bm\\bm\\)**\\(\\bm\\bm\\)**\\(\\bm\\bm\\)**\\(\\bm\\bm\\)**\\(\\bm\\bm\\)**\\(\\bm\\bm\\)**\\(\\bm\\bm\\)**\\(\\bm\\bm\\)**\\(\\bm\\bm\\)**\\(\\bm\\bm\\)**\\(\\bm\\bm\\)**\\(\\bm\\bm\\)**\\(\\bm\\bm\\)**\\(\\bm\\bm\\)**\\(\\bm\\bm\\)**\\(\\bm\\bm\\)**\\(\\bm\\bm\\)**\\(\\bm\\bm\\)**\\(\\bm\\bm\\)**\\(\\bm\\bm\\)**\\(\\bm\\bm\\)**\\(\\bm\\bm\\)**\\(\\bm\\bm\\)**\\(\\bm\\bm\\)**\\(\\bm\\bm\\)**\\(\\bm\\bm\\)**\\(\\bm\\bm\\)**\\(\\bm\\bm\\)**\\(\\bm\\bm\\)**\\(\\bm\\bm\\)**\\(\\bm\\bm\\)**\\(\\bm\\bm\\)**\\(\\bm\\bm\\)**\\(\\bm\\bm\\)**\\(\\bm\\bm\\)**\\(\\bm\\bm\\)**\\(\\bm\\bm\\)**\\(\\bm\\bm\\)**\\(\\bm\\bm\\)**\\(\\bm\\bm\\)**\\(\\bm\\bm\\)**\\(\\bm\\bm\\)**\\(\\bm\\bm\\)**\\(\\bm\\bm\\)**\\(\\bm\\bm\\)**\\(\\bm\\bm\\)**\\(\\bm\\)**\\(\\bm\\bm\\)**\\(\\bm\\bm\\)**\\(\\bm\\bm\\)**\\(\\bm\\bm\\)**\\(\\bm\\bm\\)**\\(\\bm\\bm\\)**\\(\\bm\\bm\\)**\\(\\bm\\bm\\)**\\(\\bm\\bm\\)**\\(\\bm\\bm\\)**\\(\\bm\\bm\\)**\\(\\bm\\bm\\)**\\(\\bm\\bm\\)**\\(\\bm\\bm\\)**\\(\\bm\\bm\\)**\\(\\bm\\bm\\)**\\(\\bm\\bm\\)**\\(\\bm\\bm\\)**\\(\\bm\\bm\\)**\\(\\bm\\bm\\)**\\(\\bm\\bm\\)**\\(\\bm\\bm\\)**\\(\\bm\\bm\\)**\\(\\bm\\bm\\)**\\(\\bm\\bm\\)**\\(\\bm\\bm\\)**\\(\\bm\\bm\\)**\\(\\bm\\bm\\)**\\(\\bm\\bm\\)**\\(\\bm\\bm\\)**\\(\\bm\\bm\\)**\\(\\bm\\bm\\)**\\(\\bm\\bm\\)**\\(\\bm\\bm\\)**\\(\\bm\\bm\\)**\\(\\bm\\bm\\)**\\(\\bm\\bm\\)**\\(\\bm\\bm\\)**\\(\\bm\\bm\\)**\\(\\bm\\bm\\)**\\(\\bm\\bm\\)**\\(\\bm\\bm\\)**\\(\\bm\\bm\\)**\\(\\bm\\bm\\)**\\(\\bm\\bm\\)**\\(\\bm\\bm\\)**\\(\\bm\\bm\\)**\\(\\bm\\bm\\)**\\(\\bm\\bm\\)**\\(\\bm\\bm\\)**\\(\\bm\\bm\\)**\\(\\bm\\bm\\)**\\(\\bm\\bm\\)**\\(\\bm\\mathbf{\\bm\\mathbf{\\mathbf{\\mathbf{\\mathbf{\\mathbf{\\mathbf{\\mathbf{\\mathbf{\\mathbf{\\mathbf{\\mathbf{\\mathbf{\\mathbf{\\mathbf{\\mathbf{\\mathbf{\\mathbf{\\bmbm{\\mathbf{\\mathbf{\\bm\n\n**Necessity of each component for DFRD.** We report the test performance of DFRD after divesting some modules and losses in Table 5. Here, EMA indicates the exponential moving average copy of the generator on the server. We can evidently observe that removing the EMA generator leads to a significant drop in \\(G.acc\\), which implies that it can generate effective synthetic data for the global model. The reason why the EMA generator works is that it avoids catastrophic forgetting of the global model and ensures the stability of the global model trained in heterogeneous FL. We display synthetic data in Appendix I that further corroborates the above claims. Meanwhile, we perform the leave-one-out test to explore the contributions of \\(\\mathcal{L}_{tran}\\) and \\(\\mathcal{L}_{div}\\) to DFRD separately, and further report the test results of removing them simultaneously. From Table 5, deleting either \\(\\mathcal{L}_{tran}\\) or \\(\\mathcal{L}_{div}\\) adversely affects the performance of DFRD. In addition, their joint absence further exacerbates the degradation of \\(G.acc\\). This suggests that \\(\\mathcal{L}_{tran}\\) and \\(\\mathcal{L}_{div}\\) are vital for the training of the generator. Interestingly, \\(\\mathcal{L}_{div}\\) benefits more to the global model than \\(\\mathcal{L}_{tran}\\). We speculate that the diversity of synthetic data is more desired by the global model under the premise of ensuring the fidelity of synthetic data by optimizing \\(\\mathcal{L}_{fid}\\).\n\n**Varying \\(\\beta_{tran}\\) and \\(\\beta_{div}\\).** We explore the impacts of \\(\\beta_{tran}\\) and \\(\\beta_{div}\\) on SVHN and CIFAR-10. We select \\(\\beta_{tran}\\) and \\(\\beta_{div}\\) from \\(\\{0.25,0.50,0.75,1.00,1.25,1.50\\}\\). From Fig. 4, we can see that DFRD maintains stable test performance among all selections of \\(\\beta_{tran}\\) and \\(\\beta_{div}\\) over SVHN. At the same time, \\(G.acc\\) fluctuates slightly with the increases of \\(\\beta_{tran}\\) and \\(\\beta_{div}\\) on CIFAR-10. Besides, we observe that the worst \\(G.acc\\) in Fig. 4 outperforms the baseline with the best \\(G.acc\\) in Table 2. The above results indicate that DFRD is not sensitive to choices of \\(\\beta_{tran}\\) and \\(\\beta_{div}\\) over a wide range.\n\n**Varying \\(\\alpha\\) and \\(\\lambda\\).** In order to delve into the effect of the EMA generator on DFRD in more details, we perform grid testing on the choices of control parameters \\(\\alpha\\) and \\(\\lambda\\) over SVHN and CIFAR-10. We set \\(\\alpha\\in\\{0.25,0.50,0.75,1.00,1.25,1.50\\}\\) and \\(\\lambda\\in\\{0.1,0.3,0.5,0.7,0.9\\}\\). It can be observed from Fig. 5 that high global test accuracies on SVHN are mainly located in the region of \\(\\alpha<1.25\\) and \\(\\lambda>0.5\\), while on CIFAR-10 they are mainly located in the region of \\(\\alpha>0.25\\) and \\(\\lambda<0.9\\). According to the above results, we deem that the appropriate \\(\\alpha\\) and \\(\\lambda\\) in a specific task is essential for the utility of the EMA generator. Notably, high local test accuracies mainly sit in regions that are complementary to those of high global test accuracies, suggesting that pursuing high \\(G.acc\\) and \\(L.acc\\) simultaneously seems to be a dilemma. How to ensure high \\(G.acc\\) and \\(L.acc\\) simultaneously in the field of FL is an attractive topic that is taken as our future work.\n\n## 5 Conclusion\n\nIn this paper, we propose a new FL method called DFRD, which aims to learn a robust global model in the data-heterogeneous and model-heterogeneous scenarios with the aid of DFKD. To ensure the utility, DFRD considers a conditional generator and thoroughly studies its training in terms of _fidelity_,\n\nFigure 4: Test accuracy (%) with varying \\(\\beta_{tran}\\) and \\(\\beta_{div}\\).\n\nFigure 5: Test accuracy (%) with varying (\\(\\alpha\\),\\(\\lambda\\)).\n\n\\begin{table}\n\\begin{tabular}{c c|c|c} \\hline \\hline  & SVHN & CIFAR-10 & CIFAR-100 \\\\ \\hline baseline & **34.78\\(\\pm\\)1.49** & **25.57\\(\\pm\\)37** & **28.08\\(\\pm\\)58** \\\\  & **(15.99\\(\\pm\\)1.49)** & **(16.74\\(\\pm\\)4.27)** & **(13.03\\(\\pm\\)3.46)** \\\\ \\hline -EMA & 26.97\\(\\pm\\)1.28 & 19.80\\(\\pm\\)2.25 & 24.57\\(\\pm\\)0.50 \\\\  & (14.17\\(\\pm\\)3.09) & (16.55\\(\\pm\\)0.95) & (12.23\\(\\pm\\)0.07) \\\\ \\hline -\\(\\mathcal{L}_{tran}\\) & 29.30\\(\\pm\\)2.55 & 22.97\\(\\pm\\)31 & 27.28\\(\\pm\\)0.46 \\\\  & (14.24\\(\\pm\\)1.40) & (16.33\\(\\pm\\)1.16) & (12.79\\(\\pm\\)0.40) \\\\ -\\(\\mathcal{L}_{div}\\) & 27.68\\(\\pm\\)0.85 & 22.12\\(\\pm\\)1.08 & 26.94\\(\\pm\\)1.60 \\\\  & (14.26\\(\\pm\\)3.09) & (16.64\\(\\pm\\)1.15) & (12.81\\(\\pm\\)0.40) \\\\ -\\(\\mathcal{L}_{tran}\\), -\\(\\mathcal{L}_{div}\\) & 20.32\\(\\pm\\)1.03 & 21.97\\(\\pm\\)2.48 & 25.50\\(\\pm\\)0.51 \\\\  & (13.65\\(\\pm\\)1.13) & (16.52\\(\\pm\\)1.39) & (12.30\\(\\pm\\)0.10) \\\\ \\hline \\hline \\end{tabular}\n\\end{table}\nTable 5: Impact of each component in DFRD.\n\ntransferability_ and _diversity_. Additionally, DFRD maintains an EMA generator to augment the global model. Furthermore, we propose dynamic weighting and label sampling to accurately extract the knowledge of local models. At last, we conduct extensive experiments to verify the superiority of DFRD. Due to space constraints, we discuss in detail the **limitations** and **broader impacts** of our work in Appendixes J and K, respectively.\n\n## 6 Acknowledgments\n\nThis work has been supported by the National Natural Science Foundation of China under Grant No.U1911203, and the National Natural Science Foundation of China under Grant No.62377012.\n\n## References\n\n* [1] Cordts, M., Omran, M., Ramos, S., Rehfeld, T., Enzweiler, M., Benenson, R., Franke, U., Roth, S., & Schiele, B. (2016) The cityscapes dataset for semantic urban scene understanding. _In Proceedings of the IEEE conference on computer vision and pattern recognition_, pp. 3213-3223.\n* [2] Krizhevsky, A., Sutskever, I., & Hinton, G. E. (2017) Imagenet classification with deep convolutional neural networks. Communications of the ACM **60**(6): 84-90.\n* [3] Chen, Z., Li, B., Xu, J., Wu, S., Ding, S., & Zhang, W. (2022) Towards practical certifiable patch defense with vision transformer. _In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition_, pp. 15148-15158.\n* [4] Lim, W. Y. B., Luong, N. C., Hoang, D. T., Jiao, Y., Liang, Y. C., Yang, Q., Niyato, D., & Miao, C. (2020) Federated learning in mobile edge networks: A comprehensive survey. _IEEE Communications Surveys & Tutorials_**22**(3): 2031-2063.\n* [5] Nguyen, D. C., Ding, M., Pathirana, P. N., Seneviratne, A., Li, J., & Poor, H. V. (2021) Federated learning for internet of things: A comprehensive survey. _IEEE Communications Surveys & Tutorials_**23**(3): 1622-1658.\n* [6] Li, Y., Tao, X., Zhang, X., Liu, J., & Xu, J. (2021) Privacy-preserved federated learning for autonomous driving. _IEEE Transactions on Intelligent Transportation Systems_**23**(7): 8423-8434.\n* [7] Voigt, P., & Von dem Bussche, A. (2017) The eu general data protection regulation (gdpr). _A Practical Guide, 1st Ed., Cham: Springer International Publishing_**10**(3152676): 10-5555.\n* [8] Kairouz, P., McMahan, H. B., Avent, B., Bellet, A., Bennis, M., Bhagoji, A. N., et al. (2021). Advances and open problems in federated learning. _Foundations and Trends(r) in Machine Learning_**14**(1-2): 1-210.\n* [9] Konecny, J., McMahan, B., & Ramage, D. (2015) Federated optimization: Distributed optimization beyond the datacenter. _arXiv preprint arXiv:1511.03575_.\n* [10] Konecny, J., McMahan, H. B., Yu, F. X., Richtarik, P., Suresh, A. T., & Bacon, D. (2016). Federated learning: Strategies for improving communication efficiency. _arXiv preprint arXiv:1610.05492_.\n* [11] Li, T., Sahu, A. K., Talwalkar, A., & Smith, V. (2020) Federated learning: Challenges, methods, and future directions. _IEEE signal processing magazine_**37**(3): 50-60.\n* [12] Zhao, Y., Li, M., Lai, L., Suda, N., Civin, D., & Chandra, V. (2018) Federated learning with non-iid data. _arXiv preprint arXiv:1806.00582_.\n* [13] Li, Q., Diao, Y., Chen, Q., & He, B. (2022) Federated learning on non-iid data silos: An experimental study. _In 2022 IEEE 38th International Conference on Data Engineering (ICDE)_, pp. 965-978.\n* [14] McMahan, B., Moore, E., Ramage, D., Hampson, S., & y Arcas, B. A. (2017) Communication-efficient learning of deep networks from decentralized data. _In Artificial intelligence and statistics_, pp. 1273-1282.\n* [15] Li, T., Sahu, A. K., Zaheer, M., Sanjabi, M., Talwalkar, A., & Smith, V. (2020). Federated optimization in heterogeneous networks. _Proceedings of Machine learning and systems_**2**: 429-450.\n* [16] Karimireddy, S. P., Kale, S., Mohri, M., Reddi, S., Stich, S., & Suresh, A. T. (2020) Scaffold: Stochastic controlled averaging for federated learning. _In International Conference on Machine Learning_, pp. 5132-5143.\n\n* [17] Acar, D. A. E., Zhao, Y., Navarro, R. M., Mattina, M., Whatmough, P. N., & Saligrama, V. (2021) Federated learning based on dynamic regularization. _arXiv preprint arXiv:2111.04263_.\n* [18] Kim, J., Kim, G., & Han, B. (2022) Multi-level branched regularization for federated learning. _In International Conference on Machine Learning_, pp. 11058-11073.\n* [19] Mendieta, M., Yang, T., Wang, P., Lee, M., Ding, Z., & Chen, C. (2022) Local learning matters: Rethinking data heterogeneity in federated learning. _In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition_, pp. 8397-8406.\n* [20] Lee, G., Jeong, M., Shin, Y., Bae, S., & Yun, S. Y. (2022) Preservation of the Global Knowledge by Not-True Distillation in Federated Learning. _Advances in Neural Information Processing Systems_.\n* [21] Zhang, J., Li, Z., Li, B., Xu, J., Wu, S., Ding, S., & Wu, C. (2022) Federated learning with label distribution skew via logits calibration. _In International Conference on Machine Learning_, pp. 26311-26329.\n* [22] Ignatov, A., Timofte, R., Chou, W., Wang, K., Wu, M., Hartley, T., & Van Gool, L. (2018) Ai benchmark: Running deep neural networks on android smartphones. _In Proceedings of the European Conference on Computer Vision (ECCV) Workshops_, pp. 0-0.\n* [23] Makhija, D., Han, X., Ho, N., & Ghosh, J. (2022) Architecture agnostic federated learning for neural networks. _In International Conference on Machine Learning_, pp. 14860-14870.\n* [24] Hong, J., Wang, H., Wang, Z., & Zhou, J. (2022) Efficient split-mix federated learning for on-demand and in-situ customization. _arXiv preprint arXiv:2203.09747_.\n* [25] Afonin, A., & Karimireddy, S. P. (2021) Towards model agnostic federated learning using knowledge distillation. _arXiv preprint arXiv:2110.15210_.\n* [26] Cho, Y. J., Manoel, A., Joshi, G., Sim, R., & Dimitriadis, D. (2022) Heterogeneous ensemble knowledge transfer for training large models in federated learning. _arXiv preprint arXiv:2204.12703_.\n* [27] Fang, X., & Ye, M. (2022) Robust federated learning with noisy and heterogeneous clients. _In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition_, pp. 10072-10081.\n* [28] Horvath, S., Laskaridis, S., Almeida, M., Leontiadis, I., Venieris, S., & Lane, N. (2021) Fjord: Fair and accurate federated learning under heterogeneous targets with ordered dropout. _Advances in Neural Information Processing Systems_**34**: 12876-12889.\n* [29] Diao, E., Ding, J., & Tarokh, V. (2020) HeteroFL: Computation and communication efficient federated learning for heterogeneous clients. _arXiv preprint arXiv:2010.01264_.\n* [30] Caldas, S., Konecny, J., McMahan, H. B., & Talwalkar, A. (2018) Expanding the reach of federated learning by reducing client resource requirements. _arXiv preprint arXiv:1812.07210_.\n* [31] Alam, S., Liu, L., Yan, M. & Zhang, M. (2022) FedRolex: Model-Heterogeneous Federated Learning with Rolling Sub-Model Extraction. _Advances in neural information processing systems_.\n* [32] Tang, Z., Zhang, Y., Shi, S., He, X., Han, B., & Chu, X. (2022). Virtual homogeneity learning: Defending against data heterogeneity in federated learning. _In International Conference on Machine Learning_, pp. 21111-21132.\n* [33] Tan, Y., Long, G., Liu, L., Zhou, T., Lu, Q., Jiang, J., & Zhang, C. (2022) Fedproto: Federated prototype learning across heterogeneous clients. _In Proceedings of the AAAI Conference on Artificial Intelligence_**36**(8): 8432-8440.\n* [34] Luo, K., Li, X., Lan, Y., & Gao, M. (2023) GradMA: A Gradient-Memory-based Accelerated Federated Learning with Alleviated Catastrophic Forgetting. _arXiv preprint arXiv:2302.14307_.\n* [35] Chen, H., Wang, Y., Xu, C., Yang, Z., Liu, C., Shi, B., Xu C., Xu C. & Tian, Q. (2019) Data-free learning of student networks. _In Proceedings of the IEEE/CVF International Conference on Computer Vision_, pp. 3514-3522.\n* [36] Micaelli, P., & Storkey, A. J. (2019) Zero-shot knowledge transfer via adversarial belief matching. _Advances in Neural Information Processing Systems_**32**.\n* [37] Yurochkin, M., Agarwal, M., Ghosh, S., Greenewald, K., Hoang, N., & Khazaeni, Y. (2019, May) Bayesian nonparametric federated learning of neural networks. _In International conference on machine learning_, pp. 7252-7261.\n\n* [38] Wang, H., Yurochkin, M., Sun, Y., Papailiopoulos, D., & Khazaeni, Y. (2020) Federated learning with matched averaging. _arXiv preprint arXiv:2002.06440_.\n* [39] Li, Q., He, B., & Song, D. (2021) Model-contrastive federated learning. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 10713-10722.\n* [40] Lin, T., Kong, L., Stich, S. U., & Jaggi, M. (2020) Ensemble distillation for robust model fusion in federated learning. _Advances in Neural Information Processing Systems_**33**: 2351-2363.\n* [41] Li, X., Jiang, M., Zhang, X., Kamp, M., & Dou, Q. (2021) Fedbn: Federated learning on non-iid features via local batch normalization. _arXiv preprint arXiv:2102.07623_.\n* [42] Fallah, A., Mokhtari, A., & Ozdaglar, A. (2020) Personalized federated learning: A meta-learning approach. _In Advances in Neural Information Processing Systems_.\n* [43] Pan, S. J., & Yang, Q. (2010) A survey on transfer learning. _IEEE Transactions on knowledge and data engineering_**22**(10): 1345-1359.\n* [44] Li, D., & Wang, J. (2019) Fedmd: Heterogenous federated learning via model distillation. _arXiv preprint arXiv:1910.03581_.\n* [45] He, C., Annavaram, M., & Avestimehr, S. (2020) Group knowledge transfer: Federated learning of large cnns at the edge. _Advances in Neural Information Processing Systems_**33**:14068-14080.\n* [46] Zhang, S., Liu, M., & Yan, J. (2020) The diversified ensemble neural network. _Advances in Neural Information Processing Systems_**33**:16001-16011.\n* [47] Zhu, Z., Hong, J., & Zhou, J. (2021) Data-free knowledge distillation for heterogeneous federated learning. _In International Conference on Machine Learning_, pp. 12878-12889.\n* [48] Zhang, L., Shen, L., Ding, L., Tao, D., & Duan, L. Y. (2022) Fine-tuning global model via data-free knowledge distillation for non-iid federated learning. _In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition_, pp. 10174-10183.\n* [49] Zhang, J., Chen, C., Li, B., Lyu, L., Wu, S., Ding, S., Shen, C. & Wu, C. (2022) DENSE: Data-Free One-Shot Federated Learning. _In Advances in Neural Information Processing Systems_.\n* [50] Heinbaugh, C. E., Luz-Ricca, E., & Shao, H. (2023) Data-Free One-Shot Federated Learning Under Very High Statistical Heterogeneity. _In The Eleventh International Conference on Learning Representations_.\n* [51] Fang, G., Song, J., Wang, X., Shen, C., Wang, X., & Song, M. (2021) Contrastive model inversion for data-free knowledge distillation. _arXiv preprint arXiv:2105.08584_.\n* [52] Yin, H., Molchanov, P., Alvarez, J. M., Li, Z., Mallya, A., Hoiem, D., Jha N. K., & Kautz, J. (2020) Dreaming to distill: Data-free knowledge transfer via deepinversion. _In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition_, pp. 8715-8724.\n* [53] Do, K., Le, T. H., Nguyen, D., Nguyen, D., Harikumar, H., Tran, T., Rana, S., & Venkatesh, S. (2022) Momentum Adversarial Distillation: Handling Large Distribution Shifts in Data-Free Knowledge Distillation. _Advances in Neural Information Processing Systems_**35**: 10055-10067.\n* [54] Binici, K., Pham, N. T., Mitra, T., & Leman, K. (2022) Preventing catastrophic forgetting and distribution mismatch in knowledge distillation via synthetic data. _In Proceedings of the IEEE/CVF winter conference on applications of computer vision_, pp. 663-671.\n* [55] Binici, K., Aggarwal, S., Pham, N. T., Leman, K., & Mitra, T. (2022) Robust and resource-efficient data-free knowledge distillation by generative pseudo replay. _In Proceedings of the AAAI Conference on Artificial Intelligence_**36**(6):6089-6096.\n* [56] Patel, G., Mopuri, K. R., & Qiu, Q. (2023) Learning to Retain while Acquiring: Combating Distribution-Shift in Adversarial Data-Free Knowledge Distillation. _In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition_, pp. 7786-7794.\n* [57] Odena, A., Olah, C., & Shlens, J. (2017) Conditional image synthesis with auxiliary classifier gans. _In International conference on machine learning_, pp. 2642-2651.\n* [58] Kodali, N., Abernethy, J., Hays, J., & Kira, Z. (2017) On convergence and stability of gans. _arXiv preprint arXiv:1705.07215_.\n\n* [59] Yoo, J., Cho, M., Kim, T., & Kang, U. (2019) Knowledge extraction with no observable data. _Advances in Neural Information Processing Systems_**60**.\n* [60] Luo, L., Sandler, M., Lin, Z., Zhmoginov, A., & Howard, A. (2020) Large-scale generative data-free distillation. _arXiv preprint arXiv:2012.05578_.\n* [61] Nayak, G. K., Mopuri, K. R., Shaj, V., Radhakrishnan, V. B., & Chakraborty, A. (2019) Zero-shot knowledge distillation in deep networks. _In International Conference on Machine Learning_, pp. 4743-4751.\n* [62] Wang, Z. (2021) Data-free knowledge distillation with soft targeted transfer set synthesis. _In Proceedings of the AAAI Conference on Artificial Intelligence_**35**(11): 10245-10253.\n* [63] Liu, R., Wu, F., Wu, C., Wang, Y., Lyu, L., Chen, H., & Xie, X. (2022) No one left behind: Inclusive federated learning over heterogeneous devices. _In Proceedings of the 28th ACM SIGKDD Conference on Knowledge Discovery and Data Mining_, pp. 3398-3406.\n* [64] Kim, M., Yu, S., Kim, S., & Moon, S. M. (2023) DepthFL: Depthwise Federated Learning for Heterogeneous Clients. _In The Eleventh International Conference on Learning Representations_.\n* [65] Singhal, K., Sidahmed, H., Garrett, Z., Wu, S., Rush, J., & Prakash, S. (2021) Federated reconstruction: Partially local federated learning. _Advances in Neural Information Processing Systems_**34**:11220-11232.\n* [66] Zhang, H., Bosch, J., & Olsson, H. H. (2021) Real-time end-to-end federated learning: An automotive case study. _In 2021 IEEE 45th Annual Computers, Software, and Applications Conference_, pp. 459-468.\n* [67] Li, Z., He, Y., Yu, H., Kang, J., Li, X., Xu, Z., & Niyato, D. (2022) Data heterogeneity-robust federated learning via group client selection in industrial iot. _IEEE Internet of Things Journal_**9**(18): 17844-17857.\n* [68] Zeng, S., Li, Z., Yu, H., Zhang, Z., Luo, L., Li, B., & Niyato, D. (2023) Hfedms: Heterogeneous federated learning with memorable data semantics in industrial metaverse. _IEEE Transactions on Cloud Computing_.\n* [69] Xiao, H., Rasul, K., & Vollgraf, R. (2017) Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. _arXiv preprint arXiv:1708.07747_.\n* [70] Netzer, Y., Wang, T., Coates, A., Bissacco, A., Wu, B., & Ng, A. Y. (2011) Reading digits in natural images with unsupervised feature learning.\n* [71] Krizhevsky, A., & Hinton, G. (2009) Learning multiple layers of features from tiny images.\n* [72] He, K., Zhang, X., Ren, S., & Sun, J. (2016) Deep residual learning for image recognition. _In Proceedings of the IEEE conference on computer vision and pattern recognition_, pp. 770-778.\n* [73] Bossard, L., Guillaumin, M., & Van Gool, L. (2014) Food-101-mining discriminative components with random forests. _In Computer Vision-ECCV 2014: 13th European Conference_, pp. 446-461.\n* [74] Dwork, C. (2008) Differential privacy: A survey of results. _In Theory and Applications of Models of Computation_, pp. 1-19.\n* [75] Geyer, R. C., Klein, T., & Nabi, M. (2017) Differentially private federated learning: A client level perspective. _arXiv preprint arXiv:1712.07557_.\n* [76] Cheng, A., Wang, P., Zhang, X. S., & Cheng, J. (2022) Differentially Private Federated Learning with Local Regularization and Sparsification. _In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition_, pp. 10122-10131.\n* [77] Ma, J., Naas, S. A., Sigg, S., & Lyu, X. (2022) Privacy-preserving federated learning based on multi-key homomorphic encryption. _International Journal of Intelligent Systems_**37**(9): 5880-5901.\n* [78] Zhang, L., Xu, J., Vijayakumar, P., Sharma, P. K., & Ghosh, U. (2022). Homomorphic encryption-based privacy-preserving federated learning in iot-enabled healthcare system. _IEEE Transactions on Network Science and Engineering_.", "response": "## Review\n\n### Summary\n\nThis paper presents a novel method called DFRD for robust and privacy-constrained Federated Learning (FL) amidst data and model heterogeneity. The authors propose a data-free knowledge distillation approach that utilizes a conditional generator to simulate the local models' training space, mitigating catastrophic forgetting. Experiments on several image classification datasets demonstrate that DFRD outperforms existing methods. While the method shows promise, concerns regarding hyperparameter stability, performance metrics, and experimental clarity were raised. Overall, the paper contributes valuable insights into addressing challenges in heterogeneous FL environments.\n\n### Strengths\n\n- The writing is clear and easy to follow.\n- The proposed method is novel, addressing important issues in Federated Learning.\n- Empirical evaluation is rigorous, with extensive ablation studies conducted.\n- The methodology appears practical, maintaining a consistent spatial footprint across communication cycles.\n- The authors effectively justify the necessity of loss functions and validate their effectiveness in experiments.\n\n### Weaknesses\n\n- The optimal hyperparameters for different datasets are unstable.\n- The quality of synthetic images does not meet the proposed fidelity and diversity goals.\n- Some components, like the EMA strategy, may lack novelty in the context of prior work.\n- Reported learning curves indicate models do not converge after 100 communication rounds, raising concerns about the results.\n- Overall accuracy in experiments is low, raising doubts about the practicality of the method.\n\n### Questions\n\n- Why does the method need to compute KL divergence on both G and \\tilde{G} when EMA is already used?\n- Could the authors clarify the role of the parameters w, \\rho, and \\delta in data heterogeneity?\n- What is the rationale behind selecting FedFTG and DENSE as the most relevant methods compared to others?\n- Can the authors provide a clearer explanation of the methodology behind their observations regarding transferability and decision boundaries?\n- How does the choice of hyperparameters impact the EMA method's performance, and what are the implications of different values?\n\n### Soundness\n\n**Score:** 3\n\n**Description:** Good - The paper demonstrates a solid technical foundation with a few concerns regarding stability and practical performance.\n\n### Presentation\n\n**Score:** 3\n\n**Description:** Good - The writing is generally clear, but some figures and tables could be improved for better comprehension.\n\n### Contribution\n\n**Score:** 3\n\n**Description:** Good - While the paper introduces a new method and methodology, some aspects of its novelty are marginal and could benefit from deeper theoretical justification.\n\n### Rating\n\n**Score:** 6\n\n**Description:** Weak Accept - The paper is technically solid with moderate-to-high impact potential, but it requires some improvements regarding clarity and experimental validation.\n\n### Paper Decision\n\n**Decision:** Accept (poster)\n\n**Reasons:** The paper presents a significant contribution to the field of Federated Learning by addressing critical challenges associated with data and model heterogeneity. While there are weaknesses related to performance stability and clarity, the overall originality and rigor of the experimental work justify acceptance. The paper's findings hold practical relevance, making it suitable for presentation.\n"}
{"query": "You are a highly experienced, conscientious, and fair academic reviewer, please help me review this paper. The review should be organized into nine sections: \n1. Summary: A summary of the paper in 100-150 words.\n2. Strengths/Weaknesses/Questions: The Strengths/Weaknesses/Questions of paper, which should be listed in bullet points, with each point supported by specific examples from the article where possible.\n3. Soundness/Contribution/Presentation: Rate the paper's Soundness/Contribution/Presentation, and match this score to the corresponding description from the list below and provide the result. The possible scores and their descriptions are: \n    1 poor\n    2 fair\n    3 good\n    4 excellent\n4. Rating: Give this paper an appropriate rating, match this rating to the corresponding description from the list below and provide the result. The possible Ratings and their descriptions are: \n    1 strong reject\n    2 reject, significant issues present\n    3 reject, not good enough\n    4 possibly reject, but has redeeming facets\n    5 marginally below the acceptance threshold\n    6 marginally above the acceptance threshold\n    7 accept, but needs minor improvements \n    8 accept, good paper\n    9 strong accept, excellent work\n    10 strong accept, should be highlighted at the conference   \n5. Paper Decision: It must include the Decision itself(Accept or Reject) and the reasons for this decision, based on the criteria of originality, methodological soundness, significance of results, and clarity and logic of presentation.\n\nHere is the template for a review format, you must follow this format to output your review result:\n**Summary:**\nSummary content\n\n**Strengths:**\n- Strength 1\n- Strength 2\n- ...\n\n**Weaknesses:**\n- Weakness 1\n- Weakness 2\n- ...\n\n**Questions:**\n- Question 1\n- Question 2\n- ...\n\n**Soundness:**\nSoundness result\n\n**Presentation:**\nPresentation result\n\n**Contribution:**\nContribution result\n\n**Rating:**\nRating result\n\n**Paper Decision:**\n- Decision: Accept/Reject\n- Reasons: reasons content\n\n\nPlease ensure your feedback is objective and constructive. The paper is as follows:\n\n# Birth of a Transformer: A Memory Viewpoint\n\nAlberto Bietti\n\nFlatiron Institute\n\n&Vivien Cabannes\n\nFAIR, Meta\n\n&Diane Bouchacourt\n\nFAIR, Meta\n\n&Herve Jegou\n\nFAIR, Meta\n\n&Leon Bottou\n\nFAIR, Meta\n\nWork done while at FAIR, Meta.FAIR, Meta.\n\nrequire complex nonlinear dynamics [15; 33; 45]. This simplification also allows us to introduce a natural model for individual weight matrices as _associative memories_, which store input-output or key-value pairs of embeddings through their outer products. Random high-dimensional embeddings are particularly well-suited to this viewpoint thanks to their near-orthogonality. We provide a detailed empirical study of the training dynamics, by measuring how quickly each weight matrix learns to behave as the desired associative memory, studying how this is affected by data-distributional properties, and investigate the order in which layers are learned: the model first finds the right output associations from the current token and from uniform attention patterns, then the attention heads learn to focus on the correct key-value pairs. We then present theoretical insights on this top-down learning process through population gradient dynamics. Despite its simplicity, our setup already provides useful insights on the internal structure of transformer language models and its evolution throughout training, paving the way for a better understanding of LLMs. We hope that our insights may lead to future research and improvements for LLM practitioners, _e.g._, for optimization algorithms, data pre-processing and selection, interpretability, fine-tuning, and model editing.\n\nIn summary, we make the following contributions:\n\n* We introduce a new synthetic setup to study global vs in-context learning: sequences follow bigram language models, where some bigrams change across sequences and others do not.\n* We view the transformer's weight matrices as associative memories that learn to store specific pairs of embeddings, and use this to derive a simplified but more interpretable model for our task.\n* We empirically study the training dynamics with careful probing: global bigrams are learned first, then the induction head is formed by learning appropriate memories in a top-down fashion.\n* We give theoretical insights on training dynamics, showing how a few top-down gradient steps on the population loss can recover the desired associative memories by finding signal in noisy inputs.\n\nRelated work.After the success of transformer language models for in-context learning was found [6], several works have studied how in-context learning may arise in various contexts [8; 38; 43; 48; 57]. Multiple recent papers have introduced synthetic tasks in order to better understand and interpret transformers [9; 33; 39; 61]. Several works have attempted to understand internal mechanisms in transformers that are responsible for certain behaviors, an area known as \"mechanistic interpretability\" [16; 17; 35; 39; 40; 54]. Memory and neural networks have a long history of connections [5; 18; 19; 21; 26; 27; 30; 34; 50; 55; 56]. The associative memories we consider bear similarity to [29; 56], though we use continuous input/outputs. The reader may also be interested in Fast Weight programmers [46; 47]. The use of random vectors for storing memories is related to [23]. Our approach to probing based on memory recall is related to techniques in [13; 17], though motivated differently. [14; 32; 36] study statistical and approximation properties of transformers, highlighting benefits of sparse attention patterns, but do not consider training dynamics. [25; 31; 49; 51] provide theoretical analyses of learning dynamics in transformers and other attention models, but consider different data setups and focus on single-layer architectures, while we focus on two-layer models and take a different viewpoint based on associative memories.\n\n## 2 Background\n\nThis section provides background on transformer architectures and induction head mechanisms.\n\nTransformer architecture.Transformers [52] operate on sequences of embeddings by alternating self-attention operations and token-wise feed-forward layers. We focus on decoder-only, auto-regressive architectures with a causal attention mask, which are commonly used in large language models trained for next-token prediction [6; 11; 41; 42]. We ignore normalization layers in order to simplify the architecture, since its stability benefits are not as crucial in the small models we consider. Given an input sequence of tokens \\(z_{1:T}\\in[N]^{T}\\) of length \\(T\\), where \\(N\\) is the vocabulary size, the transformer operates as follows:\n\n* **Token embeddings**: each discrete token is mapped to a \\(d\\)-dimensional embedding via an embedding map \\(W_{E}\\in\\mathbb{R}^{d\\times N}\\). We will denote the embeddings of tokens \\(z_{t}\\) by \\(x_{t}:=w_{E}(z_{t})\\), where \\(w_{E}(j)\\) is the \\(j\\)-th column of \\(W_{E}\\).\n* **Positional embeddings**: the positional embeddings \\(p_{t}=w_{P}(t)\\in\\mathbb{R}^{d}\\) are added to each token embedding depending on its position in the sequence, leading to the following input embeddings: \\[x_{t}:=x_{t}+p_{t}=w_{E}(z_{t})+w_{P}(t).\\] (1)* **Attention blocks**: given an input sequence \\(x_{1:T}\\in\\mathbb{R}^{d\\times T}\\) of embeddings, the causal attention block computes, for \\(W_{K},W_{Q},W_{V},W_{O}\\in\\mathbb{R}^{d\\times d}\\) (key, query, value, output), and for each \\(t\\), \\[x_{t}^{\\prime}:=W_{O}W_{V}x_{1:t}\\sigma(x_{1:t}^{\\top}W_{K}^{\\top}W_{Q}x_{t}) \\in\\mathbb{R}^{d},\\] (2) where \\(\\sigma\\) takes the softmax of its elements, leading to an attention of the \"values\" \\(W_{V}x_{t}\\) with weights proportional to \\(\\exp((W_{K}x_{s})^{\\top}(W_{Q}x_{t}))\\). Note that the attention operation usually considers multiple \"heads\" that each projects the input to a lower dimension. Here we stick to a single head for simplicity, since it will be sufficient for our purposes. Rewriting (2) on each \\(t\\) as \\(x_{1:T}^{\\prime}=\\mathcal{A}(x_{1:T};W_{K},W_{Q},W_{V},W_{O})\\), the \\(\\ell\\)-th layer of the transformer applies attention with layer-specific parameters along with a residual connection as follows:2 Footnote 2: We omit layer indices for simplicity of notation, and use the assignment operator \\(:=\\) instead. \\[x_{1:T}:=x_{1:T}+\\mathcal{A}(x_{1:T};W_{K}^{\\ell},W_{Q}^{\\ell},W_{V}^{\\ell},W_ {O}^{\\ell})\\]\n* **Feed-forward blocks**: feed-forward blocks operate on individual token embeddings after each attention block, typically by applying a one-hidden-layer MLP to each token, denoted \\(\\mathcal{F}(\\cdot;W_{F})\\), with a residual connection: at layer \\(\\ell\\), we have \\[x_{t}:=x_{t}+\\mathcal{F}(x_{t};W_{F}).\\] Our simplified setup will linear feed-forward layers: \\(\\mathcal{F}(x_{t};W_{F})=W_{F}x_{t}\\).\n* **Unembedding**: After the last transformer layer, the embeddings are mapped back to the vocabulary space \\(\\mathbb{R}^{N}\\) through a linear \"unembedding\" layer \\(W_{U}=[w_{U}(1),\\dots,w_{U}(N)]^{\\top}\\in\\mathbb{R}^{N\\times d}\\), where we refer to the \\(w_{U}(j)\\) as \"output embeddings\". The output of this layer is then fed into a cross-entropy loss for predicting of \\(z_{t+1}\\) from each \\(x_{t}\\).\n\nWe will sometimes refer to the representations \\(x_{t}\\) for a given token \\(t\\) throughout layers as its _residual stream_[16], since they consist of sums of embeddings and layer outputs due to residual connections.\n\nInduction head mechanism.Induction heads [16, 40] are a particular type of mechanism (or \"circuit\") in transformers that allows basic in-context prediction of the form \\([\\cdots,\\mathtt{a},\\mathtt{b},\\cdots,\\mathtt{a}]\\rightarrow\\mathtt{b}\\). These were found to be ubiquitous in transformer language models, playing a key role in enabling various forms of in-context learning. The basic mechanism consist of two attention heads in separate layers (see Figure 1 for an illustration): (i) the first is a _previous token head_ which attends to the previous token using positional information and copies its embedding to the next token; (ii) the second is the _induction head_ itself, which attends using the output of the previous token head, and outputs the original token. Our work focuses on this basic copy mechanism, but we note that richer behaviors are possible, particularly when combining multiple such mechanisms (_e.g._, [54]).\n\nFigure 1: **Induction head mechanism**. Induction heads are a two-layer mechanism that can predict \\(b\\) from a context \\([\\dots,a,b,\\dots,a]\\). The first layer is a _previous token head_, which attends to the previous token based on positional embeddings (\\(p_{t}\\to p_{t-1}\\)) and copies it after a remapping (\\(w_{E}(a)\\to w_{1}(a):=W_{O}^{1}W_{V}^{1}w_{E}(a)\\)). The second layer is the _induction head_, which attends based on the output of the previous token head (\\(w_{E}(a)\\to w_{1}(a)\\)) and outputs the attended token, remapped to output embeddings (\\(w_{E}(b)\\to w_{U}(b)\\)). Boxes in the diagram represent different embeddings in superposition on each token’s residual stream (we omit some irrelevant ones for clarity, _e.g._, positional embeddings in upper layers), and attention and output associations are shown with the associative memory viewpoint presented in Section 4.\n\n## 3 Synthetic Setup\n\nIn this section, we introduce our synthetic data setup, which allows us to carefully study how the induction head mechanism develops during training, and how transformers learn to use information from the context vs simple associations from the training data.\n\nBigram data model.Our model for sequences consists of a generic bigram language model (_i.e._, Markov chain), but where the transitions for a few _trigger tokens_ denoted \\(q_{k}\\) are modified in each sequence to always be followed by some _output tokens_\\(o_{k}\\). Let \\(K\\) be the number of trigger tokens, and fix the following distributions over the vocabulary \\([N]\\): \\(\\pi_{b}(\\cdot|i)\\), \\(\\pi_{u}\\), \\(\\pi_{o}(\\cdot|i)\\) and \\(\\pi_{q}\\), for \\(i\\in[N]\\). \\(\\pi_{b}(\\cdot|i)\\) are the global bigram conditionals, \\(\\pi_{u}\\) the global unigram distribution, while \\(\\pi_{o}\\) is used to sample output tokens at each sequence. The triggers are either fixed to some predefined set of tokens \\(Q\\), or sampled from \\(\\pi_{q}\\). Each sequence \\(z_{1:T}^{n}\\) is generated as follows:\n\n* (optional) Sample \\(q_{1},\\dots,q_{K}\\sim\\pi_{q}\\), i.i.d. without replacement (_random triggers_)\n* Sample \\(o_{k}\\sim\\pi_{o}(\\cdot|q_{k})\\), i.i.d. with replacement.\n* Sample \\(z_{1}^{n}\\sim\\pi_{u}\\) and \\(z_{t}^{n}|z_{t-1}^{n}\\sim p_{n}(\\cdot|z_{t-1}^{n})\\) for \\(t=2,\\dots,T\\), where \\[p_{n}(j|i)=\\begin{cases}\\pi_{b}(j|i),&\\text{ if }i\\notin\\{q_{k}\\}_{k}\\\\ \\mathbbm{1}\\{j=o_{k}\\},&\\text{ if }i=q_{k}.\\end{cases}\\]\n\nExperimental setup and initial experiment.Our experiments take \\(\\pi_{u}\\) and \\(\\pi_{b}\\) to be unigram and bigram character-level distributions estimated from the tiny Shakespeare dataset, with vocabulary size \\(N=65\\). We generally sample triggers from \\(\\pi_{q}=\\pi_{u}\\) or fix them to the \\(K\\) most frequent tokens. We sample uniform outputs \\(o_{k}\\) in most cases, but also experiment with \\(\\pi_{o}=\\pi_{b}\\) in Section 5.\n\nAs a preliminary experiment, we train a two-layer vanilla transformer with single-head attention layers and MLP feed-forward layers, following the training setup described in Section 5. On our synthetic data, with fixed (resp. random) triggers and uniform outputs, the model achieves over 99% accuracy (resp. 95%) on output tokens after the first occurrence, versus around 55% for one layer. This gap may be related to the difficulty of modeling three-way interactions with a single attention layer [44]. We visualize attention maps on test sequences in Figure 2, which shows that the model has learned an induction head mechanism. The sequence in the middle figure has \\((q_{k},o_{k})\\in\\{(a,b),(t,s)\\}\\). For fixed triggers, the induction head is only active for the triggers used in training, which suggests the presence of a \"memory\" in the attention layer. For random triggers, it is active on every repeated token, so that the model then needs to disambiguate between in-context and global predictions. For instance, the model may choose to use the retrieved token when it is unlikely to be sampled from the global bigram distribution, something which we found to often be the case in practice.\n\n## 4 The Associative Memory Viewpoint\n\nIn this section, we present our associative memory view on transformers: with nearly orthogonal embeddings, the weight matrices behave as associative memories which store pairs of embeddings as\n\nFigure 2: **Induction head behavior in attention maps** observed on a 2-layer transformer trained on two variants of our synthetic dataset. Each row shows the attention pattern for predicting the next token. (left) The first layer head always attends to the previous token. (center) For fixed triggers \\(Q=\\{a,t\\}\\), the second layer head mainly attends to tokens following such triggers. (right) For random triggers, the induction head mechanism is active for any repeated token (here the only trigger is \\(L\\)). Red and green boxes highlight tokens following previous occurrences of the query, with red boxes corresponding to “correct” output tokens \\(o_{k}\\) following trigger tokens \\(q_{k}\\).\n\na weighted sum of their outer products. We then introduce a simplified transformer model with fixed random embeddings that will yield a precise understanding of learning dynamics using this viewpoint.\n\n### Weight matrices as associative memories\n\nWhile intermediate representations in the transformer consist of high-dimensional vectors in residual streams, they are often \"collapsed\" down to scalar measurements by testing against other representations, using operations of the form \\(v_{j}^{\\top}Wu_{i}\\) for some matrix \\(W\\). For instance, \\(u_{i}\\) and \\(v_{j}\\) could be key and query vectors in an attention head, or input and output embeddings for predicting the next token. If \\((u_{i})_{i}\\) and \\((v_{j})_{j}\\) are orthonormal (or nearly-orthonormal) sets of embeddings, a natural way to store desired input-output associations \\(i,j\\) is through the following associative memory:\n\n\\[W=\\sum_{i,j}\\alpha_{ij}v_{j}u_{i}^{\\top}, \\tag{3}\\]\n\nso that the scores \\(v_{j}^{\\top}Wu_{i}\\approx\\alpha_{ij}\\) may be used to assess the relevance of the \\((i,j)\\) pair, _e.g._, as part of a softmax operation in attention or next token prediction.\n\nRandom embeddings.A simple way to ensure that embeddings \\((u_{i})_{i}\\) and \\((v_{j})_{j}\\) are nearly-orthonormal is to set them to be random high-dimensional vectors, such as Gaussian vectors with variance \\(1/d\\) in \\(d\\) dimensions. Indeed, these are known to satisfy [23; 53]\n\n\\[u_{i}^{\\top}u_{i}\\approx 1\\quad\\text{ and }\\quad u_{i}^{\\top}u_{j}\\approx O \\left(\\frac{1}{\\sqrt{d}}\\right),\\]\n\nso that (3) is a reasonable way to define an associative memory, without requiring an explicit activation function as employed in end-to-end memory networks [50]. We may also easily create a \"remapping\" of an existing embedding \\(u_{i}\\) by multiplying it by a random matrix \\(W_{0}\\in\\mathbb{R}^{d\\times d}\\) with Gaussian entries of variance \\(1/d\\), which is commonly used for initializing neural network parameters. The new **remapped embedding**\\(W_{0}u_{i}\\) is near-unit norm, and is near-orthogonal to \\(u_{i}\\) in addition to the other \\(u_{j}\\). Note that this fact implies that attention scores at initialization are near-uniform. See Appendix A for more details.\n\nLearning associative memories.We now show that learning associations of input-output embeddings via gradient descent leads to a weighted associative memory of a form similar to (3).\n\n**Lemma 1** (Gradients and associative memories).: _Let \\(p\\) be a data distribution over input-output tokens, and consider the following loss, where the input and output embeddings \\(W_{E}\\) and \\(W_{U}\\) are fixed:_\n\n\\[L(W)=\\mathbb{E}_{(z,y)\\sim p}[\\ell(y,W_{U}Ww_{E}(z))], \\tag{4}\\]\n\n_with \\(\\ell\\) the cross-entropy loss. The gradients of the population loss \\(L\\) then take the form_\n\n\\[\\nabla_{W}L(W)=\\sum_{k=1}^{N}\\mathbb{E}_{z}[(\\hat{p}_{W}(y=k|z)-p(y=k|z))w_{U} (k)w_{E}(z)^{\\top}], \\tag{5}\\]\n\n_where \\(\\hat{p}_{W}(y\\!=\\!k|x)=\\sigma(W_{U}Ww_{E}(z))_{k}\\) are the model's predicted probabilities. Running gradient descent (with or without weight decay) from initialization \\(W_{0}\\) then leads to estimates of the following form, for some \\(\\alpha_{0}\\) and \\(\\alpha_{ij}\\) that vary with the number of iterations:_\n\n\\[\\hat{W}=\\alpha_{0}W_{0}+\\sum_{i,j}\\alpha_{ij}w_{U}(j)w_{E}(i)^{\\top}. \\tag{6}\\]\n\nNote that (4) is a convex problem in \\(W\\), thus with appropriate step-size and large enough number of steps (with no weight decay) we can expect gradient descent to be close to the global minimum. At the optimum, if the embeddings are nearly orthogonal, then (5) implies \\(\\hat{p}_{W}(y=k|z)\\approx p(y=k|z)\\). We remark that if \\(W_{0}\\) is a Gaussian random matrix, as if often the case for neural network layers, the first term in (6) plays a minor role: testing \\(W_{0}\\) against an input-output pair \\((i,j)\\) with \\(\\alpha_{ij}\\neq 0\\) will concentrate around zero when \\(d\\) is large, while the \\((i,j)\\) term in the sum will concentrate around \\(\\alpha_{ij}\\). We also note that the gradient updates described above correspond to a so-called maximal feature learning regime similar to \\(\\mu\\)P updates in intermediate layers of deep networks [58; 59].\n\nHandling superposition.In Lemma 1, we assumed that inputs to the matrix \\(W\\) are embeddings of a single token. Yet, in transformer models, the inputs to weight matrices are often sums, or _superpositions_ of embeddings. For instance, the initial representations of each token are sums of token and positional embeddings, and representations at later layers are sums of the outputs of each previous block, due to residual connections. Outputs of attention layers are also weighted sums of potentially many embeddings, at least initially when attention patterns are spread out. By linearity, associative memories of the form (6) simply operate individually on each embedding of a superposition, and return a new superposition (up to additional noise due to near-orthogonality). In practice, we will see that learned memories often focus on a single embedding and filter out the rest as noise when irrelevant (see also Section 6). We note that linearity can also be limiting, since it makes it difficult to map sets to specific output embeddings: \\(u_{\\{i,j\\}}:=u_{i}+u_{j}\\) needs to map to \\(Wu_{i}+Wu_{j}\\), and thus cannot map to a new embedding \\(v_{\\{i,j\\}}\\). Such mappings of sets thus require non-linear associative memories, for instance by leveraging a sparse decoding of which elements are actually present (_e.g._, using compressed sensing), or by using MLPs with non-linear activations [15; 30].\n\n### A simplified two-layer transformer architecture\n\nWe consider a simpler two-layer transformer which is more interpretable with the memory viewpoint, and will help us analyze learning dynamics both empirically and theoretically.\n\n* We freeze input, output and positional embeddings (\\(W_{E},W_{U},W_{P}\\)) to their random initialization throughout training. This brings us to the Gaussian random vector setup presented above.\n* We fix \\(W^{1}_{Q}=W^{2}_{Q}=I_{d}\\), so that \\(W^{1}_{K}\\) and \\(W^{2}_{K}\\) play the role of both key and query matrices. This changes the gradient dynamics, but simplifies the model by avoiding the redundancy in (2). The pre-softmax attention scores then take the form \\(x_{q}^{\\top}W^{\\ell}_{K}x_{k}\\), with \\(x_{q}\\) (resp. \\(x_{k}\\)) the query (resp. key) embeddings, which now directly resembles an associative memory lookup.\n* We freeze \\(W^{1}_{V}\\), \\(W^{1}_{Q}\\), and \\(W^{2}_{V}\\) to random initialization. These play the role of _remapping_ attended tokens into new tokens, since for random \\(W\\) and large \\(d\\), \\(Wx\\) is nearly orthogonal to \\(x\\) and to any other random embeddings independent of \\(x\\).\n* We train \\(W^{2}_{Q}\\), since the outputs of the induction head need to be mapped back into appropriate output embeddings in order to predict the output tokens \\(o_{k}\\) correctly.\n* We use a single linear feedforward layer after the second attention block, with weight matrix \\(W_{F}\\).\n\nThis is plausibly the layer responsible for learning the global bigram distributions.\n\nWe remark that while this model freezes some parameters at initialization, it is richer than a \"lazy\" or neural tangent kernel approximation [10; 22; 24] since the model is still highly non-linear in its parameters and, as we will see, induces rich non-linear learning dynamics.\n\nSolving the bigram problem with associative memories.We now show how the above architecture can solve the synthetic bigram problem from Section 3 with well-chosen weight matrices. While this is only a hypothetical model, we show in Section 5 that it is surprisingly faithful to the learned model.\n\nRecall that due to residual connections, the inputs to the weight matrices typically consist of superpositions of various embeddings including token embeddings, positional embeddings, or \"remapped\" versions thereof. These may be viewed as sets, as illustrated in Figure 1, and associative memories can easily ignore certain elements of the set, _e.g._, ignore token embeddings by only focusing on positional embeddings. The induction head mechanism can be obtained by setting:\n\n\\[W^{1}_{K}=\\sum_{t=2}^{T}p_{t}p_{t-1}^{\\top},\\quad W^{2}_{K}=\\sum_{k\\in Q}w_{E} (k)(W^{1}_{O}W^{1}_{V}w_{E}(k))^{\\top},\\quad W^{2}_{O}=\\sum_{k=1}^{N}w_{U}(k)( W^{2}_{V}w_{E}(k))^{\\top}, \\tag{7}\\]\n\nwhere \\(Q\\) is the set of triggers when they are fixed, or the support of \\(\\pi_{q}\\) when they are random. In words, the first attention layer matches a token to the previous tokens using positional embeddings. The second layer matches the trigger token to a remapping of itself by \\(W^{1}_{O}W^{1}_{V}\\), and the output matches a remapping of the input token by \\(W^{2}_{V}\\) to the corresponding output token. We remark that one can easily make the attention patterns more peaked on the correct associations by rescaling \\(W^{1}_{K}\\) and \\(W^{2}_{K}\\). The global bigram statistics can be encoded in the feed-forward layer as follows:\n\n\\[W_{F}=\\sum_{i=1}^{N}\\sum_{j=1}^{N}\\log\\pi_{b}(j|i)w_{U}(j)w_{E}(i)^{\\top}. \\tag{8}\\]The question remains of how the model could trade-off predictions from the induction head and from the feed-forward layer, which are added together due to residual connections. With fixed triggers \\(Q\\), we may simply remove all \\(i\\in Q\\) from the summation in (8), so that the model exclusively relies on the attention head for all triggers (indeed, the output of \\(W_{O}^{2}\\) is in the span of output embeddings, which are nearly orthogonal to the row space of \\(W_{F}\\)). When the triggers can vary across different sequences, choosing between the induction head and the feed-forward layer is more ambiguous as it depends on context, and \\(W_{F}\\) may try to learn more complex mappings that also use the outputs of \\(W_{O}^{2}\\). In practice, we observe that the model often prefers the induction head, unless its output agrees with one of the top predictions from the global bigram, in which case it tends to prefer those.\n\nBeyond the simplified architecture.While our simplified architecture already captures the relevant aspects for the bigram model, it lacks some of the components that appear in standard transformers, such as non-linear MLPs, trained embeddings, layer normalization, and joint learning of a factorization \\(W_{K}^{\\top}W_{Q}\\) (potentially with low rank matrices \\(W_{K},W_{Q}\\in\\mathbb{R}^{d_{h}\\times d}\\) with \\(d_{h}<d\\) as in multi-head attention), instead of a single matrix \\(W_{K}\\). In practice, transformers also involve many more layers, as well as multiple heads at each self-attention layer. In Appendix D, we discuss how our memory viewpoint naturally extends to such architectural components, and we illustrate in Appendix E that they empirically lead to similar observations. Nonetheless, we focus on our simpler architecture in the main paper due to simplicity of exposition and better interpretability thanks to a clear identifiability of the role of each matrix, which is lost in models with more heads and layers.\n\n## 5 Empirical Study\n\nIn this section, we present our empirical analysis of learning dynamics on the bigram data defined in Section 3, for the simplified architecture defined in Section 4.2. See Appendix E for additional results. Our code is available at [https://github.com/albietz/transformer-birth](https://github.com/albietz/transformer-birth).\n\nExperimental setup.We train our models using mini-batch SGD with momentum, where each batch consists of 512 fresh sequences of length \\(T=256\\) sampled from our synthetic model. We use a fixed learning rate and weight decay. Hyperparameters are given in Appendix E. Unless otherwise noted, we use \\(d=128\\), random triggers with \\(\\pi_{q}=\\pi_{u}\\) and uniform output tokens. The reported accuracies and losses are computed over each fresh batch before it is used for optimization, and are averaged over relevant tokens: \"in-context accuracy/loss\" numbers only consider predictions of output tokens on triggers starting at the second occurrence (the first is non-deterministic), while \"global loss\" refers to average loss on non-trigger tokens.\n\nMemory recall probes.In addition to loss and accuracy, we consider metrics to check whether individual matrices have learned the desired associative memories: for a desired target memory \\(W_{*}=\\sum_{(i,j)\\in\\mathcal{M}}v_{j}u_{i}^{\\top}\\), the corresponding recall metric is computed from the empirical estimate \\(\\hat{W}\\) as\n\n\\[R(\\hat{W},W_{*})=\\frac{1}{|\\mathcal{M}|}\\sum_{(i,j)\\in\\mathcal{M}}\\mathbbm{1} \\{\\arg\\max_{j^{\\prime}}v_{j^{\\prime}}^{\\top}\\hat{W}u_{i}=j\\}. \\tag{9}\\]\n\nWe use this for each matrix in (7) as target, and additionally test the previous token matrix \\(W_{K}^{1}\\) on smaller time windows. For the final feed-forward layer, we measure the average KL divergence between the predicted softmax distribution using only \\(W_{F}\\) and the global bigram distribution \\(\\pi_{b}\\):\n\n\\[d_{KL}(W_{F},\\pi_{b}):=\\frac{1}{N}\\sum_{k=1}^{N}d_{KL}(\\sigma(W_{U}W_{F}w_{E} (k)),\\pi_{b}(\\cdot|k)). \\tag{10}\\]\n\nEmergence of the induction head via top-down learning.We begin our study by only training to minimize the loss on _trigger-output_ token predictions after their first occurrence. This should be predictable with 100% accuracy using the two-layer induction head mechanism according to Section 4. We also remove the feed-forward layer, in order to focus on the learning of attention matrices \\(W_{K}^{1}\\), \\(W_{K}^{2}\\) and \\(W_{O}^{2}\\) in isolation.\n\nFigure 3 studies the effect of freezing different layers until iteration 300 on the training dynamics. By looking at memory recall probes, we see that training key-query matrices does not lead to any\n\nlearning unless \\(W_{O}^{2}\\) is learned first, and that \\(W_{O}^{2}\\) can learn the correct associations even when trained by itself with key-value matrices at random initialization. Recall that the attention weights are essentially uniform when \\(W_{K}\\) are at random initialization, so that training \\(W_{O}^{2}\\) alone resembles a bag-of-words models that aggregates representations throughout the sequence. While such a model has poor prediction accuracy, it is nevertheless sufficient to recover the correct associations in \\(W_{O}^{2}\\) (a similar observation was made in [49] in a different setup).\n\nThen, these associations enable learning key-query matrices that focus the attention on relevant tokens, by storing relevant key-query pairs in the form of associative memories, which eventually recovers the desired induction head behavior and leads to near-perfect accuracy. The two rightmost plots suggest that the second layer is learned before the first, in the sense that \\(W_{K}^{2}\\) is easier to learn when \\(W_{K}^{1}\\) is frozen compared to the reverse, yet learning them together seems beneficial, possibly due to helpful feedback loops [1]. We also observe that \\(W_{K}^{1}\\) fits previous token associations for early positions much faster than later positions (purple vs gray line). This is likely due to the fact that it should be enough for the previous token head to attend to the first appearance of each trigger \\(q_{k}\\), which is typically early in the sequence, so that most of the gradient will focus on early positions.\n\nOverall, this provides a fine-grained understanding of the learning dynamics of induction heads. In Section 6, we analyze how a few gradient steps in a top-down fashion may suffice to recover appropriate associative memories in high dimension and with enough data. See Appendix E for additional experiments, including on the role of dimensionality.\n\nGlobal vs in-context learning.Figure 4(left/right) shows that when training all layers jointly, the global bigram statistics tend to be learned more quickly than the induction head, as seen from the quick drop in loss and KL in early iterations. The \\(W_{O}^{2}\\) probe also seems to improve quickly initially, but only leads to mild improvements to in-context predictions. The full learning of the in-context mechanism takes longer, likely due to slower dynamics of the key-query matrices. We also observe a tension between \\(W_{O}^{2}\\) and \\(W_{F}\\) later in training, leading to slight degradations of our probe metrics. This may be due to the fact that the input to \\(W_{F}\\) now contains additional signal from the induction head which may be leveraged for better predictions, in particular for disambiguation in the case of random triggers, so that our guess of memories in Section 4.2 may no longer be accurate.\n\nRole of the data distribution.We can see in Figure 4(left) that changes to the data distribution can have a significant effect on the speed of learning the in-context mechanism. We observe that the following may slow down in-context learning: (i) a smaller number of triggers \\(K\\), (ii) using only rare fixed triggers, and (iii) using random triggers instead of fixed triggers. By inspecting the individual memory probes (see Figure 5 in Appendix E), we hypothesize that (i) and (ii) are due to slow learning of \\(W_{O}^{2}\\), while (iii) is more related to slow learning of key-query matrices. This is reasonable since (i-ii) reduce the number of overall output tokens in the data, while (iii) increases the number of possible trigger tokens that should be stored in \\(W_{K}^{2}\\), thus increasing the data requirements in order to learn the full associative memory. We also show in Figure 4(center) that changing the output token distribution to bigram distributions at training time reduces the in-context accuracy\n\nFigure 3: **Learning the induction head alone: in-context accuracy (top) and recall probes (bottom) with some layers frozen until iteration 300. The output matrix \\(W_{O}^{2}\\) can and must be learned before the key-query matrices, but does not suffice for good accuracy. It is easier to learn \\(W_{K}^{2}\\) before \\(W_{K}^{1}\\), and \\(W_{K}^{1}\\) stores initial context positions (\\(t<64\\)) much faster than late positions.**\n\nwhen using out-of-distribution output tokens, while training on uniform outputs performs well on both distributions. This highlights that using a more diverse training distribution can lead to models with better generalization accuracy, with little additional training cost.\n\nAdditional experiments.In Appendix E, we provide additional experimental results for varying dimensionality, more complex architectures and training methods, as well as more fine-grained visualizations of the memory associations.\n\n## 6 Theoretical Insights on Learning Dynamics\n\nIn this section, we provide theoretical insights on how gradients near initialization may allow the emergence of induction heads, and how this behavior is affected by data-distributional properties.\n\nFinding signal in noisy inputs.In Lemma 1, we showed how gradient dynamics on a simple classification task with fixed embeddings of the inputs and outputs lead to associative memories. We now show that when inputs consist of superpositions of multiple embeddings, as is the case in the transformer residual streams, gradients may learn associative memories that filter out irrelevant components of these superpositions, focusing on useful signal instead.\n\n**Lemma 2** (Gradient associative memory with noisy inputs).: _Let \\(p\\) be a data distribution on \\((x,y)\\in\\mathbb{R}^{d}\\times[N]\\), and consider the following classification problem, with fixed output embeddings \\(W_{U}\\):_\n\n\\[L(W)=\\mathbb{E}_{(x,y)\\sim p}[\\ell(y,W_{U}Wx)].\\]\n\n_The gradients take the following form: denoting \\(\\mu_{k}:=\\mathbb{E}[x|y=k]\\) and \\(\\hat{\\mu}_{k}:=\\mathbb{E}_{x}[\\frac{\\hat{p}w(k|x)}{p(y=k)}x]\\),_\n\n\\[\\nabla_{W}L(W)=\\sum_{k=1}^{N}p(y=k)w_{U}(k)(\\hat{\\mu}_{k}-\\mu_{k})^{\\top}.\\]\n\nThe key takeaway from this lemma is that with enough data (here infinite data), the associative memory arising from gradients can learn to filter out noise from inputs, since it only depends on its expectations or conditional expectations. In particular, \\(\\mu_{k}\\) can isolate relevant parts of \\(x\\) that are predictive of a label \\(k\\), and thus can lead to the right associations.\n\nAn illustrative example.To gain more intuition about this result, consider the following example: we would like to predict \\(y\\) from \\(x=w_{E}(y)+p_{t}\\), where \\(p_{t}\\) is a positional embedding at a random position \\(t\\in[T]\\), which we would like to ignore. Further assume that \\(y\\) is uniformly distributed with \\(p(y=k)=1/N\\), and consider the matrix obtained after one population gradient step with step-size \\(\\eta\\) starting from an initialization \\(W_{0}=0\\) (so that \\(\\hat{p}_{W_{0}}(k|x)=1/N\\)):\n\n\\[W_{1}=\\frac{\\eta}{N}\\sum_{k=1}^{N}w_{U}(k)(\\mu_{k}-\\bar{\\mu})^{\\top},\\]\n\nFigure 4: **Global vs in-context learning and data-distributional effects. (left) Loss on global (dashed) vs in-context (solid) tokens throughout training, for fixed or random trigger tokens \\(q_{k}\\). The red curves fixes the trigger \\(q_{1}\\) to the most frequent token, while the fixed triggers in blue curves are less common. (center) In-context accuracy with different training and test distributions \\(\\pi_{o}\\) for output tokens. Uniform leads to better generalization than global bigrams \\(\\pi_{b}\\). (right) Probe metrics throughout training: \\(W_{O}^{2}\\) and \\(W_{F}\\) eventually compete and deviate from our natural estimates.**\n\nwith \\(\\bar{\\mu}=\\mathbb{E}[x]\\). We show in Appendix B that when \\(d\\) is large enough to ensure near-orthonormal embeddings, we have\n\n\\[w_{U}(k)^{\\top}W_{1}(w_{E}(y)+p_{t})\\approx\\frac{\\eta}{N}\\;\\mathbb{1}\\{k=y\\}+O \\left(\\frac{1}{N^{2}}\\right),\\]\n\nso that for large enough \\(N\\) and \\(T\\), we obtain a near-perfect classifier that ignores the positional embedding, after just one gradient step (but a highly idealized one). Understanding how this translates to the finite dimension and finite sample regime is an important theoretical question that we leave for future work (see [7] for an initial step in that direction). We note that data models related to the above have been useful to study gradient dynamics of neural networks on continuous data [2, 25, 28]. Using a single gradient step to learn representations has also been fruitful in other contexts [3, 12].\n\nLearning the induction head with gradients.We may extend the arguments above to show how a few gradient steps can learn the induction head mechanism. We show the following in Appendix B.3.\n\n**Theorem 3** (Learning induction head via three gradient steps, informal).: _In a simplified setup, the induction head mechanism as constructed in (7) can be learned via sequential gradient steps on the population loss from random initialization, on \\(W^{2}_{O}\\), then \\(W^{2}_{K}\\), followed by \\(W^{1}_{K}\\)._\n\nTo show this result, we use Lemma 2 in a similar manner to the illustrative example above to show how training \\(W^{2}_{O}\\) by itself at initialization, _i.e._, when the attention patterns are near-uniform, can recover the desired associative memory. This is possible because when predicting an output token at later occurrences of a trigger, the same output token is guaranteed to be present in the context, while other tokens need not appear more relative to other sequences. See also Figure 9 in Appendix E for numerical experiments verifying this for finite data and dimension. Once \\(W^{2}_{O}\\) has learned the correct associations, we show that the gradient with respect to the key-value matrix \\(W^{2}_{K}\\) at zero initialization can leverage the correctness of \\(W^{2}_{O}\\) to find the right associative memory that focuses the attention on correct triggers. Finally, by linearizing the second-layer attention around \\(W^{2}_{K}=0\\), we show how gradients w.r.t. \\(W^{1}_{K}\\) may learn correct associations for the previous token head.\n\n## 7 Discussion\n\nIn this paper, we studied the question of how transformers develop in-context learning abilities, using a simplified setup that allows a fine-grained understanding the model and its training dynamics. While our model already captures rich phenomena at play in the bigram task we consider, more elaborate models are likely needed to understand transformers trained on more complex tasks like language modeling. This includes learning embeddings that are more adapted to the data and more structured (_e.g._, word embeddings [37, 31], or grokking [33, 39]), factorized key-query and value-output matrices that may induce additional regularization effects [20], and non-linear feedforward layers, which may provide richer associative memories between sets of embeddings. Understanding how transformers leverage such aspects to learn in richer settings is an important next step.\n\n## Acknowledgments and Disclosure of Funding\n\nThe authors thank Sainbayar Sukhbaatar and Shubham Toshniwal for helpful discussions.\n\n## References\n\n* [1] Z. Allen-Zhu and Y. Li. Backward feature correction: How deep learning performs deep learning. _arXiv preprint arXiv:2001.04413_, 2020.\n* [2] Z. Allen-Zhu and Y. Li. Towards understanding ensemble, knowledge distillation and self-distillation in deep learning. In _Proceedings of the International Conference on Learning Representations (ICLR)_, 2023.\n* [3] J. Ba, M. A. Erdogdu, T. Suzuki, Z. Wang, D. Wu, and G. Yang. High-dimensional asymptotics of feature learning: How one gradient step improves the representation. _Advances in Neural Information Processing Systems (NeurIPS)_, 2022.\n\n* [4] J. L. Ba, J. R. Kiros, and G. E. Hinton. Layer normalization. _arXiv preprint arXiv:1607.06450_, 2016.\n* [5] T. Bricken and C. Pehlevan. Attention approximates sparse distributed memory. _Advances in Neural Information Processing Systems (NeurIPS)_, 2021.\n* [6] T. Brown, B. Mann, N. Ryder, M. Subbiah, J. D. Kaplan, P. Dhariwal, A. Neelakantan, P. Shyam, G. Sastry, A. Askell, et al. Language models are few-shot learners. In _Advances in Neural Information Processing Systems (NeurIPS)_, 2020.\n* [7] V. Cabannes, E. Dohmatob, and A. Bietti. Scaling laws for associative memories. _arXiv preprint arXiv:2310.02984_, 2023.\n* [8] S. Chan, A. Santoro, A. Lampinen, J. Wang, A. Singh, P. Richemond, J. McClelland, and F. Hill. Data distributional properties drive emergent in-context learning in transformers. In _Advances in Neural Information Processing Systems (NeurIPS)_, 2022.\n* [9] F. Charton. What is my math transformer doing?-three results on interpretability and generalization. _arXiv preprint arXiv:2211.00170_, 2022.\n* [10] L. Chizat, E. Oyallon, and F. Bach. On lazy training in differentiable programming. _Advances in Neural Information Processing Systems (NeurIPS)_, 2019.\n* [11] A. Chowdhery, S. Narang, J. Devlin, M. Bosma, G. Mishra, A. Roberts, P. Barham, H. W. Chung, C. Sutton, S. Gehrmann, et al. Palm: Scaling language modeling with pathways. _arXiv preprint arXiv:2204.02311_, 2022.\n* [12] A. Damian, J. Lee, and M. Soltanolkotabi. Neural networks can learn representations with gradient descent. In _Conference on Learning Theory (COLT)_, 2022.\n* [13] G. Dar, M. Geva, A. Gupta, and J. Berant. Analyzing transformers in embedding space. _arXiv preprint arXiv:2209.02535_, 2022.\n* [14] B. L. Edelman, S. Goel, S. Kakade, and C. Zhang. Inductive biases and variable creation in self-attention mechanisms. In _Proceedings of the International Conference on Machine Learning (ICML)_, 2022.\n* [15] N. Elhage, T. Hume, C. Olsson, N. Schiefer, T. Henighan, S. Kravec, Z. Hatfield-Dodds, R. Lasenby, D. Drain, C. Chen, R. Grosse, S. McCandlish, J. Kaplan, D. Amodei, M. Wattenberg, and C. Olah. Toy models of superposition. _Transformer Circuits Thread_, 2022.\n* [16] N. Elhage, N. Nanda, C. Olsson, T. Henighan, N. Joseph, B. Mann, A. Askell, Y. Bai, A. Chen, T. Conerly, N. DasSarma, D. Drain, D. Ganguli, Z. Hatfield-Dodds, D. Hernandez, A. Jones, J. Kernion, L. Lovitt, K. Ndousse, D. Amodei, T. Brown, J. Clark, J. Kaplan, S. McCandlish, and C. Olah. A mathematical framework for transformer circuits. _Transformer Circuits Thread_, 2021.\n* [17] M. Geva, J. Bastings, K. Filippova, and A. Globerson. Dissecting recall of factual associations in auto-regressive language models. _arXiv preprint arXiv:2304.14767_, 2023.\n* [18] M. Geva, R. Schuster, J. Berant, and O. Levy. Transformer feed-forward layers are key-value memories. In _Conference on Empirical Methods in Natural Language Processing (EMNLP)_, 2021.\n* [19] A. Graves, G. Wayne, and I. Danihelka. Neural turing machines. _arXiv preprint arXiv:1410.5401_, 2014.\n* [20] S. Gunasekar, B. E. Woodworth, S. Bhojanapalli, B. Neyshabur, and N. Srebro. Implicit regularization in matrix factorization. _Advances in Neural Information Processing Systems (NIPS)_, 2017.\n* [21] J. J. Hopfield. Neural networks and physical systems with emergent collective computational abilities. _Proceedings of the national academy of sciences_, 79(8):2554-2558, 1982.\n\n* [22] J. Hron, Y. Bahri, J. Sohl-Dickstein, and R. Novak. Infinite attention: Nngp and ntk for deep attention networks. In _Proceedings of the International Conference on Machine Learning (ICML)_, 2020.\n* [23] A. Iscen, T. Furon, V. Gripon, M. Rabbat, and H. Jegou. Memory vectors for similarity search in high-dimensional spaces. _IEEE transactions on big data_, 4(1):65-77, 2017.\n* [24] A. Jacot, F. Gabriel, and C. Hongler. Neural tangent kernel: Convergence and generalization in neural networks. _Advances in Neural Information Processing Systems (NeurIPS)_, 2018.\n* [25] S. Jelassi, M. Sander, and Y. Li. Vision transformers provably learn spatial structure. In _Advances in Neural Information Processing Systems (NeurIPS)_, 2022.\n* [26] Y. Jiang and C. Pehlevan. Associative memory in iterated overparameterized sigmoid autoencoders. In _Proceedings of the International Conference on Machine Learning (ICML)_, 2020.\n* [27] A. Joulin and T. Mikolov. Inferring algorithmic patterns with stack-augmented recurrent nets. _Advances in Neural Information Processing Systems (NIPS)_, 2015.\n* [28] S. Karp, E. Winston, Y. Li, and A. Singh. Local signal adaptivity: Provable feature learning in neural networks beyond kernels. _Advances in Neural Information Processing Systems (NeurIPS)_, 2021.\n* [29] T. Kohonen. Correlation matrix memories. _IEEE Transactions on Computers_, 1972.\n* [30] D. Krotov and J. J. Hopfield. Dense associative memory for pattern recognition. _Advances in Neural Information Processing Systems (NIPS)_, 2016.\n* [31] Y. Li, Y. Li, and A. Risteski. How do transformers learn topic structure: Towards a mechanistic understanding. In _Proceedings of the International Conference on Machine Learning (ICML)_, 2023.\n* [32] B. Liu, J. T. Ash, S. Goel, A. Krishnamurthy, and C. Zhang. Transformers learn shortcuts to automata. In _Proceedings of the International Conference on Learning Representations (ICLR)_, 2023.\n* [33] Z. Liu, O. Kitouni, N. S. Nolte, E. Michaud, M. Tegmark, and M. Williams. Towards understanding grokking: An effective theory of representation learning. In _Advances in Neural Information Processing Systems (NeurIPS)_, 2022.\n* [34] R. McEliece, E. Posner, E. Rodemich, and S. Venkatesh. The capacity of the hopfield associative memory. _IEEE transactions on Information Theory_, 33(4):461-482, 1987.\n* [35] K. Meng, D. Bau, A. Andonian, and Y. Belinkov. Locating and editing factual associations in gpt. In _Advances in Neural Information Processing Systems (NeurIPS)_, 2022.\n* [36] W. Merrill, A. Sabharwal, and N. A. Smith. Saturated transformers are constant-depth threshold circuits. _Transactions of the Association for Computational Linguistics_, 10:843-856, 2022.\n* [37] T. Mikolov, K. Chen, G. Corrado, and J. Dean. Efficient estimation of word representations in vector space. In _Proceedings of the International Conference on Learning Representations (ICLR)_, 2013.\n* [38] S. Min, X. Lyu, A. Holtzman, M. Artetxe, M. Lewis, H. Hajishirzi, and L. Zettlemoyer. Rethinking the role of demonstrations: What makes in-context learning work? In _Conference on Empirical Methods in Natural Language Processing (EMNLP)_, 2022.\n* [39] N. Nanda, L. Chan, T. Liberum, J. Smith, and J. Steinhardt. Progress measures for grokking via mechanistic interpretability. In _Proceedings of the International Conference on Learning Representations (ICLR)_, 2023.\n\n* [40] C. Olsson, N. Elhage, N. Nanda, N. Joseph, N. DasSarma, T. Henighan, B. Mann, A. Askell, Y. Bai, A. Chen, T. Conerly, D. Drain, D. Ganguli, Z. Hatfield-Dodds, D. Hernandez, S. Johnston, A. Jones, J. Kernion, L. Lovitt, K. Ndousse, D. Amodei, T. Brown, J. Clark, J. Kaplan, S. McCandlish, and C. Olah. In-context learning and induction heads. _Transformer Circuits Thread_, 2022.\n* [41] A. Radford, J. Wu, R. Child, D. Luan, D. Amodei, I. Sutskever, et al. Language models are unsupervised multitask learners. _Technical report, OpenAI_, 2019.\n* [42] J. W. Rae, S. Borgeaud, T. Cai, K. Millican, J. Hoffmann, F. Song, J. Aslanides, S. Henderson, R. Ring, S. Young, et al. Scaling language models: Methods, analysis & insights from training gopher. _arXiv preprint arXiv:2112.11446_, 2021.\n* [43] Y. Razeghi, R. L. Logan IV, M. Gardner, and S. Singh. Impact of pretraining term frequencies on few-shot reasoning. In _Conference on Empirical Methods in Natural Language Processing (EMNLP)_, 2022.\n* [44] C. Sanford, D. Hsu, and M. Telgarsky. Representational strengths and limitations of transformers. In _Advances in Neural Information Processing Systems (NeurIPS)_, 2023.\n* [45] A. M. Saxe, J. L. McClelland, and S. Ganguli. Exact solutions to the nonlinear dynamics of learning in deep linear neural networks. _arXiv preprint arXiv:1312.6120_, 2013.\n* [46] I. Schlag, K. Irie, and J. Schmidhuber. Linear transformers are secretly fast weight programmers. In _Proceedings of the International Conference on Machine Learning (ICML)_, 2021.\n* [47] J. Schmidhuber. Learning to control fast-weight memories: An alternative to dynamic recurrent networks. _Neural Computation_, 4(1):131-139, 1992.\n* [48] S. Shin, S.-W. Lee, H. Ahn, S. Kim, H. Kim, B. Kim, K. Cho, G. Lee, W. Park, J.-W. Ha, et al. On the effect of pretraining corpora on in-context learning by a large-scale language model. In _North American Chapter of the Association for Computational Linguistics (NAACL)_, 2022.\n* [49] C. Snell, R. Zhong, D. Klein, and J. Steinhardt. Approximating how single head attention learns. _arXiv preprint arXiv:2103.07601_, 2021.\n* [50] S. Sukhbaatar, A. Szlam, J. Weston, and R. Fergus. End-to-end memory networks. _Advances in Neural Information Processing Systems (NIPS)_, 2015.\n* [51] Y. Tian, Y. Wang, B. Chen, and S. Du. Scan and snap: Understanding training dynamics and token composition in 1-layer transformer. In _Advances in Neural Information Processing Systems (NeurIPS)_, 2023.\n* [52] A. Vaswani, N. Shazeer, N. Parmar, J. Uszkoreit, L. Jones, A. N. Gomez, L. Kaiser, and I. Polosukhin. Attention is all you need. In _Advances in Neural Information Processing Systems (NIPS)_, 2017.\n* [53] R. Vershynin. _High-dimensional probability: An introduction with applications in data science_, volume 47. Cambridge university press, 2018.\n* [54] K. Wang, A. Variengien, A. Conmy, B. Shlegeris, and J. Steinhardt. Interpretability in the wild: a circuit for indirect object identification in gpt-2 small. In _Proceedings of the International Conference on Learning Representations (ICLR)_, 2023.\n* [55] J. Weston, S. Chopra, and A. Bordes. Memory networks. In _Proceedings of the International Conference on Learning Representations (ICLR)_, 2015.\n* [56] D. J. Willshaw, O. P. Buneman, and H. C. Longuet-Higgins. Non-holographic associative memory. _Nature_, 222(5197):960-962, 1969.\n* [57] S. M. Xie, A. Raghunathan, P. Liang, and T. Ma. An explanation of in-context learning as implicit bayesian inference. In _Proceedings of the International Conference on Learning Representations (ICLR)_, 2022.\n\n* [58] G. Yang and E. J. Hu. Tensor programs iv: Feature learning in infinite-width neural networks. In _Proceedings of the International Conference on Machine Learning (ICML)_, 2021.\n* [59] G. Yang, E. J. Hu, I. Babuschkin, S. Sidor, X. Liu, D. Farhi, N. Ryder, J. Pachocki, W. Chen, and J. Gao. Tensor programs v: Tuning large neural networks via zero-shot hyperparameter transfer. In _Advances in Neural Information Processing Systems (NeurIPS)_, 2021.\n* [60] B. Zhang and R. Sennrich. Root mean square layer normalization. In _Advances in Neural Information Processing Systems (NeurIPS)_, 2019.\n* [61] Y. Zhang, A. Backurs, S. Bubeck, R. Eldan, S. Gunasekar, and T. Wagner. Unveiling transformers with lego: a synthetic reasoning task. _arXiv preprint arXiv:2206.04301_, 2022.\n\nAssociative Memories with Random Vectors\n\nIn this section, we provide basic properties of associative memories based on outer products of random Gaussian embeddings, as described in Section 4.\n\nWe consider embeddings \\(u_{k}\\in\\mathbb{R}^{d}\\) with i.i.d. Gaussian \\(\\mathcal{N}(0,\\frac{1}{d})\\) entries.\n\nWe recall a few facts:\n\n* (_Norm_) We have \\(u_{i}^{\\top}u_{i}\\approx 1\\). This is standard from the concentration of random vectors in high dimension (see, _e.g._, [53, Theorem 3.1.1]).\n* (_Near-orthogonality_) For \\(i\\neq j\\), we have \\(u_{i}^{\\top}u_{j}=O(1/\\sqrt{d})\\). To see this, denoting \\(u_{i}=d^{-1/2}(\\tilde{u}_{ik})_{k}\\), where \\(\\tilde{u}_{ik}\\) are the unnormalized entries of \\(u_{i}\\), note that we have \\[\\sqrt{d}u_{i}^{\\top}u_{j}=\\frac{1}{\\sqrt{d}}\\sum_{k=1}^{d}\\tilde{u}_{ik} \\tilde{u}_{jk}\\to\\mathcal{N}(0,1),\\] by the central limit theorem, since for each \\(k\\), the quantities \\(\\tilde{u}_{ik}\\tilde{u}_{jk}\\) are zero-mean, unit-variance, i.i.d. random variables.\n* (_Remapping: norm_) If \\(W\\) is a Gaussian random matrix with i.i.d. \\(\\mathcal{N}(0,\\frac{1}{d})\\) entries, then for any fixed \\(x\\) we have \\(\\|Wx\\|\\approx\\|x\\|\\). This follows from Johnson-Lindenstrauss (see, _e.g._, [53, Lemma 5.3.2 and Exercise 5.3.3]). In particular, if \\(x\\) is a normalized Gaussian embedding as above, then \\(\\|Wx\\|\\approx 1\\).\n* (_Remapping: near-orthogonality_) Consider a random vector \\(x=\\frac{1}{\\sqrt{d}}\\tilde{x}\\) and a random matrix \\(W=\\frac{1}{\\sqrt{d}}\\tilde{W}\\), where the entries of \\(\\tilde{x}\\) and \\(\\tilde{W}\\) are i.i.d. \\(\\mathcal{N}(0,1)\\). Then \\(x\\) and \\(Wx\\) are nearly orthogonal. To see this, note that \\(\\mathbb{E}[x^{\\top}Wx]=\\mathbb{E}[x^{\\top}\\,\\mathbb{E}[W]x]=0\\), and the variance is \\[\\mathbb{E}(x^{\\top}Wx)^{2} =\\mathbb{E}\\sum_{i,j}x_{i}^{2}W_{ij}^{2}x_{j}^{2}=\\frac{1}{d^{3}} \\,\\mathbb{E}\\sum_{i}\\tilde{x}_{i}^{4}\\tilde{W}_{ii}^{2}+\\mathbb{E}\\sum_{i\\neq j }\\tilde{x}_{i}^{2}\\tilde{x}_{j}^{2}\\tilde{W}_{ij}^{2}\\] \\[=\\frac{1}{d^{3}}\\left(dM_{4}M_{2}+\\frac{d(d-1)}{2}M_{2}^{3} \\right)=O\\left(\\frac{1}{d}\\right),\\] where \\(M_{2}\\) and \\(M_{4}\\) denote the second and fourth moments of the standard Gaussian, respectively. Then, Chebyshev's inequality implies that \\(|x^{\\top}Wx|=O(1/\\sqrt{d})\\) with high probability.\n\nEnsuring appropriate memory lookups then requires such properties to hold for many embeddings and pairs of embeddings, with errors that are small enough to ensure correct associations. This may be achieved with careful union bounds or more powerful concentration results. We do not attempt to do this in a precise manner in this paper, and will generally assume \\(d\\) large enough to satisfy the desired associative memory behaviors, noting that a precise analysis is an important direction for future work (see [7] for a first step in this direction).\n\n## Appendix B Theoretical Insights on Gradient Dynamics\n\nIn this section, we provide additional details on the theoretical insights from Section 6, including details on the illustrative example (Section B.1), derivations of gradients w.r.t. key-query matrices at initialization (Section B.2), as well as a study of how the induction head mechanism may develop in a simplified setup, using a sequence of single layer-wise gradient steps in a top-down manner (Section B.3).\n\n### Details on illustrative example\n\nConsider the example discussed in Section 6: we would like to predict \\(y\\) from \\(x=w_{E}(y)+p_{t}\\), where \\(p_{t}\\) is a positional embedding at a random position \\(t\\in[T]\\), which we would like to ignore. Further assume that \\(y\\) is uniformly distributed (\\(p(y=k)=1/N\\)) and consider the matrix obtained after one population gradient step with step-size \\(\\eta\\) starting from an initialization \\(W_{0}=0\\) (sothat \\(\\hat{p}_{W_{0}}(k|x)=1/N\\)):\n\n\\[W_{1}=\\frac{\\eta}{N}\\sum_{k=1}^{N}w_{U}(k)(\\mu_{k}-\\bar{\\mu})^{\\top}, \\tag{11}\\]\n\nwith \\(\\bar{\\mu}=\\mathbb{E}[x]\\).\n\nNote that we have \\(\\mu_{k}=w_{E}(k)+\\frac{1}{T}\\sum_{t}p_{t}\\) and \\(\\bar{\\mu}=\\frac{1}{N}\\sum_{k}w_{E}(k)+\\frac{1}{T}\\sum_{t}p_{t}\\), so that (11) becomes\n\n\\[W_{1}=\\frac{\\eta}{N}\\sum_{k=1}^{N}w_{U}(k)(w_{E}(k)-\\bar{w}_{E})^{\\top}, \\tag{12}\\]\n\nwith \\(\\bar{w}_{E}:=\\frac{1}{N}\\sum_{k=1}^{N}w_{E}(k)\\). When \\(d\\) is large enough to ensure near-orthonormal embeddings, we have for any \\(y\\) and \\(t\\),\n\n\\[W_{1}(w_{E}(y)+p_{t})\\approx\\frac{\\eta}{N}w_{U}(y)+O\\left(\\frac{1}{N^{2}} \\right).\\]\n\nThis implies\n\n\\[w_{U}(k)^{\\top}W_{1}(w_{E}(y)+p_{t})\\approx\\frac{\\eta}{N}\\,\\mathbb{1}\\left\\{k= y\\right\\}+O\\left(\\frac{1}{N^{2}}\\right),\\]\n\nas claimed in the main text. The classifier \\(\\check{y}=\\arg\\max_{k}w_{U}(k)^{\\top}W_{1}(w_{E}(y)+p_{t})\\) then has essentially perfect accuracy, and has learned to ignore the spurious positional embeddings, which are simply exogenous noise.\n\n### Gradients on key-query matrices at initialization\n\nWe now derive expressions for population gradients of the attention key-query matrices at zero initialization, noting that random initialization behaves similarly to zero initialization. Although the optimization problems involving these matrices are non-convex, these gradients at initialization lead to associative memories, similar to Lemma 2. When output matrices of the previous layer already encode the desired associations, these gradients can lead to associative memories that focus the attention on the correct key-value pairs.\n\nWe begin with the following lemma, which gives the gradient of the loss w.r.t. \\(W=W_{K}^{2}\\) at zero initialization. For simplicity, we drop the \\(d^{-1/2}\\) factor from the softmax, which only changes gradients by a multiplicative factor, and thus does not change its form.\n\n**Lemma 4** (Gradient of second attention layer).: _Consider the following loss for predicting the next token \\(y\\) from an attention layer with inputs \\(X=[x_{1},\\ldots,x_{T}]\\), and value-output matrix \\(\\Phi_{2}:=W_{O}^{2}W_{V}^{2}\\):_\n\n\\[L(W)=\\mathbb{E}_{(X,y)}[\\ell(y,\\xi(X))],\\qquad\\xi(X)=W_{U}\\Phi_{2}X\\sigma(X^{ \\top}Wx_{T}), \\tag{13}\\]\n\n_with \\(\\ell\\) the cross-entropy loss and \\(\\sigma(u)_{t}=\\frac{e^{u_{t}}}{\\sum_{e}e^{u_{s}}}\\) for \\(u\\in\\mathbb{R}^{T}\\) is the softmax._\n\n_The gradient at \\(W=0\\) is given by_\n\n\\[\\nabla_{W}L(W)\\big{|}_{W=0} =\\sum_{k=1}^{N}\\mathbb{E}_{(X,y)}\\left[(\\hat{p}_{W}(k|X)-\\mathbb{1 }\\{y\\!=\\!k\\})\\frac{1}{T}\\sum_{t=1}^{T}w_{U}(k)^{\\top}\\Phi_{2}x_{t}\\cdot(x_{t} -\\bar{x}_{1:T})x_{T}^{\\top}\\right]\\] \\[=\\frac{1}{T}\\sum_{k=1}^{N}\\sum_{t=1}^{T}\\mathbb{E}_{X}[\\hat{p}_{W }(k|X)w_{U}(k)^{\\top}\\Phi_{2}x_{t}\\cdot(x_{t}-\\bar{x}_{1:T})x_{T}^{\\top}]\\] \\[\\quad-\\frac{1}{T}\\sum_{k=1}^{N}\\sum_{t=1}^{T}p(y=k)\\,\\mathbb{E}_{ X}[w_{U}(k)^{\\top}\\Phi_{2}x_{t}\\cdot(x_{t}-\\bar{x}_{1:T})x_{T}^{\\top}\\mid y=k]\\]\n\n_with \\(\\bar{x}_{1:T}=\\frac{1}{T}\\sum_{t=1}^{T}x_{t}\\)._\n\nNow we consider the gradient w.r.t. \\(W=W_{K}^{1}\\) at zero initialization, and consier a simplification of the second layer attention to its linearization around \\(W_{K}^{2}=0\\). We will see that this still provides first-order information that is sufficient for \\(W_{K}^{1}\\) to be learned.\n\n**Lemma 5** (Gradient of first attention layer).: _Consider the following loss for predicting the next token \\(y\\) from a stack of two attention layers, with all parameters fixed except for \\(W=W_{K}^{1}\\), the key-query matrix at the first attention layer:_\n\n\\[L(W)=\\mathbb{E}_{(X,y)}[\\ell(y,\\xi(X))],\\qquad\\xi(X)=W_{U}\\Phi_{2}X\\bar{\\sigma} (Z(W)^{\\top}W_{2}x_{T}). \\tag{14}\\]\n\n_Here, \\(\\bar{\\sigma}(u_{1:T})_{t}=\\frac{1}{T}(1+u_{t}-\\frac{1}{T}\\sum_{s=1}^{T}u_{s})\\) is the linearization of the softmax around \\(0\\), and \\(Z(W)=[z_{1}(W),\\ldots,z_{T}(W)]\\) with_\n\n\\[z_{t}(W)=\\sum_{s=1}^{t}\\Phi_{1}x_{s}\\sigma(p_{1:t}^{\\top}Wp_{t})_{s},\\]\n\n_and \\(\\Phi_{\\ell}=W_{O}^{\\ell}W_{V}^{\\ell}\\) for \\(\\ell=1,2\\)._\n\n_The gradient at \\(W=0\\) is given by_\n\n\\[\\nabla_{W}L(W)\\big{|}_{W=0}\\] \\[\\quad=\\sum_{k=1}^{N}\\mathbb{E}_{X}\\left[\\hat{p}_{W}(k|X)\\frac{1}{ T}\\sum_{t=1}^{T}w_{U}(k)^{\\top}\\Phi_{2}x_{t}\\cdot\\frac{1}{t}\\sum_{s=1}^{t}( \\Phi_{1}x_{s})^{\\top}W_{2}x_{T}(p_{s}-\\bar{p}_{1:t})p_{t}^{\\top}\\right]\\] \\[\\quad\\quad-\\sum_{k=1}^{N}p(y=k)\\,\\mathbb{E}_{X}\\left[\\frac{1}{T} \\sum_{t=1}^{T}w_{U}(k)^{\\top}\\Phi_{2}x_{t}\\cdot\\frac{1}{t}\\sum_{s=1}^{t}(\\Phi_ {1}x_{s})^{\\top}W_{2}x_{T}(p_{s}-\\bar{p}_{1:t})p_{t}^{\\top}|y=k\\right]\\] \\[\\quad\\quad-\\sum_{k=1}^{N}\\mathbb{E}_{X}\\left[\\hat{p}_{W}(k|X)w_{U }(k)^{\\top}\\Phi_{2}\\bar{x}_{1:T}\\cdot\\frac{1}{T}\\sum_{t=1}^{T}\\frac{1}{t}\\sum_{ s=1}^{t}(\\Phi_{1}x_{s})^{\\top}W_{2}x_{T}(p_{s}-\\bar{p}_{1:t})p_{t}^{\\top}\\right]\\] \\[\\quad\\quad+\\sum_{k=1}^{N}p(y=k)\\,\\mathbb{E}_{X}\\left[w_{U}(k)^{ \\top}\\Phi_{2}\\bar{x}_{1:T}\\cdot\\frac{1}{T}\\sum_{t=1}^{T}\\frac{1}{t}\\sum_{s=1}^ {t}(\\Phi_{1}x_{s})^{\\top}W_{2}x_{T}(p_{s}-\\bar{p}_{1:t})p_{t}^{\\top}|y=k\\right]\\]\n\n### Learning the induction head mechanism\n\nIn this section, we analyze the training dynamics of the induction head mechanism, in the following simplified setup: we consider a single trigger (\\(K=1\\)), and assume that \\(\\pi_{u}\\), \\(\\pi_{q}\\), \\(\\pi_{o}\\) and \\(\\pi_{b}(\\cdot|i)\\) are uniform over \\([N]\\) for any \\(i\\).\n\nTo further simplify the analysis, we consider a loss that only considers sequences of length \\(T\\) where the last input \\(z_{T}\\) is the _second occurrence_ of the trigger token, and the label \\(y=z_{T+1}\\) is the corresponding output token. We note that this may be easily extended to later occurrences of the trigger. This is similar to the setup of Figure 1, where the loss is only taken on triggers after the second occurrence: in that case, the loss may be written as a weighted sum of the one we consider here, weighted by the probability of the second (or later) trigger appearing at the given position \\(T\\).\n\nIn practice, when the loss is on all tokens and \\(W_{F}\\) is also learned, we may expect that \\(W_{F}\\) quickly learns the global bigram statistics, as we saw empirically in Section 5. Indeed, the current token embedding, which is included in the input superposition, has strong predictive signal compared to the attention layers, which initially mainly appear as noise. This is then similar to the setup of Lemma 1, which provides recovery of bigram statistics when \\(d\\) is large (though we note that the other information from attention layers in the inputs may eventually be used and bias away from perfect recovery, see Figure 4(right)). Once such global estimates are obtained, the expected loss will be mainly dominated by trigger tokens, leading to the setup above.\n\nFor simplicity, we thus drop the feed-forward layer \\(W_{F}\\) in the remainder of this section, focusing on the learning of \\(W_{O}^{2}\\), \\(W_{K}^{2}\\) and \\(W_{K}^{1}\\), in this top-down order. We will consider zero-initialization a single gradient steps, noting that random initialization should lead to similar associative memory behaviors when the dimension is large enough, since it leads to a remapping of input embeddings which is near-orthogonal to any output embedding (see Appendix A).\n\n#### b.3.1 Learning \\(W_{O}^{2}\\)\n\nWe begin by studying the learning of the second output matrix \\(W_{O}^{2}\\). In the above data model, we may consider a loss as in Lemma 2 with input-outputs \\((x,y)\\), where \\(y\\) is the output token of the sequence,and \\(x\\) depends on the random sequence \\(z_{1:T}\\) as\n\n\\[x=\\frac{1}{T}\\sum_{t=1}^{T}W_{V}^{2}(w_{E}(z_{t})+\\varepsilon_{t}),\\]\n\nwhere \\(\\varepsilon_{t}=p_{t}+\\frac{1}{t}\\sum_{s=1}^{t}\\Phi_{1}(w_{E}(z_{s})+p_{s})\\) with \\(\\Phi_{1}=W_{O}^{1}W_{V}^{1}\\), is a \"noise\" vector from the residual streams, containing positional embeddings as well as an average attention output from the first layer. In practice, the logit predictions are of the form \\(W_{U}(W_{O}^{2}x+\\varepsilon_{T})\\) due to residual connections, but we ignore the term \\(W_{U}\\varepsilon_{T}\\) for simplicity, noting that it is near-zero when \\(d\\) is large.\n\nAfter a gradient step on \\(W_{O}^{2}\\) with step-size \\(\\eta\\), starting from zero-initialization (so that \\(\\hat{p}(k|x)=p(y=k)=1/N\\) for all \\(x\\)), Lemma 2 yields\n\n\\[W_{O}^{2}=\\frac{\\eta}{N}\\sum_{k=1}^{N}w_{U}(k)(\\mathbb{E}[x|y=k]-\\mathbb{E}[x] )^{\\top}. \\tag{15}\\]\n\nNow, consider the random variables \\(q\\) (trigger token), \\(o\\) (output token), \\(t_{o}\\) (position of the first occurrence of the output token). In our simplified data model, \\(q\\) and \\(t_{o}\\) have the same distribution regardless of the conditioning on \\(y=k\\), while \\(o\\) is equal to \\(k\\) when \\(y=k\\), while it is uniform in \\([N]\\) without this condition. The sequence \\(z_{1:T}\\) has the same distribution in either \\(p(\\cdot)\\) or \\(p(\\cdot|y=k)\\), except for the token \\(z_{t_{o}}\\).\n\nThen, we may write:\n\n\\[\\mathbb{E}[x|y=k]-\\mathbb{E}[x] =\\frac{1}{T}\\left(\\mathbb{E}[W_{V}^{2}w_{E}(z_{t_{o}})|y=k]- \\mathbb{E}[W_{V}^{2}w_{E}(z_{t_{o}})]\\right)\\] \\[\\quad+\\frac{1}{T}\\left(\\mathbb{E}\\left[\\sum_{t=t_{o}}^{T}W_{V}^{2 }\\varepsilon_{t}|y=k\\right]-\\mathbb{E}\\left[\\sum_{t=t_{o}}^{T}W_{V}^{2} \\varepsilon_{t}\\right]\\right),\\]\n\nsince \\(\\varepsilon_{t}\\) is independent of \\(o\\) when \\(t<t_{o}\\). Noting that \\(\\varepsilon_{t}\\) only depends on \\(z_{t_{o}}\\) and thus on \\(y\\) through the first layer attention, we have\n\n\\[\\mathbb{E}[x|y=k]-\\mathbb{E}[x] =\\frac{1}{T}W_{V}^{2}(w_{E}(k)-\\bar{w}_{E})\\] \\[\\quad+\\frac{1}{T}\\left(\\mathbb{E}\\left[\\sum_{t=t_{o}}^{T}\\frac{1} {t}W_{V}^{2}\\Phi_{1}w_{E}(z_{t_{o}})|y=k\\right]-\\mathbb{E}\\left[\\sum_{t=t_{o} }^{T}\\frac{1}{t}W_{V}^{2}\\Phi_{1}w_{E}(z_{t_{o}})\\right]\\right)\\] \\[=\\frac{1}{T}W_{V}^{2}(w_{E}(k)-\\bar{w}_{E})+\\frac{\\tau}{T}W_{V}^ {2}\\Phi_{1}(w_{E}(k)-\\bar{w}_{E}),\\]\n\nwhere \\(\\tau:=\\mathbb{E}\\left[\\sum_{t=t_{o}}^{T}\\frac{1}{t}\\right]\\), and \\(\\bar{w}_{E}=\\frac{1}{N}\\sum_{k=1}^{N}w_{E}(k)\\). Thus, (15) becomes\n\n\\[W_{O}^{2}=\\frac{\\eta}{NT}\\sum_{k=1}^{N}w_{U}(k)(W_{V}^{2}(w_{E}(k)-\\bar{w}_{E} ))^{\\top}+\\frac{\\eta\\tau}{NT}\\sum_{k=1}^{N}w_{U}(k)(W_{V}^{2}\\Phi_{1}(w_{E}(k )-\\bar{w}_{E}))^{\\top}, \\tag{16}\\]\n\nso that when \\(d\\) is learn enough to ensure near-orthonormal embeddings, we have\n\n\\[w_{U}(k)^{\\top}W_{O}^{2}W_{V}^{2}w_{E}(j) \\approx\\frac{\\eta}{NT}\\,1\\{k=j\\}+O\\left(\\frac{\\eta}{N^{2}T}\\right)\\] \\[w_{U}(k)^{\\top}W_{O}^{2}W_{V}^{2}\\Phi_{1}w_{E}(j) \\approx\\frac{\\eta\\tau}{NT}\\,1\\{k=j\\}+O\\left(\\frac{\\eta\\tau}{N^{2} T}\\right),\\]\n\nwhere the \\(O(\\cdot)\\) terms are due to the \\(\\bar{w}_{E}\\) elements. The first line yields a behavior that matches desired associative memory in (7) of Section 4.2 when \\(N\\) is large. The second line shows additional spurious associations that are stored in \\(W_{O}^{2}\\) due to the output of the first layer attention, but which may be \"cleaned up\" once the attention layers start focusing on the correct tokens.\n\nFinally, we note that despite the recovery of these useful associations after one gradient step, the predictions with this estimate \\(W_{O}^{2}\\) are still near-random, since in the bag-of-words setup with average attention, the output token cannot be distinguished from any other token in the sequence in our model (except perhaps the trigger token, which is guaranteed to appear twice, but does not provide any signal to infer the output token, since the two are independent).\n\n#### b.3.2 Learning \\(W^{2}_{k}\\)\n\nNow assume that \\(W^{2}_{O}\\) is as in (16). As argued above, the predictions \\(\\hat{p}(k|x)\\) are essentially random \\(1/N\\) in our model for \\(W^{2}_{K}=0\\), so that after one gradient step on \\(W^{2}_{K}\\) with learning rate \\(\\eta\\), Lemma 4 yields:\n\n\\[W^{2}_{K}=\\frac{\\eta}{TN}\\sum_{k,t}\\left(\\mathbb{E}[w_{U}(k)^{\\top}\\Phi_{2}x_{t }\\cdot(x_{t}-\\bar{x})x_{T}^{\\top}\\mid y=k]-\\mathbb{E}[w_{U}(k)^{\\top}\\Phi_{2}x _{t}\\cdot(x_{t}-\\bar{x})x_{T}^{\\top}]\\right), \\tag{17}\\]\n\nwhere \\(x_{t}\\) are the inputs to the second attention layer, given by\n\n\\[x_{t} =x_{t,0}+x_{t,1} \\tag{18}\\] \\[x_{t,0} =w_{E}(z_{t})+p_{t}\\] (19) \\[x_{t,1} =\\frac{1}{t}\\sum_{s=1}^{t}\\Phi_{1}(w_{E}(z_{s})+p_{s}). \\tag{20}\\]\n\nFrom now on, we consider a simplified architecture where only \\(x_{t,0}\\) are fed as queries and values, while only \\(x_{t,1}\\) are fed as keys. Using the fact that trigger tokens \\(q\\) are sampled uniformly (_i.e._, \\(\\pi_{q}=1/N\\)), we have\n\n\\[W^{2}_{K} =\\frac{\\eta}{TN}\\sum_{k=1}^{N}\\sum_{t=1}^{T}\\left(\\mathbb{E}[A_{t,k}\\mid y=k]-\\mathbb{E}_{X}[A_{t,k}]\\right) \\tag{21}\\] \\[=\\frac{\\eta}{TN^{2}}\\sum_{k=1}^{N}\\sum_{t=1}^{T}\\sum_{j=1}^{N} \\left(\\mathbb{E}[A_{t,k}\\mid y=k,q=j]-\\mathbb{E}[A_{t,k}\\mid q=j]\\right) \\tag{22}\\]\n\nwhere\n\n\\[A_{t,k}=w_{U}(k)^{\\top}\\Phi_{2}x_{t,0}\\cdot(x_{t,1}-\\bar{x}_{1})x_{T,0}^{\\top}, \\tag{23}\\]\n\nwith \\(\\bar{x}_{1}=\\frac{1}{T}\\sum_{t}x_{t,1}\\). Now, note that we have \\(w_{U}(k)^{\\top}\\Phi_{2}x_{t,0}\\approx\\alpha\\,\\mathbb{1}\\{z_{t}=k\\}\\) with \\(\\alpha=\\eta/TN\\) by (16), and \\(x_{T,0}=w_{E}(q)+p_{T}\\). This yields\n\n\\[W^{2}_{K}\\approx\\frac{\\alpha\\eta}{TN^{2}}\\sum_{j=1}^{N}\\sum_{k=1}^{N}\\Delta_{k,j}(w_{E}(j)+p_{T})^{\\top}, \\tag{24}\\]\n\nwith\n\n\\[\\Delta_{k,j} :=\\mathbb{E}\\left[\\sum_{t=1}^{T}\\mathbb{1}\\{z_{t}=k\\}(x_{t,1}- \\bar{x}_{1})|y=k,q=j\\right]-\\mathbb{E}\\left[\\sum_{t=1}^{T}\\mathbb{1}\\{z_{t}=k \\}(x_{t,1}-\\bar{x}_{1})|q=j\\right]\\] \\[=\\Delta^{o}_{k,j}+\\Delta^{q}_{k,j}+\\Delta^{r}_{k,j},\\]\n\nwhere the three terms split the sum inside the expectation\n\n\\[\\Delta^{o}_{k,j} :=\\mathbb{E}\\left[\\mathbb{1}\\{z_{t_{o}}=k\\}(x_{t_{o},1}-\\bar{x}_{ 1})|y=k,q=j\\right]-\\mathbb{E}\\left[\\mathbb{1}\\{z_{t_{o}}=k\\}(x_{t_{o},1}-\\bar{ x}_{1})|q=j\\right]\\] \\[\\Delta^{q}_{k,j} :=\\mathbb{E}\\left[\\sum_{t\\in\\mathcal{T}_{q}}\\mathbb{1}\\{z_{t}\\!= \\!k\\}(x_{t,1}-\\bar{x}_{1})|y\\!=\\!k,q\\!=\\!j\\right]-\\mathbb{E}\\left[\\sum_{t\\in \\mathcal{T}_{q}}\\mathbb{1}\\{z_{t}\\!=\\!k\\}(x_{t,1}-\\bar{x}_{1})|q\\!=\\!j\\right]\\] \\[\\Delta^{r}_{k,j} :=\\mathbb{E}\\left[\\sum_{t\\in\\mathcal{T}_{r}}\\mathbb{1}\\{z_{t}\\!= \\!k\\}(x_{t,1}-\\bar{x}_{1})|y\\!=\\!k,q\\!=\\!j\\right]-\\mathbb{E}\\left[\\sum_{t\\in \\mathcal{T}_{r}}\\mathbb{1}\\{z_{t}\\!=\\!k\\}(x_{t,1}-\\bar{x}_{1})|q\\!=\\!j\\right],\\]\n\nwhere \\(\\mathcal{T}_{q}=\\{t_{o}-1,T\\}\\) and \\(\\mathcal{T}_{r}=[T]\\setminus\\{t_{o},t_{o}-1,T\\}\\) (recall that \\(t_{o}\\) is a random variable, corresponding to the first occurrence of the output token, so that these sets are random).\n\nWe will now show that \\(\\Delta^{o}_{k,j}\\) carries the desired signal for the appropriate induction head associative memory, while \\(\\Delta^{q}_{k,j}\\) and \\(\\Delta^{r}_{k,j}\\) are negligible, for \\(N\\) large enough.\n\n**Controlling \\(\\Delta_{k,j}^{o}\\).** For \\(t=t_{o}\\), noting that \\(z_{t_{o}}=y\\), we have\n\n\\[\\Delta_{k,j}^{o} =\\mathbb{E}\\left[\\mathbb{1}\\left\\{y=k\\right\\}(x_{t_{o},1}-\\bar{x}_{ 1})|y=k,q=j\\right]-\\mathbb{E}\\left[\\mathbb{1}\\{y=k\\}(x_{t_{o},1}-\\bar{x}_{1})|q =j\\right]\\] \\[=\\left(1-\\frac{1}{N}\\right)\\mathbb{E}\\left[x_{t_{o},1}-\\bar{x}_{1 }|y=k,q=j\\right]\\] \\[=\\frac{N-1}{N}\\left(\\bar{p}+\\sum_{i=1}^{N}a_{k,j,i}\\Phi_{1}w_{E}(i )\\right),\\]\n\nwith \\(a_{k,j,i}\\approx\\left(\\Phi_{1}w_{E}(i)\\right)^{\\top}\\mathbb{E}\\left[x_{t_{o},1 }-\\bar{x}_{1}|y=k,q=j\\right]\\) thanks to near-orthonormality, and\n\n\\[\\bar{p}=\\mathbb{E}_{t_{o}}\\left[\\frac{1}{t_{o}}\\sum_{s=1}^{t_{o}}p_{s}-\\frac{ 1}{T}\\sum_{t=1}^{T}\\frac{1}{t}\\sum_{s=1}^{t}p_{s}\\right]\\]\n\nis a spurious positional mixture.\n\nWe then distinguish the following cases:\n\n* If \\(j\\neq k\\) and \\(i=j\\), since the trigger token \\(j\\) only appears at positions \\(t_{o}-1\\) and \\(T\\), we have \\[a_{k,j,i}\\approx\\mathbb{E}_{t_{o}}\\left[\\frac{1}{t_{o}}-\\frac{1}{T}\\sum_{t=t_{ o}-1}^{T}\\frac{1}{t}-\\frac{1}{T^{2}}\\right]=:\\gamma_{T}.\\] We may expect \\(t_{o}\\) to be concentrated around \\(T/2\\), in which case \\(\\gamma_{T}\\gtrsim\\frac{2}{T}-\\frac{1}{T^{2}}\\geq\\frac{C}{T}>0\\) for \\(T\\) larger than a small constant.\n* If \\(j=k=i\\), the two occurrences of the trigger happen one after the other, so it must be that \\(t_{o}=T\\). Then \\[a_{k,j,i}\\approx\\frac{2}{T}-\\frac{1}{T(T-1)}-\\frac{2}{T^{2}}=\\frac{2}{T}+O \\left(\\frac{1}{T^{2}}\\right),\\] for \\(T\\) larger than a small constant.\n* If \\(i\\neq j=k\\), all tokens up to position \\(t_{o}-2=T-2\\) are i.i.d. uniform in \\([N]\\setminus\\{j\\}\\), so that \\[a_{k,j,i}\\approx\\frac{T-2}{T(N-1)}-\\frac{1}{T}\\left((T-2)\\cdot\\frac{1}{N-1}+ \\frac{T-2}{(T-1)(N-1)}+\\frac{T-2}{T(N-1)}\\right)=O\\left(\\frac{1}{N}\\right)\\]\n* If \\(i\\neq j\\) and \\(j\\neq k\\), all tokens except at positions \\(t_{o}-1\\), \\(t_{o}\\) and \\(T\\) (we have \\(t_{o}<T\\)) are uniform in \\([N]\\setminus\\{j\\}\\). The triggers do not contribute anything to \\(a_{k,j,i}\\) since \\(i\\neq j\\), and the output token may be also randomized by taking the average over \\(k\\in[N]\\setminus\\{j\\}\\). We thus obtain \\[\\frac{1}{N-1}\\sum_{k\\neq j}a_{k,j,i}\\approx O\\left(\\frac{1}{N}\\right).\\]\n\nIn summary, we obtain\n\n\\[\\frac{1}{N}\\sum_{k=1}^{N}a_{k,j,i}\\approx\\begin{cases}O\\left(\\frac{1}{N} \\right),&\\text{ if }i\\neq j\\\\ \\Omega\\left(\\frac{1}{T}\\right),&\\text{ if }i=j.\\end{cases}\\]\n\nThus, when \\(N\\) is large, while \\(T\\) is moderate, the above sum leads to more signal in the \\(i=j\\) terms compared to \\(i\\neq j\\). In particular, this yields\n\n\\[(\\Phi_{1}w_{E}(i))^{\\top}\\left(\\frac{1}{N}\\sum_{k=1}^{N}\\Delta_{k,j}^{o} \\right)\\approx\\begin{cases}O\\left(\\frac{1}{N}\\right),&\\text{ if }i\\neq j\\\\ \\Omega\\left(\\frac{1}{T}\\right),&\\text{ if }i=j,\\end{cases}\\]\n\nso that this component in (24) acts precisely like the desired associative memory in (7).\n\nIt remains to show that the other components are negligible compared to this. It then suffices to show:\n\n\\[\\frac{1}{N}\\sum_{k=1}^{N}(\\Delta_{k,j}^{q}+\\Delta_{k,j}^{r})\\approx o\\left( \\frac{1}{T}\\right).\\]\n\n**Controlling \\(\\Delta^{q}_{k,j}\\).**  For \\(t\\in\\mathcal{T}_{q}\\), note that we always have \\(z_{t}=j\\) in the expectations, so that \\(\\Delta^{q}_{k,j}=0\\) unless \\(k=j\\). For \\(k=j\\), we have \\(\\Delta^{q}_{k,j}=O(1)\\), so that\n\n\\[\\frac{1}{N}\\sum_{k=1}^{N}\\Delta^{q}_{k,j}=O\\left(\\frac{1}{N}\\right).\\]\n\n**Controlling \\(\\Delta^{r}_{k,j}\\).**  Using that \\(\\|x_{t,1}-\\bar{x}_{1}\\|\\leq C=O(1)\\) for all \\(t\\), we provide the following crude bound via the triangle inequality and Holder inequality:\n\n\\[\\Delta^{r}_{k,j} =\\mathbb{E}\\left[\\sum_{t\\in\\mathcal{T}_{r}}\\mathbbm{1}\\{z_{t}\\!=\\! k\\}(x_{t,1}-\\bar{x}_{1})|y\\!=\\!k,q\\!=\\!j\\right]-\\mathbb{E}\\left[\\sum_{t\\in \\mathcal{T}_{r}}\\mathbbm{1}\\{z_{t}\\!=\\!k\\}(x_{t,1}-\\bar{x}_{1})|q\\!=\\!j\\right]\\] \\[\\|\\Delta^{r}_{k,j}\\| \\leq C\\left(\\mathbb{E}\\left[\\sum_{t\\in\\mathcal{T}_{r}}\\mathbbm{1 }\\{z_{t}\\!=\\!k\\}|y\\!=\\!k,q\\!=\\!j\\right]+\\mathbb{E}\\left[\\sum_{t\\in\\mathcal{T} _{r}}\\mathbbm{1}\\{z_{t}\\!=\\!k\\}|q\\!=\\!j\\right]\\right)\\leq\\frac{2CT}{N},\\]\n\nsince \\(z_{t}\\) is independent of \\(y\\) given \\(t\\in\\mathcal{T}_{r}\\) and thus is uniform in \\([N]\\setminus\\{j\\}\\), and \\(|\\mathcal{T}_{r}|\\leq T\\). We note, however, that \\(\\Delta^{r}_{k,j}\\) may be controlled much more finely by leveraging the similarities between the distributions of \\(z_{t},t\\in\\mathcal{T}_{r}\\) with or without conditioning on \\(y\\).\n\nOverall, we have shown that up to some spurious positional embeddings, \\(W^{2}_{K}\\) behaves as the desired associative memory from (7) when \\(N\\) is large enough, satisfying:\n\n\\[(\\Phi_{1}w_{E}(i))^{\\top}W^{2}_{K}w_{E}(j)\\approx\\frac{\\alpha\\eta}{TN}\\left\\{ \\Omega\\left(\\frac{1}{T}\\right)\\mathbbm{1}\\{i=j\\}+O\\left(\\frac{T}{N}\\right)\\right\\} \\tag{25}\\]\n\nWe note that one may then amplify the gap between correct and incorrect associations by having a large enough step-size, which then makes the softmax more peaked and hence the attention more sparse and focused on correct associations.\n\n#### b.3.3 Learning \\(W^{1}_{k}\\)\n\nWe now assume that \\(W^{2}_{O}\\) and \\(W^{2}_{K}\\) have learned the correct associations, and consider one gradient step away from zero-initialization on \\(W^{1}_{K}\\). Note that when \\(W^{1}_{K}=0\\), the predictions of the model are still often near random chance. Indeed, the second layer attention will attend to all tokens starting at the first occurrence of the trigger, since all such tokens contain \\(\\Phi_{1}w_{E}(q)\\) in their average attention, which activates the second-layer attention head. Then the output is likely to predict the trigger itself, which will be an incorrect prediction most of the time.\n\nWe may thus consider \\(\\hat{p}(k|X)=1/N\\) at this stage as well. We also consider a simplified architecture where the first layer attention only uses positional embeddings in the key-query matrix, and only token embeddings in the value-output matrix. In particular, we have \\(x_{t}=w_{E}(z_{t})\\). Lemma 5 then gives the following form for \\(W^{1}_{K}\\) after one gradient step of step-size \\(\\eta\\):\n\n\\[W^{1}_{K} =\\frac{\\eta}{N}\\sum_{k=1}^{N}\\mathbb{E}_{X}\\left[\\frac{1}{T}\\sum_ {t=1}^{T}w_{U}(k)^{\\top}\\Phi_{2}x_{t}\\cdot\\frac{1}{t}\\sum_{s=1}^{t}(\\Phi_{1}x _{s})^{\\top}W^{2}_{K}x_{T}(p_{s}-\\bar{p}_{1:t})p^{\\top}_{t}|y=k\\right]\\] \\[\\quad-\\frac{\\eta}{N}\\sum_{k=1}^{N}\\mathbb{E}_{X}\\left[\\frac{1}{T} \\sum_{t=1}^{T}w_{U}(k)^{\\top}\\Phi_{2}x_{t}\\cdot\\frac{1}{t}\\sum_{s=1}^{t}(\\Phi_ {1}x_{s})^{\\top}W^{2}_{K}x_{T}(p_{s}-\\bar{p}_{1:t})p^{\\top}_{t}\\right]\\] \\[\\quad-\\frac{\\eta}{N}\\sum_{k=1}^{N}\\mathbb{E}_{X}\\left[w_{U}(k)^{ \\top}\\Phi_{2}\\bar{x}_{1:T}\\cdot\\frac{1}{T}\\sum_{t=1}^{T}\\frac{1}{t}\\sum_{s=1} ^{t}(\\Phi_{1}x_{s})^{\\top}W^{2}_{K}x_{T}(p_{s}-\\bar{p}_{1:t})p^{\\top}_{t}|y=k\\right]\\] \\[\\quad+\\frac{\\eta}{N}\\sum_{k=1}^{N}\\mathbb{E}_{X}\\left[w_{U}(k)^{ \\top}\\Phi_{2}\\bar{x}_{1:T}\\cdot\\frac{1}{T}\\sum_{t=1}^{T}\\frac{1}{t}\\sum_{s=1} ^{t}(\\Phi_{1}x_{s})^{\\top}W^{2}_{K}x_{T}(p_{s}-\\bar{p}_{1:t})p^{\\top}_{t}\\right].\\]\n\nNote that since \\(W^{2}_{O}\\) and \\(W^{2}_{K}\\) already captured the desired associations at this stage, we have\n\n\\[w_{U}(k)^{\\top}\\Phi_{2}x_{t}\\approx\\alpha\\,\\mathbbm{1}\\{z_{t}=k\\}\\quad\\text{ and }\\quad(\\Phi_{1}x_{s})^{\\top}W^{2}_{K}x_{T}\\approx\\alpha^{\\prime}\\,\\mathbbm{1}\\,\\{z_{s}=z_{T}\\},\\]for some \\(\\alpha,\\alpha^{\\prime}>0\\). Recall that in our model, we have \\(z_{T}=q\\) with probability one (\\(q\\) is the trigger token), and that \\(q\\) only appears twice: once at position \\(t_{q}:=t_{o}-1<T\\) and once at position \\(T\\). We then have, for any \\(t>1\\),\n\n\\[W^{1}_{K}p_{t}\\approx\\frac{\\eta\\alpha\\alpha^{\\prime}}{NTt}\\sum_{k=1}^{N}(A_{t,k }-B_{t,k}-C_{t,k}+D_{t,k}),\\]\n\nwith\n\n\\[A_{t,k} =\\mathbb{E}[\\mathbbm{1}\\{z_{t}\\!=\\!k\\}\\;\\mathbbm{1}\\{t_{q}\\leq t \\}(p_{t_{q}}-\\bar{p}_{1:t})|y\\!=\\!k] \\tag{26}\\] \\[B_{t,k} =\\mathbb{E}[\\mathbbm{1}\\{z_{t}\\!=\\!k\\}\\;\\mathbbm{1}\\{t_{q}\\leq t \\}(p_{t_{q}}-\\bar{p}_{1:t})]\\] (27) \\[C_{t,k} =\\mathbb{E}[r_{k}\\;\\mathbbm{1}\\{t_{q}\\leq t\\}(p_{t_{q}}-\\bar{p}_ {1:t})|y\\!=\\!k]\\] (28) \\[D_{t,k} =\\mathbb{E}[r_{k}\\;\\mathbbm{1}\\{t_{q}\\leq t\\}(p_{t_{q}}-\\bar{p}_ {1:t})], \\tag{29}\\]\n\nwhere \\(r_{k}:=\\frac{1}{T}\\sum_{t=1}^{T}\\mathbbm{1}\\{z_{t}=k\\}\\). We have\n\n\\[A_{t,k} =\\mathbb{E}[\\mathbbm{1}\\{z_{t}\\!=\\!k\\}(\\mathbbm{1}\\{t_{q}=t-1\\}+ \\mathbbm{1}\\{t_{q}\\in[t-2]\\cup\\{t\\}\\})(p_{t_{q}}-\\bar{p}_{1:t})|y\\!=\\!k]\\] \\[=\\mathbb{P}(t_{q}=t-1|y=k)(p_{t-1}-\\bar{p}_{1:t})+\\frac{1}{N}\\sum _{s\\in[t-2]\\cup\\{t\\}}\\mathbb{P}(t_{q}=s|y=k)(p_{s}-\\bar{p}_{1:t})\\] \\[=\\mathbb{P}(t_{q}=t-1)(p_{t-1}-\\bar{p}_{1:t})+\\frac{1}{N}\\sum_{s \\in[t-2]\\cup\\{t\\}}\\mathbb{P}(t_{q}=s)(p_{s}-\\bar{p}_{1:t})\\] \\[=\\mathbb{P}(t_{q}=t-1)(p_{t-1}-\\bar{p}_{1:t})+O\\left(\\frac{1}{N} \\right),\\]\n\nsince the distribution of \\(t_{q}\\) is the same regardless of \\(y\\). We proceed similarly for the other quantities and obtain the following:\n\n\\[B_{t,k} =O\\left(\\frac{1}{N}\\right)\\] \\[C_{t,k} =\\frac{\\mathbb{P}(t_{q}=t-1)}{T}(p_{t-1}-\\bar{p}_{1:t})+O\\left( \\frac{1}{N}\\right)\\] \\[D_{t,k} =O\\left(\\frac{1}{N}\\right).\\]\n\nThis yields the following associative memory behavior, for \\(t>1\\):\n\n\\[p_{s}^{\\top}W^{1}_{K}p_{t}\\approx\\frac{\\eta\\alpha\\alpha^{\\prime}(T-1)}{T^{2}t }\\left\\{\\mathbb{P}(t_{q}=t-1)\\left(\\mathbbm{1}\\{s=t-1\\}-\\frac{1}{t}\\;\\mathbbm{ 1}\\{s\\in[t]\\}\\right)+O\\left(\\frac{1}{N}\\right)\\right\\},\\]\n\nwhich matches the desired \"previous token head\" behavior from (7) when \\(N\\) is large. As in the case of \\(W^{2}_{K}\\), we may then \"saturate\" the softmax by choosing a large enough step-size.\n\n## Appendix C Other Proofs\n\n### Proof of Lemma 1\n\nProof.: Recall the form of the cross-entropy loss for classification with \\(K\\) classes:\n\n\\[\\ell(y,\\xi)=-\\sum_{k=1}^{N}\\mathbbm{1}\\left\\{y=k\\right\\}\\log\\frac{e^{\\xi_{k}} }{\\sum_{j}e^{\\xi_{j}}}.\\]\n\nIts derivatives take the form\n\n\\[\\frac{\\partial\\ell}{\\partial\\xi_{k}}(y,\\xi)=s(\\xi)_{k}-\\mathbbm{1}\\{y=k\\},\\]\n\nwith \\(s(\\xi)_{k}=\\frac{e^{\\xi_{k}}}{\\sum_{j}e^{\\xi_{j}}}\\) the softmax.\n\nThe gradient of \\(L\\) is then given by\n\n\\[\\nabla_{W}L(W) =\\mathbb{E}_{(z,y)}\\left[\\sum_{k=1}^{N}\\frac{\\partial\\ell}{\\partial \\xi_{k}}(y,W_{U}Ww_{E}(z))\\nabla_{W}(w_{U}(k)^{\\top}Ww_{E}(z))\\right]\\] \\[=\\mathbb{E}_{(z,y)}\\left[\\sum_{k=1}^{N}(\\hat{p}_{W}(k|z)-\\mathbb{1 }\\{y=k\\})w_{U}(k)w_{E}(z)^{\\top}\\right]\\] \\[=\\sum_{k=1}^{N}\\mathbb{E}_{z}[\\mathbb{E}_{y}[(\\hat{p}_{W}(k|z)- \\mathbb{1}\\{y=k\\})w_{U}(k)w_{E}(z)^{\\top}\\mid z]]\\] \\[=\\sum_{k=1}^{N}\\mathbb{E}_{z}[(\\hat{p}_{W}(k|z)-\\mathbb{E}_{y}[ \\mathbb{1}\\{y=k\\}|z])w_{U}(k)w_{E}(z)^{\\top}],\\]\n\nwhich yields the desired result. \n\n### Proof of Lemma 2\n\nProof.: Using similar steps as the proof of Lemma 1, we have\n\n\\[\\nabla_{W}L(W) =\\mathbb{E}_{(x,y)}\\left[\\sum_{k=1}^{N}\\frac{\\partial\\ell}{ \\partial\\xi_{k}}(y,W_{U}Wx)\\nabla_{W}(w_{U}(k)^{\\top}Wx)\\right]\\] \\[=\\mathbb{E}_{(x,y)}\\left[\\sum_{k=1}^{N}(\\hat{p}_{W}(k|x)-\\mathbb{1 }\\{y=k\\})w_{U}(k)x^{\\top}\\right]\\] \\[=\\sum_{k=1}^{N}w_{U}(k)\\,\\mathbb{E}_{x}[\\hat{p}_{W}(k|x)x]^{\\top} -\\sum_{k=1}^{N}\\mathbb{E}_{y}[\\mathbb{1}\\{y=k\\}w_{U}(k)\\,\\mathbb{E}[x|y]^{\\top}]\\] \\[=\\sum_{k=1}^{N}w_{U}(k)\\,\\mathbb{E}_{x}[\\hat{p}_{W}(k|x)x]^{\\top} -\\sum_{k,j=1}^{N}p(y=j)\\,\\mathbb{1}\\{j=k\\}w_{U}(k)\\,\\mathbb{E}[x|y=j]^{\\top}\\] \\[=\\sum_{k=1}^{N}p(y=k)w_{U}(k)(\\hat{\\mu}_{k}-\\mu_{k})^{\\top},\\]\n\nwith \\(\\hat{\\mu}_{k}=p(y=k)^{-1}\\,\\mathbb{E}_{x}[\\hat{p}_{W}(k|x)x]\\) and \\(\\mu_{k}=\\mathbb{E}[x|y=k]\\). \n\n### Proof of Lemma 4\n\nProof.: To better isolate the role of keys from values, we denote the keys that are fed into the matrix \\(W\\) by \\(Z=[z_{1},\\ldots,z_{T}]\\in\\mathbb{R}^{d\\times T}\\), while the query is simply \\(x_{T}\\). In practice we have \\(Z=X\\), and both are superpositions of potentially multiple embeddings (if \\(W\\) is part of the second attention layer, these are the token embedding, positional embedding, and the output of the first attention layer).\n\nThe gradient of the loss at \\(W=0\\) writes:\n\n\\[\\nabla_{W}L(W)\\big{|}_{W=0}=\\mathbb{E}_{(X,Z,y)}\\left[\\sum_{k=1}^{ N}\\frac{\\partial\\ell}{\\partial\\xi_{k}}(y,\\xi)\\cdot\\nabla_{W}(w_{U}(k)^{\\top} \\Phi_{2}X\\sigma(Z^{\\top}Wx_{T}))\\big{|}_{W=0}\\right] \\tag{30}\\] \\[=\\mathbb{E}_{(X,Z,y)}\\left[\\sum_{k=1}^{N}(\\hat{p}_{W}(k|X,Z)- \\mathbb{1}\\{y=k\\})\\cdot\\nabla_{W}(w_{U}(k)^{\\top}\\Phi_{2}X\\sigma(Z^{\\top}Wx_{ T}))\\big{|}_{W=0}\\right]. \\tag{31}\\]\n\nWe have\n\n\\[\\nabla_{W}(w_{U}(k)^{\\top}\\Phi_{2}X\\sigma(Z^{\\top}Wx_{T}))\\big{|}_ {W=0} =\\sum_{t=1}^{T}w_{U}(k)^{\\top}\\Phi_{2}x_{t}\\cdot\\nabla_{W}(\\sigma (Z^{\\top}Wx_{T})_{t})\\] \\[=\\frac{1}{T}\\sum_{t=1}^{T}w_{U}(k)^{\\top}\\Phi_{2}x_{t}\\cdot(z_{t} -\\bar{z}_{1:T})x_{T}^{\\top},\\]where \\(\\bar{z}_{1:T}=\\frac{1}{T}\\sum_{t}z_{t}\\), and we used the fact that\n\n\\[\\frac{\\partial}{\\partial u_{s}}\\sigma(u)_{t}\\big{|}_{u=0}=\\frac{1}{T}\\,\\mathbbm{1 }\\{t=s\\}-\\frac{1}{T^{2}}. \\tag{32}\\]\n\nThe gradient (31) now writes\n\n\\[\\nabla_{W}L(W)\\big{|}_{W=0}=\\sum_{k=1}^{N}\\mathbb{E}_{(X,Z)}[(\\hat{\\rho}_{W}(k| X,Z)-\\mathbbm{1}\\{y=k\\})\\frac{1}{T}\\sum_{t=1}^{T}w_{U}(k)^{\\top}\\Phi_{2}x_{t} \\cdot(z_{t}-\\bar{z}_{1:T})x_{T}^{\\top}],\\]\n\nand the result follows. \n\n### Proof of Lemma 5\n\nProof.: The linearization of the second layer softmax around zero takes the following form:\n\n\\[\\bar{\\sigma}(Z^{\\top}W_{2}x_{T})_{t}=\\frac{1}{T}(1+z_{t}^{\\top}W_{2}x_{T}-\\bar{ z}_{1:T}^{\\top}W_{2}x_{T}),\\]\n\nwith \\(z_{t}=\\sum_{s=1}^{t}\\Phi_{1}x_{s}\\sigma(p_{1:t}^{\\top}Wp_{t})_{s}\\) the output of the first attention layer.\n\n\\[\\xi_{k} =\\sum_{t=1}^{T}w_{U}(k)^{\\top}\\Phi_{2}x_{t}\\bar{\\sigma}(Z^{\\top}W_ {2}x_{T})\\] \\[=\\frac{1}{T}\\sum_{t=1}^{T}w_{U}(k)^{\\top}\\Phi_{2}x_{t}+\\frac{1}{T} \\sum_{t=1}^{T}w_{U}(k)^{\\top}\\Phi_{2}x_{t}\\sum_{s=1}^{t}(\\Phi_{1}x_{s})^{\\top} W_{2}x_{T}\\sigma(p_{1:t}^{\\top}Wp_{t})_{s}\\] \\[\\quad-w_{u}(k)^{\\top}\\Phi_{2}\\bar{x}_{1:T}\\cdot\\frac{1}{T}\\sum_{t= 1}^{T}\\sum_{s=1}^{t}(\\Phi_{1}x_{s})^{\\top}W_{2}x_{T}\\sigma(p_{1:t}^{\\top}Wp_{t })_{s}.\\]\n\nThen,\n\n\\[\\nabla_{W}L(W)\\big{|}_{W=0} \\tag{33}\\] \\[\\quad=\\mathbb{E}_{(X,y)}\\left[\\sum_{k=1}^{N}\\frac{\\partial\\ell}{ \\partial\\xi_{k}}(y,\\xi)\\cdot\\nabla_{W}\\xi_{k}\\big{|}_{W=0}\\right]\\] (34) \\[\\quad=\\mathbb{E}_{(X,y)}\\left[\\sum_{k=1}^{N}\\frac{\\partial\\ell}{ \\partial\\xi_{k}}(y,\\xi)\\frac{1}{T}\\sum_{t=1}^{T}w_{U}(k)^{\\top}\\Phi_{2}x_{t} \\frac{1}{t}\\sum_{s=1}^{t}(\\Phi_{1}x_{s})^{\\top}W_{2}x_{T}(p_{s}-\\bar{p}_{1:t}) p_{t}^{\\top}\\right]\\] (35) \\[\\quad-\\mathbb{E}_{(X,y)}\\left[\\sum_{k=1}^{N}\\frac{\\partial\\ell}{ \\partial\\xi_{k}}(y,\\xi)w_{U}(k)^{\\top}\\Phi_{2}\\bar{x}_{1:T}\\cdot\\frac{1}{T} \\sum_{t=1}^{T}\\frac{1}{t}\\sum_{s=1}^{t}(\\Phi_{1}x_{s})^{\\top}W_{2}x_{T}(p_{s}- \\bar{p}_{1:t})p_{t}^{\\top}\\right], \\tag{36}\\]\n\nusing (32). The result follows by using \\(\\frac{\\partial\\ell}{\\partial\\xi_{k}}(y,\\xi)=\\hat{p}(k|\\xi)-\\mathbbm{1}\\{y=k\\}\\).\n\n## Appendix D Beyond our Simplified Architecture\n\nWhile the simplified architecture presented in Section 4.2 is sufficient to capture the desired induction behavior for our bigram task, transformer architectures used in practice typically involve more components, as well as more heads and layers. In this section, we discuss how our memory viewpoint extends to such architectures.\n\nFactorizations.In practice, transformers typically involve products of matrices, potentially with a low-rank bottleneck. For instance, our key-query matrices \\(W_{K}\\) should instead be considered as a product \\(W_{K}^{\\top}W_{Q}\\), and the output-value matrices \\(W_{O}\\) and \\(W_{V}\\) are typically jointly optimized.\n\nConsider an associative memory of the form:\n\n\\[W_{*}=\\sum_{i}y_{i}x_{i}^{\\top}\\in\\mathbb{R}^{d\\times d},\\]\n\nwhere \\((x_{i})_{i}\\) and \\((y_{i})_{i}\\) are appropriate collections of near-orthonormal embeddings.\n\nWe now argue that a similar associative memory can be achieved with the factorization \\(W=\\frac{d}{2d^{\\prime}}UV\\), where \\(U\\in\\mathbb{R}^{d\\times d^{\\prime}}\\), \\(V\\in\\mathbb{R}^{d^{\\prime}\\times d}\\) with \\(d^{\\prime}\\leq d\\) (for instance \\(d^{\\prime}\\) could be the dimension of attention heads), are given by:3\n\nFootnote 3: When \\(d^{\\prime}\\) is the head dimension, the \\(\\frac{d}{d^{\\prime}}\\) scaling can be interpreted as the correct multiplier to use in attention logits, which plays a similar role to the \\(\\frac{d}{d^{\\prime}}\\) multiplier in the \\(\\mu\\)P scaling [59], for our setup where the variance of the random entries of input embeddings is \\(1/d\\) instead of \\(1\\) as in [59].\n\n\\[U =U_{0}+\\sum_{i}y_{i}(V_{0}x_{i})^{\\top}\\] \\[V =V_{0}+\\sum_{i}(U_{0}^{\\top}y_{i})x_{i}^{\\top},\\]\n\nwhere \\(U_{0}\\) and \\(V_{0}\\) are random matrices with \\(\\mathcal{N}(0,\\frac{1}{d})\\) entries. These matrices are similar to those that would arise from a single gradient step individually on \\(U\\) and \\(V\\) from initializations \\(U_{0}\\) and \\(V_{0}\\), as in Lemma 1. To see why \\(W\\) behaves like \\(W_{*}\\), note that we have\n\n\\[UV=U_{0}V_{0}+\\sum_{i}y_{i}(V_{0}x_{i})^{\\top}V_{0}+\\sum_{i}(U_{0}^{\\top}y_{i} )x_{i}^{\\top}+\\sum_{i,j}y_{i}(V_{0}x_{i})^{\\top}(U_{0}^{\\top}y_{j})x_{j}^{\\top}.\\]\n\nIt is also easy to check using central limit arguments (similar to remapping in Appendix A) that \\(\\tilde{x}_{i}:=\\sqrt{\\frac{d}{d^{\\prime}}}V_{0}x_{i}\\in\\mathbb{R}^{d^{\\prime}}\\) and \\(\\tilde{y}_{i}:=\\sqrt{\\frac{d}{d^{\\prime}}}U_{0}^{\\top}y_{i}\\) are all nearly-orthonormal embeddings. Thus, we have\n\n\\[2y_{k}^{\\top}Wx_{l} =\\tilde{y}_{k}^{\\top}\\tilde{x}_{l}+\\sum_{i}y_{k}^{\\top}y_{i} \\tilde{x}_{i}^{\\top}\\tilde{x}_{l}+\\sum_{i}\\tilde{y}_{k}^{\\top}\\tilde{y}_{i}x_ {i}^{\\top}x_{l}+\\sum_{i,j}y_{k}^{\\top}y_{i}\\tilde{x}_{i}^{\\top}\\tilde{y}_{j}x_ {j}^{\\top}x_{l}\\] \\[\\approx 0+\\mathbbm{1}\\{k=l\\}+\\mathbbm{1}\\{k=l\\}+0,\\]\n\nwhere the first and last term vanish due to the cross-terms \\(\\tilde{y}_{i}^{\\top}\\tilde{x}_{i^{\\prime}}\\) which vanish for any \\(i,i^{\\prime}\\). Thus, \\(W\\) and \\(W_{*}\\) encode the same associations, when \\(d\\) and \\(d^{\\prime}\\) are large enough to ensure near-orthogonality.\n\nLayer-normalization.Normalization layers [4, 60] are typically used in transformers to improve training stability [52], and are applied on each token representation, either after [52] or before [6] each block. It may be seen as an operation of the form4\n\nFootnote 4: RMSNorm [60] would use the variance instead of the norm in the denominator, leading to an additional \\(\\sqrt{d}\\) factor in the numerator. Here we use the norm, which is more natural when embeddings have near-unit norm, in contrast to the \\(\\approx\\sqrt{d}\\) norm for the standard parameterization.\n\n\\[LN(x)=\\frac{x}{\\|x\\|},\\]\n\napplied to the input or output of a given block.\n\nIn order to obtain a basic understanding of the role of layer-norm in our associative memory setup, we may consider the setup Lemma 1, with a normalization applied after the linear operation, leading to the population loss:\n\n\\[L(W)=\\mathbb{E}_{(x,y)\\sim p}[\\ell(y,W_{U}LN(Wx))]. \\tag{37}\\]\n\nThe gradients then take the form\n\n\\[\\nabla_{W}L(W)=\\sum_{k=1}^{N}\\mathbb{E}_{x}\\left[\\frac{\\hat{p}_{W}(y=k|x)-p(y =k|x)}{\\|Wx\\|}\\left(I-\\frac{(Wx)(Wx)^{\\top}}{\\|Wx\\|^{2}}\\right)w_{U}(k)x^{\\top }\\right]. \\tag{38}\\]This illustrates that in addition to weighting the updates on class \\(k\\) by the prediction error \\(\\hat{p}(k|x)-p(k|x)\\), the updates are also projected on the orthogonal of the \\(Wx\\) direction. This means that an update on the direction \\(w_{U}(k)x^{\\top}\\) will occur only to the extent that \\(Wx\\) is not already aligned with \\(w_{U}(k)\\). Thus, if an association is \"stored\" once, so that \\(Wx\\approx w_{U}(y)\\), layer-norm will avoid further updating \\(W\\) in that direction, hopefully avoiding norms that grow too much, and also encouraging frequent and infrequent tokens to be weighted similarly in the final memory (see also [7] for more discussion on this).\n\nNote that at random initialization \\(Wx\\) is nearly orthogonal to any \\(w_{U}(k)\\), so that layer-norm only starts playing a significant role later in training, and does not affect our theoretical analysis based on single gradient steps.\n\nMLP blocks.If we denote by \\((u_{i})_{i}\\) and \\((v_{i})_{i}\\) collections of near-orthonormal input and output embeddings, an MLP block may encode associations \\((i,j)\\in\\mathcal{M}\\) as follows:\n\n\\[F(x)=\\sum_{(i,j)\\in\\mathcal{M}}v_{j}\\sigma(u_{i}^{\\top}x-b),\\]\n\nwhere \\(\\sigma\\) is a non-linear activation and \\(b\\) a bias term. Then, if one assumes that \\(u_{i}^{\\top}u_{j}\\leq b\\) for \\(i\\neq j\\) and \\(\\sigma(t)=0\\) for \\(t<0\\), then this can help filter out noise that arises from near-orthogonality, and may then lead to additional storage capacity, at the cost of additional computation (see, _e.g._, [7, 30]).\n\nAn additional benefit of MLP layers discussed in Section 4 is that they may encode many-to-many associations, which is useful when multiple embeddings are in the residual stream and need to be considered jointly (_e.g._, a subject and a relation in a factual recall task [35]). This may be achieved, for instance, by considering embeddings \\(u_{\\mathcal{I}}=\\frac{1}{\\sqrt{|\\mathcal{I}|}}\\sum_{i\\in\\mathcal{I}}u_{i}\\), where \\(\\mathcal{I}\\) are sets of bounded size (_e.g._, as obtained using layer-norm over a residual stream). Then, assuming the \\(u_{i}\\) are nearly-orthonormal, we have \\(u_{\\mathcal{I}}^{\\top}u_{\\mathcal{I}}\\approx 1\\) while \\(u_{\\mathcal{I}}^{\\top}u_{\\mathcal{I}}\\lesssim 1-\\delta\\) if \\(\\mathcal{I}=\\mathcal{I}^{\\prime}\\), for some \\(\\delta\\) that depends on the maximal cardinality of these sets. Although \\(1-\\delta\\) is no-longer vanishingly small, defining\n\n\\[F(x)=\\sum_{(\\mathcal{I},\\mathcal{J})\\in\\mathcal{M}}v_{\\mathcal{J}}\\sigma(u_{ \\mathcal{I}}^{\\top}x-b),\\]\n\nthe non-linearity may still succeed at filtering out any \\(\\mathcal{I}\\) that does not correspond to the query set in \\(x\\). We leave the question of how such non-linear associative memories may arise from training dynamics to future work.\n\nMultiple heads and layers.We remark that our view of weights as associative memories applies to any parameter other than embedding/unembedding layers, and thus naturally extends to multiple heads (using the low-rank factorizations described above) and multi-layer models.\n\nIt is important to note, however, that the redundancy introduced by having more heads and layers makes it more challenging to identify which layer/head/weight will learn certain associations (see, _e.g._, Figure 11 in Appendix E). This is in contrast to our simplified architecture of Section 4.2, where we may identify the role of each matrix (up to some possible redundancy when using the feed-forward layer \\(W_{F}\\)). In practice, mechanisms may appear in different heads/layers across different training runs, which makes interpretability more challenging, and typically requires some causal identification techniques, such as mediation analysis [35, 54].\n\n## Appendix E Experiment Details and Additional Experiments\n\nIn this section, we present additional details on the experiments, as well as additional results.\n\nComputing setup.We use Pytorch and each run uses a single GPU, along with 60 CPU cores for real-time data generation. We will make our code available upon publication.\n\nHyperparameters.We now provide the hyperparameters used in each figure. The SGD step-size is denoted \\(\\eta\\). We fix the momentum parameter to \\(0.9\\) and the weight decay parameter to \\(10^{-4}\\). \\(U\\) denotes the uniform distribution over \\([N]\\).\n\n* Figure 2: \\(K=3\\), \\(\\pi_{q}=\\pi_{u}\\) (random triggers) or \\(Q\\) is the \\(K\\) most likely elements of \\(\\pi_{u}\\), \\(\\pi_{o}=U\\), \\(d=128\\), \\(d_{hidden}=4\\times 128\\) (hidden dimension of the feed-forward MLPs), \\(\\eta=0.2\\).\n* Figure 3: \\(K=5\\), \\(\\pi_{q}=\\pi_{u}\\) (random triggers), \\(\\pi_{o}=U\\), \\(d=128\\), \\(\\eta=0.2\\).\n* Figure 4(left) and Figure 5: \\(\\pi_{o}=U\\), \\(d=128\\), \\(\\eta=1\\). For random triggers we use \\(\\pi_{q}=\\pi_{u}\\). For \\(K=1\\) with fixed frequent trigger, the only trigger is the most probable token according to \\(\\pi_{u}\\), while for \\(K=5\\) with fixed rare triggers, the five triggers are the 6-th to 10-th most probable tokens according to \\(\\pi_{u}\\).\n* Figure 4(center): \\(K=3\\), \\(\\pi_{q}=\\pi_{u}\\) (random triggers), \\(\\pi_{o}=U\\) or \\(\\pi_{o}=\\pi_{b}\\) (conditioned on the trigger), \\(d=128\\), \\(\\eta=1\\).\n* Figure 4(right): \\(K=3\\), \\(\\pi_{q}=\\pi_{u}\\) (random triggers), \\(\\pi_{o}=U\\), \\(d=128\\), \\(\\eta=1\\).\n\nMemory recall probes and data-distributional properties.Figure 5 and Figure 6 show the evolution of the different memory probes for the settings considered in Figure 4(left,center). Figure 5 highlights that associative memories for the induction head are slower to learn when using few triggers (small \\(K\\)), rare fixed triggers, or random triggers (note that the probe for \\(W_{K}^{2}\\) with fixed triggers only shows recall accuracy on the set of triggers \\(Q\\), which is an easier task). Figure 6\n\nFigure 5: Memory recall probes for the setting of Figure 4(left).\n\nFigure 6: Memory recall probes for the setting of Figure 4(center).\n\nshows that using uniform output tokens can lead to better fitting of \\(W^{2}_{O}\\) and \\(W^{2}_{K}\\) compared to using output tokens sampled from bigrams. In addition to the increased diversity when using uniform outputs, this may also be due to the fact that bigram outputs are already well predicted using global statistics with just the feed-forward layer, hence the gradient signal on such well-predicted tokens may not propagate through the induction head mechanism. In contrast, the recall accuracy for \\(W^{1}_{K}\\) is comparable for both settings, since the previous token head is useful at all positions regardless of the output token distribution.\n\nVisualizing memories.Figure 7 shows visualizations of the associative memory behaviors after training. We see that diagonal elements dominate in the plots, which corresponds to correct associations lead to high'memory recall'. Nonetheless, we see that some of the diagonal elements are weaker than others, particularly for late positions in \\(W^{1}_{K}\\), and for some of the trigger tokens in \\(W^{2}_{K}\\), while the diagonal for \\(W^{2}_{O}\\) seems to be roughly uniform. We note that characters corresponding to capital letters have token index 13 to 38, while lowercase letters have index 39 to 64. The association patterns found in \\(W^{2}_{K}\\) then seem related to frequencies of appearance of triggers, whereby capital letters appear less frequently in the data, and are also less frequently chosen as triggers, compared to lowercase letters. Similarly, since the first occurrence of triggers is typically early in a sequence, it is natural that \\(W^{1}_{K}\\) learns stronger associations at earlier positions. In contrast, diagonal elements for \\(W^{2}_{O}\\) are nearly uniform, which agrees with the fact that output tokens are sampled uniformly in this setup. We refer to the follow-up work [7] for an analysis of how data frequencies affect association strength in such associative memories.\n\nEffect of dimension.Recall that our study of associative memories with random embeddings requires large dimension \\(d\\) in order to ensure near-orthogonality, and thus store input-output pairs more effectively. In Figure 8, we evaluate the recall accuracy for \\(W^{2}_{O}\\) for varying dimension, when training it by itself, and only on the output tokens (as in Figure 3). We see that higher dimension leads to faster learning of the memory, in particular \\(d=128\\) seems sufficient for fast learning after just a few iterations with a tuned learning rate. If the learning rate isn't tuned, we notice that there is a further slowdown for low dimension, is likely due to issues with the fact that our experiments use the standard parameterization of neural networks at initialization, rather than maximal update parameterizations [58]. Note that learning \\(W^{2}_{O}\\) alone is a convex optimization problem, and we hypothesize that higher dimension makes the problem better conditioned, and hence easier to learn. In Figure 9, we show \"one-step\" recall accuracies for classifying output tokens from the average\n\nFigure 8: Effect of dimension on learning \\(W^{2}_{O}\\) alone, with fixed or tuned learning rate.\n\nFigure 9: Accuracy of one-step estimate of \\(W^{2}_{O}\\) with varying dimension and number of batches used for computing expectations. Each batch consists of 32 sequences of 256 tokens for a total of 8 192 tokens, with \\(K=5\\) random triggers and uniform outputs.\n\nattention input to \\(W_{O}^{2}\\), given by\n\n\\[R_{1}=\\frac{1}{N}\\sum_{k=1}^{N}\\mathbbm{1}\\left\\{k=\\arg\\max_{k^{\\prime}}(W_{V}^{2} w_{E}(k^{\\prime}))^{\\top}(\\mu_{k}-\\mu)\\right\\},\\]\n\nwhere \\(\\mu_{k}=\\mathbb{E}[x|y=k]\\) and \\(\\mu=\\mathbb{E}[x]\\), for \\(x=\\frac{1}{t}\\sum_{s=1}^{t}W_{V}^{2}w_{E}(z_{s})\\) and \\(y=z_{t+1}\\), when \\(z_{t}\\) is a trigger token after its first occurrence. Expectations are computed over batches of data of varying sizes and in different dimensions. We call this \"one-step\" since it is related to the classifier obtained after performing a single gradient step on \\(W_{O}^{2}\\) from zero initialization (see Lemma 2 and Appendix B.3.1). The plots illustrate that this simple one-step model is already able to extract relevant signal from the noisy average attention, after a handful of batches of data, corresponding to tens of thousands of tokens, and that this gets easier as the dimension increases.\n\nMore complex architectures.Figure 10 shows training behavior for a more complex model than the simplified one considered in Section 5, namely where we train all parameters, replace the linear \\(W_{F}\\) feedforward layer by a two-layer MLP, and were (pre-)layer-normalization is added. Despite these changes, we see similar behavior for the memory recall probes (which now involve embeddings that may change over time), suggesting that the model is still identifying the same memory associations, despite the additional redundancies in parameters and modified training dynamics.\n\nFigure 11 shows the attention maps obtained when training a multi-head version of our two-layer model, with four attention heads per layer. We see that the redundancy of multiple heads creates difficulties in identifiability: only one of the first layer heads learns the previous token behavior, while the induction behavior is shared across different heads at the second layer. This illustrates the challenges of interpretability in the presence of redundant models, which then require additional work to identify which of the layers and heads are performing a given behavior, _e.g._, through interventions and causal mediation analysis [35, 54].\n\nFigure 11: Attention maps for a two-layer model with 4 attention heads. In the first layer (top), the previous token mechanism is mostly achieved by one of the four heads, while the induction behavior at the second layer (bottom) is distributed across the different heads.\n\nFigure 10: Training of a more realistic architecture with (i) ReLU MLP instead of linear layer for the second feed-forward layer, (ii) all parameters trained, including embeddings, (iii) pre-layer normalization. The loss, in-context accuracy and memory recall probes are similar to the simplified architecture (see, _e.g._, Figure 4).", "response": "## Review\n\n### Summary\n\nThis paper provides a comprehensive analysis of in-context learning (ICL) mechanisms in a simplified two-layer transformer architecture through a synthetic task that involves associative recall. The authors focus on the emergence of induction heads and associative memory in attention weight matrices, linking them to global bigram distribution learning. They present both mathematical justifications and experimental results, revealing how these mechanisms develop during training. Although the study is insightful and advances understanding of ICL in transformers, concerns are raised regarding the simplicity of the model and task, which may limit the generalizability of the findings to more complex systems.\n\n### Strengths\n\n- The problem is well-motivated and addresses a significant aspect of language models: understanding ICL.\n- The theoretical contributions are interesting and provide a non-trivial analysis of the mechanisms at play.\n- The methodology is clean, utilizing memory recall probes effectively.\n- The paper is well-written and easy to follow, with clear motivations and conclusions.\n- The experimental results provide valuable insights into the learning dynamics of transformers.\n\n### Weaknesses\n\n- The simplicity of the analyzed architecture raises questions about the applicability of the findings to more complex transformers.\n- The significance of the results could be better articulated, particularly concerning practical implications.\n- The paper lacks a thorough discussion of limitations and the potential non-generalizability of results to larger models.\n- Some design decisions, like freezing layers, may lead to biased results that do not reflect the behavior of fully trained models.\n- The overall presentation could be improved for clarity, as some sections are difficult to follow.\n\n### Questions\n\n- What are the implications of ignoring layer normalization and dropout on the findings?\n- Could the authors clarify the significance of framing induction heads as associative memory?\n- Were any experiments performed to test the effects of varying the dimensionality of the model?\n- How do the findings apply to larger transformer models, and what additional experiments could validate this?\n- Can the authors provide a proof sketch for Lemma 1 and discuss its implications for real transformers?\n\n### Soundness\n\n**Score:** 3\n\n**Description:** 3 = good; the theoretical framework and empirical results are solid, but the simplifications in the model may affect the applicability to real-world scenarios.\n\n### Presentation\n\n**Score:** 3\n\n**Description:** 3 = good; while the paper is generally well-written, there are clarity issues that could benefit from editing and restructuring for better flow.\n\n### Contribution\n\n**Score:** 3\n\n**Description:** 3 = good; the paper offers valuable insights and a useful case study, but the novelty is somewhat limited by the simplicity of the task and model.\n\n### Rating\n\n**Score:** 7\n\n**Description:** 7 = accept, but needs minor improvements; the paper is technically solid and impactful in understanding ICL in transformers, though some aspects require further exploration.\n\n### Paper Decision\n\n**Decision:** Accept\n\n**Reasons:** The paper provides significant insights into how in-context learning mechanisms emerge in transformer architectures, contributing to the understanding of language models. While there are valid concerns regarding the simplicity of the model and task, the rigorous theoretical and experimental work presented makes it a valuable contribution to the field.\n"}
{"query": "You are a highly experienced, conscientious, and fair academic reviewer, please help me review this paper. The review should be organized into nine sections: \n1. Summary: A summary of the paper in 100-150 words.\n2. Strengths/Weaknesses/Questions: The Strengths/Weaknesses/Questions of paper, which should be listed in bullet points, with each point supported by specific examples from the article where possible.\n3. Soundness/Contribution/Presentation: Rate the paper's Soundness/Contribution/Presentation, and match this score to the corresponding description from the list below and provide the result. The possible scores and their descriptions are: \n    1 poor\n    2 fair\n    3 good\n    4 excellent\n4. Rating: Give this paper an appropriate rating, match this rating to the corresponding description from the list below and provide the result. The possible Ratings and their descriptions are: \n    1 strong reject\n    2 reject, significant issues present\n    3 reject, not good enough\n    4 possibly reject, but has redeeming facets\n    5 marginally below the acceptance threshold\n    6 marginally above the acceptance threshold\n    7 accept, but needs minor improvements \n    8 accept, good paper\n    9 strong accept, excellent work\n    10 strong accept, should be highlighted at the conference   \n5. Paper Decision: It must include the Decision itself(Accept or Reject) and the reasons for this decision, based on the criteria of originality, methodological soundness, significance of results, and clarity and logic of presentation.\n\nHere is the template for a review format, you must follow this format to output your review result:\n**Summary:**\nSummary content\n\n**Strengths:**\n- Strength 1\n- Strength 2\n- ...\n\n**Weaknesses:**\n- Weakness 1\n- Weakness 2\n- ...\n\n**Questions:**\n- Question 1\n- Question 2\n- ...\n\n**Soundness:**\nSoundness result\n\n**Presentation:**\nPresentation result\n\n**Contribution:**\nContribution result\n\n**Rating:**\nRating result\n\n**Paper Decision:**\n- Decision: Accept/Reject\n- Reasons: reasons content\n\n\nPlease ensure your feedback is objective and constructive. The paper is as follows:\n\n# Large Graph Property Prediction via\n\nGraph Segment Training\n\nKaidi Cao\\({}^{1}\\), Phitchaya Mangpo Phothilimthana\\({}^{2}\\), Sami Abu-El-Haija\\({}^{2}\\), Dustin Zelle\\({}^{2}\\),\n\nYanqi Zhou\\({}^{2}\\), Charith Mendis\\({}^{3}\\), Jure Leskovec\\({}^{1}\\), Bryan Perozzi\\({}^{2}\\)\n\n\\({}^{1}\\)Stanford University, \\({}^{2}\\)Google, \\({}^{3}\\)UIUC\n\nThe work was partially completed during Kaidi Cao's internship at Google.The work was partially completed when Charith Mendis was a visiting researcher at Google.\n\n###### Abstract\n\nLearning to predict properties of a large graph is challenging because each prediction requires the knowledge of an entire graph, while the amount of memory available during training is bounded. Here we propose Graph Segment Training (GST), a general framework that utilizes a divide-and-conquer approach to allow learning large graph property prediction with a constant memory footprint. GST first divides a large graph into segments and then backpropagates through only a few segments sampled per training iteration. We refine the GST paradigm by introducing a historical embedding table to efficiently obtain embeddings for segments not sampled for backpropagation. To mitigate the staleness of historical embeddings, we design two novel techniques. First, we finetune the prediction head to fix the input distribution shift. Second, we introduce _Stale Embedding Dropout_ to drop some stale embeddings during training to reduce bias. We evaluate our complete method GST+EFD (with all the techniques together) on two large graph property prediction benchmarks: MalNet and TpuGraphs. Our experiments show that GST+EFD is both memory-efficient and fast, while offering a slight boost on test accuracy over a typical full graph training regime.\n\n## 1 Introduction\n\nGraph property prediction is a task of predicting a certain property or a characteristic of an entire graph [3]. Important applications include, predicting properties of molecules [32, 12], predicting properties of programs/code [1, 12, 6, 41] and predicting properties of organisms based on their protein-protein interaction networks [27, 42].\n\nThese popular graph property prediction tasks deal with relatively small graphs, so the scalability issue arises only from a large number of (small) graphs.\n\nHowever, graph property prediction tasks also face another scalability challenge, which arises due to the large size of each individual graph, as some graphs can have millions or even billions of nodes and edges [10]. Training typical Graph Neural Networks (GNNs) to classify such large graphs can be computationally infeasible, as the memory needed scales at least linearly with the size of the graph [40]. This presents a challenge as even most powerful GPUs, which are optimized for handling large amounts of data, only have a limited amount of memory available.\n\nPrevious efforts to improve scalability of GNNs have mostly focused on developing methods for node-level and link-level prediction tasks, which can be performed using sampled subgraphs [11, 5, 13, 38, 43, 2, 21]. However, there is a lack of research on how to train scalable models for _property prediction of large graphs_. Training a model on a sampled subgraph alone is insufficient for thesetypes of tasks, as the subgraph sampled may not contain all the necessary information to make accurate predictions for the entire graph. For example, if the task is to predict the diameter of the graph, it is unlikely that a fixed-size subgraph would contain sufficient features for the GNN to make an accurate and reliable prediction. Thus, it is essential to aggregate information from the entire graph to predict a graph property.\n\nIn this paper, we address the problem of property prediction of large graphs. We propose Graph Segment Training (GST)3, which is able to train on large graphs with constant (GPU) memory footprint.\n\nFootnote 3: Source code available at [https://github.com/kaidic/GST](https://github.com/kaidic/GST).\n\nOur approach partitions each large graph into smaller segments with a controlled size in the pre-processing phase. During the training process, a random subset of segments is selected to update the model at each step, rather than using the entire graph. This way, we need to maintain intermediate activations for only a few segments for backpropagation; embeddings for the remaining segments are created without saving their intermediate activations. The embeddings of all segments are then combined to generate an embedding for the original large graph, which is used for prediction. Therefore, each large graph has an upper bound on memory consumption during training regardless of its original size. This allows us to train the model on large graphs without running into an out-of-memory (OOM) issue, even with limited computational resources.\n\nTo accelerate the training process further, we introduce a historical embedding table to efficiently produce embeddings for graph segments that do not require gradients, as historical embeddings eliminate additional computation on such segments. However, the historical embeddings induce staleness issues during training, so we design two techniques to mitigate such issues in practice. First, we characterize the input distribution mismatch issue between training and test stages of the prediction head, and propose to finetune only the prediction head at the end of training to close the input distribution gap. Second, we identify bias in the loss function due to stale historical embeddings, and introduce _Stale Embedding Dropout_ to drop some stale embeddings during training to reduce this bias. Our final proposed method, called GST+EFD, is both memory-efficient and fast.\n\nWe evaluate our method on the following datasets: MalNet-Tiny, MalNet-Large and TpuGraphs. A typical full graph training pipeline (Full Graph Training) can only train on MalNet-Tiny, and\n\nFigure 1: (a) **Full Graph Training**: Classically, models are trained using the entire graph, meaning all nodes and edges of the graph are used to compute gradients. For large graphs, this might be computationally infeasible. (b) **Graph Segment Training**: Our solution is to partition each large graph into smaller segments and select a random subset of segments to update the model; embeddings for the remaining segments are produced without saving their intermediate activations. The embeddings of all segments are combined to generate an embedding for the original large graph, which is then used for prediction. The important benefit is that GPU memory requirement only depends on the segment size (but not the full graph size).\n\nunavoidably reaches OOM on MalNet-Large and TpuGraphs dataset. On the contrary, we empirically show that the proposed GST framework successfully trains on arbitrarily large graphs using a single NVIDIA-V100 GPU with only 16GB of memory for MalNet-Large and four GPUs for TpuGraphs dataset, while maintaining comparable performance with Full Graph Training. We finally demonstrate that our complete method GST+EFD slightly outperforms GST by another 1-2% in terms of final evaluation metric, and simultaneously being 3x faster in terms of training time.\n\n## 2 Preliminaries\n\n**Notation.** For a function \\(f(\\mathbf{a}):\\mathbb{R}^{d_{0}}\\rightarrow\\mathbb{R}^{d_{1}}\\), we use \\(D^{k}_{\\mathbf{a}}f[\\mathbf{a}_{0}]\\in\\mathbb{R}^{d_{0}\\times d_{1}}\\) to denote its \\(k\\)-th derivative of \\(f\\) with respect to \\(\\mathbf{a}\\) evaluated at value \\(\\mathbf{a}_{0}\\). Let \\(f\\circ g\\) to denote the composition of function \\(f\\) with \\(g\\). We use \\(\\bigcirc\\) to denote entry-wise product, and let \\(\\mathbf{a}\\bigodot^{2}\\stackrel{{\\Delta}}{{=}}\\mathbf{a}\\bigodot\\mathbf{a}\\). We use \\(\\bigoplus\\) to represent aggregation. \\(\\bigoplus_{j\\leq J}\\mathbf{a}_{j}\\) means aggregating the set \\(\\{\\mathbf{a}_{j}\\}_{j\\leq J}\\), where \\(j\\) indexes segments, and \\(J\\) is the number of segments in a graph. We usually drop \\(j\\) subscript from \\(\\bigoplus\\) for brevity. \\(\\mathbf{a}_{i}\\bigoplus\\mathbf{b}_{j}\\) aggregates both sets \\(\\{\\mathbf{a}_{i}\\}\\) and \\(\\{\\mathbf{b}_{j}\\}\\) together. \\(\\bigoplus\\) can be mean or sum operators when applying to vectors. We define the input graph as \\(\\mathcal{G}=\\{\\mathcal{V},\\mathcal{E}\\}\\), where \\(\\mathcal{V}=\\{v_{1},...,v_{m}\\}\\) is the node set and \\(\\mathcal{E}\\subseteq\\mathcal{V}\\times\\mathcal{V}\\) is the edge set.\n\nLet dataset \\(\\mathcal{D}=\\{(\\mathcal{G}^{(i)},y^{(i)})\\}_{i\\leq n}\\) contain \\(n\\) examples: each label \\(y^{(i)}\\) is associated with \\(\\mathcal{G}^{(i)}\\).\n\n**Graph Neural Network.** We consider a backbone graph neural network \\(F\\) that takes a graph \\(\\mathcal{G}^{(i)}\\) and generates graph embedding \\(\\mathbf{h}^{(i)}\\in\\mathbb{R}^{d_{h}}\\), followed by a final prediction head \\(F^{\\prime}\\) that takes graph embedding \\(\\mathbf{h}^{(i)}\\) and outputs final predictions: \\(\\widehat{y}^{(i)}=(F^{\\prime}\\circ F)(\\mathcal{G}^{(i)})\\). We optimize GNN with the loss function as \\(\\mathcal{L}((F^{\\prime}\\circ F)(\\mathcal{G}^{(i)}),y^{(i)})\\).\n\n**Historical Embedding Table.** We define an embedding table \\(\\mathcal{T}:\\mathcal{K}\\rightarrow\\mathbb{R}^{d_{h}}\\), where key-space \\(\\mathcal{K}\\subset\\mathbb{Z}\\times\\mathbb{Z}\\) is a tuple: (graph index \\(i\\leq n\\), segment index \\(j\\leq J\\)). We use \\(\\tilde{\\mathbf{h}}^{(i)}_{j}=\\mathcal{T}(i,j)\\) to denote embedding for graph segment \\(\\mathcal{G}^{(i)}_{j}\\): an embedding not up to date with the current backbone \\(F\\).\n\n## 3 Our Method: GST+EFD\n\n### Graph Segment Training (GST)\n\nGiven a training graph dataset \\(\\mathcal{D}_{\\text{train}}=\\{(\\mathcal{G}^{(i)},y^{(i)})\\}_{i=1}^{n}\\), a common SGD update step requires calculating the gradient:\n\n\\[\\nabla_{\\theta}\\sum_{(\\mathcal{G}^{(i)},y^{(i)})\\in\\mathcal{B}}\\mathcal{L}((F^ {\\prime}\\circ F)(\\mathcal{G}^{(i)}),y^{(i)})\\]\n\nwhere \\(\\theta\\) is trainable weights in \\(F^{\\prime}\\circ F\\), and \\(\\mathcal{B}\\) is a sampled minibatch. Graphs can differ in size (the number of nodes \\(|\\mathcal{V}^{(i)}|\\)), with some being too large to fit into the device's memory. This is because the memory required to store all intermediate activations for computing gradients is proportional to the number of nodes and edges in the graph.\n\nTo address the above issue, we propose to partition each original input graph into a collection of graph segments, _i.e._,\n\n\\[\\mathcal{G}^{(i)}\\approx\\bigoplus\\mathcal{G}^{(i)}_{j}\\quad\\text{ for }j\\in\\{1,2,\\ldots,J^{(i)}\\}\\]\n\nAn example of a partition algorithm is METIS [16]. This preprocessing step will result in a training set \\(\\mathcal{D}_{\\text{train}}=\\{(\\bigoplus_{j\\leq J^{(i)}}\\mathcal{G}^{(i)}_{j}, y^{(i)})\\}_{i=1}^{n}\\). Number of partitions \\(J^{(i)}\\) can vary across graphs, but the size of each graph segment can be bounded by a controlled size (\\(|\\mathcal{V}^{(i)}_{j}|<m_{\\text{GST}},\\forall(i,j)\\)) so that a batch of a fixed number of graph segments can always fit within the device's memory.\n\nWhen processing graph segment \\(\\mathcal{G}^{(i)}_{j}\\), we can obtain its segment embedding through the backbone: \\(\\mathbf{h}^{(i)}_{j}=F(\\mathcal{G}^{(i)}_{j})\\). The prediction head \\(F^{\\prime}\\) requires information from the whole graph to make the prediction, thus we propose to aggregate all segment embeddings to recover the full graph embedding: \\(\\mathbf{h}^{(i)}=\\bigoplus\\mathbf{h}^{(i)}_{j}\\). A simple realization of this aggregation is mean pooling. Note that naively applyingthe prediction head \\(F^{\\prime}\\) on the aggregated graph embedding -- \\(\\widehat{y}^{(i)}=F^{\\prime}(\\bigoplus\\mathbf{h}_{j}^{(i)})\\) -- would not provide any reduction in peak memory consumption, as we need to keep track of the activations of all graph segments \\(\\{\\mathcal{G}_{j}^{(i)}\\}_{j\\in J^{(i)}}\\) to perform backpropagation.\n\nThus, we propose to perform backpropagation on only a few randomly sampled graph segments \\(\\mathcal{S}^{(i)}\\subseteq\\{1,\\ldots,J^{(i)}\\}\\) and generate embeddings without requiring gradients for the rest. We hereby denote \\(\\mathbf{h}_{s}\\) and \\(\\bar{\\mathbf{h}}_{j}\\) as segment embeddings that require and do not require gradient, respectively. An entire graph embedding is then: \\(\\mathbf{h}^{(i)}\\approx\\{\\mathbf{h}_{s}^{(i)}\\}_{s\\in\\mathcal{S}^{(i)}}\\bigoplus\\{\\bar {\\mathbf{h}}_{j}^{(i)}\\}_{j\\notin\\mathcal{S}^{(i)}}\\triangleq\\mathbf{h}_{s}^{(i)} \\bigoplus\\bar{\\mathbf{h}}_{j}^{(i)}\\). We name this general pipeline as GST and summarize it in Algorithm 1.\n\n```\n0: A preprocessed training graph dataset \\(\\mathcal{D}_{\\text{train}}=\\{(\\bigoplus\\mathcal{G}_{j}^{(i)},y^{(i)})\\}_{i=1}^{n}\\). A parameterized backbone \\(F\\) and a prediction head \\(F^{\\prime}\\).\n1:for\\(t=1\\) to \\(T_{0}\\)do\n2:\\(\\mathcal{B}\\leftarrow\\text{SampleMiniBatch}(\\mathcal{D}_{\\text{train}})\\)\n3:for\\((\\mathcal{G}^{(i)},\\widehat{y}^{(i)})\\) in \\(\\mathcal{B}\\)do\n4:\\(\\{\\mathcal{G}_{s}^{(i)}\\}_{s\\in\\mathcal{S}^{(i)}}\\leftarrow\\text{SampleGraphSegments}( \\mathcal{G}^{(i)})\\)\n5:\\(\\bar{\\mathbf{h}}_{j}^{(i)}\\leftarrow\\text{ProduceEmbedding}(\\mathcal{G}_{j}^{(i)})\\) for \\(j\\notin\\mathcal{S}^{(i)}\\)\n6:\\(\\mathbf{h}_{s}^{(i)}\\gets F(\\mathcal{G}_{s}^{(i)})\\) for\\(s\\in\\mathcal{S}^{(i)}\\)\n7:endfor\n8: SGD on loss \\(\\leftarrow\\frac{1}{|\\mathcal{B}|}\\sum_{i}\\mathcal{L}\\left(F^{\\prime}(\\mathbf{h}_{s} ^{(i)}\\bigoplus\\bar{\\mathbf{h}}_{j}^{(i)}),\\widehat{y}^{(i)}\\right)\\)\n9:endfor\n```\n\n**Algorithm 1** General Framework of GST\n\nOne implementation of ProduceEmbedding\\((\\cdot)\\) in Algorithm 1 is to use the same feature encoder \\(F\\) to forward all the segments in \\(\\{\\mathcal{G}_{j}^{(i)}\\}_{j\\notin\\mathcal{S}^{(i)}}\\) without storing any intermediate activation (by stopping gradient).\n\n### GST with Historical Embedding Table\n\nCalculating \\(\\bar{\\mathbf{h}}_{j}\\) by stopping gradient guarantees an upper bound on peak memory consumption. However, since we do not need gradients for segments \\(\\{\\mathcal{G}_{i}^{(i)}\\}_{j\\notin\\mathcal{S}^{(i)}}\\), computing forward pass on these segments can be avoided to make training faster. To achieve this, we use historical embeddings acquired in previous training iterations \\(\\tilde{\\mathbf{h}}_{j}^{(i)}=\\mathcal{T}(i,j)\\). With an embedding table \\(\\mathcal{T}\\), one can implement ProduceEmbedding\\((\\cdot)\\) by fetching the corresponding embedding from the table without any computation. We update the embedding table after conducting the forward pass on a graph segment. We optimize the following loss \\(\\mathcal{L}(F^{\\prime}(\\mathbf{h}_{s}^{(i)}\\bigoplus\\tilde{\\mathbf{h}}_{j}^{(i)}),y^{( i)})\\) during training. We name the embedding version of our algorithm as GST+E.\n\nNote that GST+E has runtime advantages over GST. For each segment \\(\\mathcal{G}_{j}^{(i)}\\) that does not require gradient, GST needs to run an actual forward pass over \\(\\mathcal{G}_{j}^{(i)}\\), while GST+E only needs to fetch the embedding from a hash table. GST+E has a small overhead from writing the updated embedding of \\(\\mathcal{G}_{s}^{(i)}\\) back into the table \\(\\mathcal{T}\\), which is relatively quick and can be run in a separate thread so that it does not impede the main training algorithm until the next iteration.\n\nThe drawback of GST+E is that historical embeddings from \\(\\mathcal{T}\\) may be stale;\n\n\\(\\tilde{\\mathbf{h}}_{j}^{(i)}\\) can be the result of an out-dated feature extractor \\(F\\). This type of staleness can hurt the training in various ways. Below, we provide two techniques to mitigate the staleness issue.\n\n### Prediction Head Finetuning\n\nLet's compare the input-output distribution of the prediction head \\(F^{\\prime}\\) during the training and inference stage. We have training distribution \\(\\mathcal{P}_{\\text{train}}(\\mathbf{h},y)=\\mathcal{P}(\\mathbf{h}_{s}\\bigoplus\\tilde{ \\mathbf{h}}_{j},y)\\) and test distribution \\(\\mathcal{P}_{\\text{test}}(\\mathbf{h},y)=\\mathcal{P}(\\bigoplus\\mathbf{h}_{j},y)\\). Regardless of the innate distribution shift between the training and test stage of the dataset, we note that stale historical embeddings can further widen the gap between the training and test distributions. In this case, the minimizer of the expected training loss does not minimize the expected test loss:\n\n\\[\\operatorname*{arg\\,min}_{\\theta}\\mathbb{E}_{(\\mathbf{h},y)\\sim\\mathcal{P}(\\mathbf{h}_{s} \\bigoplus\\tilde{\\mathbf{h}}_{j},y)}\\mathcal{L}(F^{\\prime}(\\mathbf{h}),y)\\neq \\operatorname*{arg\\,min}_{\\theta}\\mathbb{E}_{(\\mathbf{h},y)\\sim\\mathcal{P}( \\bigoplus h_{j},y)}\\mathcal{L}(F^{\\prime}(\\mathbf{h}),y)\\]\n\nTo mitigate the distribution misalignment, we introduce the Prediction Head Finetuning technique. Concretely, at the end of training, we update each embedding \\(\\mathbf{h}_{j}^{(i)}\\) in the embedding table \\(\\mathcal{T}\\) by forwarding each graph segment in the training set with the most current feature encoder \\(F\\). We then finetune only the prediction head \\(F^{\\prime}\\) with all the input embeddings up-to-date. We use GST+EF to denote GST+E refined with the Prediction Head Finetuning technique.\n\nThe overhead from the finetuning is minimal, as we need to update the embedding table \\(\\mathcal{T}\\) only once. The rest of the finetuning stage does not involve the notoriously slow graph convolution because the prediction head \\(F^{\\prime}\\) is simply a multi-layer perceptron.\n\n### Stale Embedding Dropout\n\nThe finetuning technique primarily addresses the negative impact of stale embeddings on prediction head \\(F^{\\prime}\\). However, the staleness also impacts the backbone \\(F\\). Prior works studying the effects of stale historical embeddings commonly assume that historical embeddings do not become too stale, _i.e._, \\(\\|\\tilde{\\mathbf{h}}_{j}^{(i)}-\\tilde{\\mathbf{h}}_{j}^{(i)}\\|\\leq\\epsilon,\\forall(i,j)\\). Given this assumption, if the neural network \\((F^{\\prime}\\circ F)(\\cdot)\\) is \\(k\\)-Lipschitz continuous, the gradients will also be bounded and never run too far from its true estimation, _i.e._, \\(\\|\\nabla F^{\\prime}(\\tilde{\\mathbf{h}}_{j}^{(i)})-\\nabla F^{\\prime}(\\tilde{\\mathbf{h} }_{j}^{(i)})\\|\\leq k^{\\prime}\\cdot\\epsilon\\). Thus, the network can often converge to the similar local minima even when using historical embeddings.\n\nThe above assumption does not hold under our GST+E framework. The rationale is that \\(\\tilde{\\mathbf{h}}_{j}^{(i)}\\) gets updated infrequently in the embedding table \\(\\mathcal{T}\\). Suppose we iterate through every graph \\(\\mathcal{G}^{(i)}\\) in the training set \\(\\mathcal{D}_{\\text{train}}\\) for each epoch, every time we train on a graph \\(\\mathcal{G}^{(i)}\\approx\\bigoplus\\mathcal{G}_{j}^{(i)}\\), only a few graph segments \\(\\mathcal{G}_{s}^{(i)}\\) will be updated in the table \\(\\mathcal{T}\\). This implies that all the other segment embeddings of \\(\\mathcal{G}^{(i)}\\) will be at least \\(n\\)-iteration stale, with \\(n\\) being the number of graphs in the training set, and the most outdated segment embedding could be approximately \\(nJ^{(i)}/S^{(i)}\\)-iteration stale, where \\(S^{(i)}\\) is \\(|\\mathcal{G}^{(i)}|\\).\n\nThis staleness introduces an additional source of bias and variance to the stochastic optimization; the loss function calculated with historical embeddings is no longer an unbiased estimation of its true value. To mitigate the negative impact of historical embeddings on loss function estimation, we propose the second technique, _Stale Embedding Dropout_ (SED). Unlike a standard Dropout, which uniformly drops elements and weighs up the rest, we propose to drop only stale segment embeddings and weigh up only segment embeddings that are up-to-date. Concretely, assume with the keep probability \\(p\\), the weight \\(\\eta\\) for each segment is defined as:\n\n\\[\\eta^{(i)}=\\begin{cases}p+(1-p)\\frac{J^{(i)}}{S^{(i)}}&\\text{for }\\mathcal{G}_{s}^{(i)}\\\\ 0&\\text{for }\\mathcal{G}_{j}^{(i)},\\text{with prob. }(1-p)\\\\ 1&\\text{for }\\mathcal{G}_{j}^{(i)},\\text{with prob. }p\\end{cases} \\tag{1}\\]\n\nPlease refer to the theoretical analysis in the next section. By combining the two proposed techniques, we denote our final algorithm as GST+EFD, which we summarized in Algorithm 2 in Appendix B.\n\n## 4 Theoretical Analysis\n\nWe characterize the effect of stale historical embeddings by studying the difference between \\(\\mathcal{L}(F^{\\prime}(\\mathbf{h}_{s}^{(i)}\\bigoplus\\tilde{\\mathbf{h}}_{j}^{(i)}))\\) and \\(\\mathcal{L}(F^{\\prime}(\\bigoplus\\mathbf{h}_{j}^{(i)}))\\). Assume \\(\\mathcal{G}^{(i)}\\) has \\(J^{(i)}\\) segments and we perform backpropagation on \\(S^{(i)}\\) segments. We let \\(\\delta^{(i)}\\triangleq\\mathbf{h}_{s}^{(i)}\\bigoplus\\tilde{\\mathbf{h}}_{j}^{(i)}- \\bigoplus\\mathbf{h}_{j}^{(i)}\\) be the perturbation on the graph embedding.\n\nWe apply Taylor expansion around \\(\\delta^{(i)}=0\\) on the final loss to analyze the effect of this perturbation.\n\n\\[\\mathbb{E}_{s}\\mathcal{L}(F^{\\prime}(\\mathbf{h}_{s}^{(i)}\\mathbf{\\bigoplus} \\tilde{\\mathbf{h}}_{j}^{(i)}))-\\mathcal{L}(F^{\\prime}(\\mathbf{\\bigoplus}\\mathbf{h}_{j}^{(i) })) \\tag{2}\\] \\[= \\mathbb{E}_{s}\\mathcal{L}(F^{\\prime}(\\mathbf{\\bigoplus}\\mathbf{h}_{j}^{(i) }+\\delta^{(i)}))-\\mathcal{L}(F^{\\prime}(\\mathbf{\\bigoplus}\\mathbf{h}_{j}^{(i)}))\\] \\[\\approx \\sum_{j}\\mathbb{E}_{\\delta_{j}^{(i)}}\\underbrace{D_{\\mathbf{h}_{j}^{( i)}}(\\mathcal{L}\\circ F^{\\prime})[\\mathbf{h}_{j}^{(i)}]\\delta_{j}^{(i)}}_{B}+ \\underbrace{\\frac{1}{2}{\\delta_{j}^{(i)}}^{\\top}(D_{\\mathbf{h}_{j}^{(i)}}^{2}( \\mathcal{L}\\circ F^{\\prime})[\\mathbf{h}_{j}^{(i)}])\\delta_{j}^{(i)}}_{R}\\]\n\nIn the equation above, the first-order term acts as a bias term introduced by the stale historical embedding, and the second-order term acts as an additional regularizer.\n\nLet ET denote using the embedding table without applying SED. We analyze the effect of ET and SED under the Taylor Expansion in Eq. 2 by substituting different \\(\\delta^{(i)}\\). For the first term, we have\n\n\\[\\mathbb{E}_{\\delta_{j}^{(i)\\text{ET}}}[B] =C\\times\\frac{J^{(i)}-S^{(i)}}{J^{(i)}}\\mathbb{E}(\\tilde{\\mathbf{h}}_ {j}^{(i)}-\\mathbf{h}_{j}^{(i)})\\] \\[\\mathbb{E}_{\\delta_{j}^{(i)\\text{SED}}}[B] =C\\times\\frac{J^{(i)}-S^{(i)}}{J^{(i)}}\\mathbb{E}(\\tilde{\\mathbf{h}}_ {j}^{(i)}-\\mathbf{h}_{j}^{(i)})p\\]\n\nwhere \\(C\\) is a constant matrix.\n\nWe extend the above analysis to the following theorem. Please find the complete proof in Appendix A.\n\n**Theorem 4.1**.: _Under proper conditions, SED with a keep ratio \\(p\\) ensures to reduce bias term introduced by historical embeddings by a factor of \\(p\\), while introducing another regularization term._\n\nTheorem 4.1 indicates that SED can reduce the bias in the loss function introduced by the stale historical embeddings, at the cost of another regularization term, which might potentially increase total regularization. Prior works commonly make a hidden assumption that \\(\\mathbb{E}(\\tilde{\\mathbf{h}}_{j}^{(i)}-\\mathbf{h}_{j}^{(i)})\\) is so small that the negative effect may be neglected. In our setting, the historical embeddings can be very stale, so having a \\(p\\) factor helps reduce bias. It is worthwhile to check the limiting cases. If \\(p=1\\) (keeping all the stale embeddings without dropping any), both \\(\\mathbb{E}_{\\delta_{j}^{(i)\\text{SED}}}[B]\\) and \\(\\mathbb{E}_{\\delta_{j}^{(i)\\text{SED}}}[R]\\) degrade to the result of ET. If \\(p=0\\) (dropping all the stale embeddings), then the algorithm degrades to training on only \\(S^{(i)}\\) segments without aggregating other segments (which we denote as GST-One, when \\(S^{(i)}=1\\)). \\(\\mathbb{E}_{\\delta_{j}^{(i)\\text{SED}}}[B]=0\\) indicates that there is no stale bias in this case. However, the term \\(\\mathbb{E}_{\\delta_{j}^{(i)\\text{SED}}}[R]\\) could become too large so that it impedes training.\n\n## 5 Experiments\n\n### Experimental Setup\n\n**Datasets.** MalNet [10] is a large-scale graph representation learning dataset, with the goal to predict the category of a function call graph. MalNet is the largest public graph database constructed to date in terms of average graph size. Its widely-used split is called _MalNet-Tiny_, containing 5,000 graphs across balanced 5 types, with each graph containing at most 5,000 nodes. To evaluate our approach on the regime where the graph size is large, we construct an alternative split from the original MalNet dataset, which we named _MalNet-Large_. _MalNet-Large_ also contains 5,000 graphs across balanced 5 types. _MalNet-Large_'s average graph size reaches 47k with the largest graph containing 541k nodes. We will release our experimental split for _MalNet-Large_ to promote future research.\n\nTpuGraphs is an internal large scale graph regression dataset, whose goal is to predict an execution time of an XLA's HLO graph with a specific compiler configuration on a Tensor Processing Unit (TPU). XLA [28] is a production backend compiler for various machine learning frameworks, including TensorFlow, PyTorch, and JAX. In this particular dataset, the compiler configuration controls physical layouts of tensors in the graph, and the runtime is measured on TPU v3 [15]. This dataset cares more about the ranking of the configurations for each graph than the absolute runtimes, since the ultimate goal is to use a model to select the best configuration for each graph.4\nFootnote 4: We have made public a dataset [24] that closely parallels our internal dataset.\n\n### The _MalNet-Large_\n\nThe _MalNet-Large_ is a graph representation learning dataset, with the goal to predict the category of a function call graph. MalNet is the largest public graph database constructed to date in terms of average graph size. Its widely-used split is called _MalNet-Tiny_, containing 5,000 graphs across balanced 5 types, with each graph containing at most 5,000 nodes. To evaluate our approach on the regime where the graph size is large, we construct an alternative split from the original MalNet dataset, which we named _MalNet-Large_. _MalNet-Large_ also contains 5,000 graphs across balanced 5 types. _MalNet-Large_'s average graph size reaches 47k with the largest graph containing 541k nodes. We will release our experimental split for _MalNet-Large_ to promote future research.\n\nTpuGraphs is an internal large scale graph regression dataset, whose goal is to predict an execution time of an XLA's HLO graph with a specific compiler configuration on a Tensor Processing Unit (TPU). XLA [28] is a production backend compiler for various machine learning frameworks, including TensorFlow, PyTorch, and JAX. In this particular dataset, the compiler configuration controls physical layouts of tensors in the graph, and the runtime is measured on TPU v3 [15]. This dataset cares more about the ranking of the configurations for each graph than the absolute runtimes, since the ultimate goal is to use a model to select the best configuration for each graph.\n\nis similar to the dataset used in [17], but the runtime prediction is at the entire graph level rather than at the kernel (subgraph) level. TpuGraphs contains 5,153 HLO graphs and a total of 757,375 unique pairs of graphs and configurations. The average graph size is 38k, and the maximum is 615k. From the perspective of GST, a graph together with a configuration defines one \\(\\mathcal{G}^{(i)}\\) because the configuration is featurized as parts of input node features to the GNN.\n\nPlease refer to Table 4 in Appendix for detailed statistics.\n\n**Methods.** We test combinations of the following proposed techniques and some baselines. (1) Full Graph Training: we train on all graphs in their original scale without applying any partitioning beforehand. (2) GST-One: we partition the original graph into a collection of graph segments \\(\\mathcal{G}^{(i)}\\approx\\bigoplus\\mathcal{G}^{(i)}_{j}\\), but we randomly select only one segment \\(\\mathcal{G}^{(i)}_{j}\\) for each graph to train every iteration. (3) GST: following the general GST framework described in Algorithm 1, we replace \\(\\text{ProduceEmbedding}(\\cdot)\\) by using the same feature encoder \\(F\\) to forward all the segments in \\(\\{\\mathcal{G}^{(i)}_{j}\\}_{j\\notin\\mathcal{S}^{(i)}}\\) without storing any intermediate activation. We set \\(S^{(i)}=1\\) in our experiments. (4) E: we introduce an embedding table \\(\\tilde{\\mathbf{h}}^{(i)}_{j}=\\mathcal{T}(i,j)\\) to store the historical embedding of each graph segment, and we fetch the embedding from \\(\\mathcal{T}\\) if we do not need to calculate gradient for the corresponding segment. (5) F: in addition to introducing the embedding table \\(\\mathcal{T}\\), we finetune the prediction head \\(F^{\\prime}\\) with all up-to-date segment embeddings at the end of training. (6) D: we apply SED defined in Eq. 1 during training.\n\nWhen these techniques are combined, we concatenate the acronyms with a \"+\" to GST as an abbreviation. We conduct all the experiments on MalNet with a single NVIDIA-V100 GPU with 16GB of memory, and four NVIDIA-V100 GPUs (for data parallelism) with 16GB of memory for TpuGraphs. Please refer to Appendix B for additional implementation details.\n\n### Empirical Results on MalNet\n\nTo demonstrate the general applicability of our proposed GST framework, we consider three backbones, namely, GCN [19], SAGE [11], and GraphGPS [25]. GCN and SAGE are two popular GNN architectures. GraphGPS is a Graph Transformer that recently achieves state-of-the-art performance on many graph-level tasks, but is well-known for its issue on scalability. We report the top-1 test accuracy of various methods on MalNet-Tiny and MalNet-Large in Table 1. We include MalNet-Tiny in this study because its graphs are relatively small so that it is still possible to run Full Graph Training.\n\nNotably, we observe that GST slightly outperforms Full Graph Training in terms of test accuracy on MalNet-Tiny. GST has exactly the same number of weight parameters with Full Graph Training. This implies that GST potentially has a better hierarchical graph pooling mechanism that leads to better generalization. As we step from MalNet-Tiny to MalNet-Large, Full Graph Training strategy can no longer fit the large graphs on a GPU, so we report OOM in the table. GST's estimation on graph segment embeddings \\(\\tilde{\\mathbf{h}}^{(i)}_{j}\\) that do not require gradients is accurate, and thus does not suffer from staleness issues. Therefore, we use GST as an estimation for the performance of Full Graph Training on MalNet-Large.\n\nNaively training on only one graph segment (GST-One) yields inferior performance than Full Graph Training and GST, showing that it is essential to aggregate embeddings from all graph segments.\n\n\\begin{table}\n\\begin{tabular}{c|c c c|c c c} \\hline \\hline Dataset & \\multicolumn{3}{c|}{MalNet-Tiny} & \\multicolumn{3}{c}{MalNet-Large} \\\\ Backbone & GCN & SAGE & GraphGPS & GCN & SAGE & GraphGPS \\\\ \\hline Full Graph Training & 87.84\\(\\pm\\)1.37 & 88.08\\(\\pm\\)1.68 & 90.82\\(\\pm\\)0.59 & OOM & OOM & OOM \\\\ GST & 88.26\\(\\pm\\)0.80 & 88.42\\(\\pm\\)1.03 & 91.03\\(\\pm\\)0.81 & 88.35\\(\\pm\\)1.14 & 88.62\\(\\pm\\)0.82 & 91.39\\(\\pm\\)0.85 \\\\ GST-One & 71.62\\(\\pm\\)3.85 & 72.64\\(\\pm\\)4.73 & 77.63\\(\\pm\\)3.15 & 60.41\\(\\pm\\)6.29 & 57.13\\(\\pm\\)7.36 & 66.82\\(\\pm\\)4.71 \\\\ GST+E & 86.53\\(\\pm\\)1.18 & 86.82\\(\\pm\\)0.93 & 89.75\\(\\pm\\)0.89 & 48.42\\(\\pm\\)6.61 & 43.28\\(\\pm\\)7.01 & 62.47\\(\\pm\\)3.19 \\\\ GST+EF & 87.67\\(\\pm\\)0.78 & 87.83\\(\\pm\\)0.81 & 90.52\\(\\pm\\)0.71 & 84.83\\(\\pm\\)0.96 & 85.26\\(\\pm\\)0.87 & 91.33\\(\\pm\\)0.65 \\\\ GST+ED & 88.18\\(\\pm\\)0.48 & 88.50\\(\\pm\\)0.74 & 90.96\\(\\pm\\)0.68 & 82.17\\(\\pm\\)4.74 & 71.83\\(\\pm\\)6.31 & 89.46\\(\\pm\\)1.36 \\\\ GST+EFD & 88.78\\(\\pm\\)0.45 & 89.24\\(\\pm\\)0.53 & 92.46\\(\\pm\\) 0.66 & 89.67\\(\\pm\\)0.71 & 89.78\\(\\pm\\)0.68 & 92.52\\(\\pm\\)0.58 \\\\ \\hline \\hline \\end{tabular}\n\\end{table}\nTable 1: Test accuracy on MalNet-Tiny and MalNet-Large. We report the standard deviation over five runs. GST+EFD achieves better accuracy than Full Graph Training, and GST, while being much more memory efficient and computationally faster.\n\nSolely introducing the historical embedding table (GST+E) significantly deteriorates the optimization due to the staleness issue. Each of the proposed techniques (Prediction Head Finetuning and SED) individually is beneficial in combating the staleness issue. The combination of our two techniques (GST+EFD) achieves the best performance, slightly outperforming GST by another 1-2% in terms of final evaluation metric.\n\n### Empirical results on TpuGraphs dataset\n\nAs mentioned in Section 5.1, we care more about the ranking of configurations for each graph than the absolute runtimes in this dataset. Thus, we report the ordered pair accuracy (OPA) averaged over all computational graphs in Table 2. OPA metric is defined as:\n\n\\[\\text{OPA}(y,\\widehat{y})=\\frac{\\sum_{i}\\sum_{j}\\mathbb{I}[\\widehat{y}_{i}> \\widehat{y}_{j}]\\cdot\\mathbb{I}[y_{i}>y_{j}]}{\\sum_{i}\\sum_{j}\\mathbb{I}[y_{i} >y_{j}]}\\]\n\nWith some compiler domain knowledge, we found it better to predict each graph segment's runtime first and then aggregate them using sum pooling. This means the prediction head is part of \\(F\\), and \\(F^{\\prime}\\) is simply a summation function. Since there are no learnable weights in \\(F^{\\prime}\\), we omit Prediction Head Finetuning in this experiment, and GST+EFD in Table 2 excludes the finetuning stage. We observe a clear tradeoff between fitting training examples and generalization. First, GST has a much higher training OPA than the other methods, indicating accurate estimation on segments that do not require gradient is essential for training OPA. Training with one segment only or with the embedding table yields lower training OPA, and consequently lower test OPA as well. SED in GST+EFD functions as a regularization technique. Although it slightly lowers the training OPA compared to GST+E, it achieves better test OPA, even than GST, due to bias mitigation.\n\n### Ablation Studies\n\n**Effect of finetuning.** We visualize the training/test accuracy curve of GST+EFD over time in Figure 2. The staleness introduced by historical embeddings drastically hurts generalization, as shown for the first 600 epochs. We start finetuning at epoch 600, and the gap between training and test accuracy decreases by a large margin instantly.\n\n\\begin{table}\n\\begin{tabular}{c|c c} \\hline \\hline  & Train OPA & Test OPA \\\\ \\hline Full Graph Training & OOM & OOM \\\\ GST & 87.43 & 86.84 \\\\ GST-One & 67.19 & 81.03 \\\\ GST+E & 74.02 & 82.62 \\\\ GST+EFD & 71.40 & 89.20 \\\\ \\hline \\hline \\end{tabular}\n\\end{table}\nTable 2: Training and test ordered pair accuracy Table 3: Runtime analysis (average training time per iteration in ms) on MalNet-Large dataset.\n\n**Ablation study on segment dropout ratio.** To analyze the effect of the keep ratio \\(p\\) in SED, we vary its value from 0 to 1 and visualize the results in Figure 3. When \\(p=1\\), GST+EFD degrades back to using the historical embedding table without SED, as the performance decreases due to staleness. When \\(p=0\\), GST+EFD becomes GST-One, where we drop all the stale historical embeddings. This extreme case introduces too heavy regularization that impedes the model from fitting the training data, leading to a decrease in test performance ultimately. We found that \\(p=0.5\\) achieves a satisfactory tradeoff between fitting the training data and adding a proper amount of regularization.\n\n**Ablation study on segment size.** We also alter the maximum segment size and visualize the results in Figure 4. A smaller maximum segment size will result in much more number of segments. Interestingly, we found that the proposed GST+EFD is very robust to the choice of the maximum segment size, as long as the segment size is reasonably large.\n\n**Ablation study on partition algorithms.** Please refer to Appendix C.\n\n### Runtime Analysis\n\nNext, we empirically compare runtime of different variants under the proposed GST framework. We summarize an average time for one forward-backward pass during training on MalNet-Large dataset in Table 3. Since GST runs inference for the graph segments that do not require gradients, the runtime of GST is significantly higher than others'. We also found that GST+E's and GST+EFD's runtime are very close to GST-One's; this means the overhead of fetching embeddings from the embedding table \\(\\mathcal{T}\\) is minimal. Moreover, GST+EFD's runtime is slightly lower than GST+E's because in the implementation, we can skip the fetching process if an embedding is set to be dropped. This result demonstrates that our proposed GST+EFD not only is efficient in terms of memory usage but also reduces training time significantly.\n\n## 6 Related Works\n\n**Graph property prediction.** In the context of graph property prediction, a model must predict a certain characteristic associated with the whole graph. Standard graph neural networks produce node embeddings as outputs [19; 11; 34]. To create a graph embedding (a vector representing the entire graph) out of the node embeddings, pooling methods are usually applied at the end. Common approaches to this problem include simply summing up or averaging all the node embeddings [8], or introducing a \"virtual node\" connected to all the other nodes [23]. Fully-connected Graph Transformer was recently proposed with an outstanding success on existing graph property prediction benchmarks [35; 25]. However, the fully-connected attention matrix limits the applicability of Graph Transformer to only small graphs with limited number of nodes [26].\n\n**GNN with graph partitioning.** ClusterGCN [7] is designed for node-level tasks by training on graph segments. It partitions a graph into graph segments and randomly selects graph segments to form a minibatch during training. ROC [14], PipeGCN [29] and BNS-GCN [30] achieve distributed node-level GCN training through partitioning a graph into small segments such that each could be fitted into a single GPU memory, and then training multiple segments in parallel. All the above graph partitioning techniques for GNNs rely on the fact that an ego-subgraph (a subgraph centered around a node) contains sufficient information to make a prediction for the centered node. This is not true for graph property prediction tasks where we need to aggregate information from the whole graph to make an accurate prediction.\n\n**GNN with historical embeddings.** The idea of historical embeddings was first introduced in VR-GCN [4], which uses historical embeddings to control the variance of neighborhood sampling. GNNAutoScale [9] incorporates historical embeddings to recover a more accurate neighborhood estimation in a scalable fashion. Developed upon GNNAutoScale, Yu et al. [37] uses a momentum step to incorporate historical embeddings when updating feature representations to further alleviate the staleness issue. These prior works maintain a historical embedding for each node because they consider node-level tasks. As we consider graph property prediction tasks, we record a historical embedding for each graph segment rather than each node.\n\nConclusion\n\nWe study how to train a GNN model for large graph property prediction tasks. We propose Graph Segment Training (GST), a general framework for learning large graph property prediction tasks with a constant memory footprint. We further introduce a historical embedding table to efficiently produce embeddings for graph segments that do not require gradients, and design two novel techniques -- Prediction Head Finetuning and Stale Embedding Dropout -- to mitigate the staleness issue. In conclusion, we believe that our proposed method is a step toward making graph property prediction learning more practical and scalable.\n\n## Acknowledgements\n\nKC and JL acknowledge the support of DARPA under Nos. HR00112190039 (TAMI), N660011924033 (MCS); ARO under Nos. W911NF-16-1-0342 (MURI), W911NF-16-1-0171 (DURIP); NSF under Nos. OAC-1835598 (CINES), OAC-1934578 (HDR), CCF-1918940 (Expeditions), NIH under No. 3U54HG010426-04S1 (HuBMAP), Stanford Data Science Initiative, Wu Tsai Neurosciences Institute, Amazon, Docomo, GSK, Hitachi, Intel, JPMorgan Chase, Juniper Networks, KDDI, NEC, and Toshiba. CM's contributions were partially supported by ACE, one of the seven centers in JUMP 2.0, a Semiconductor Research Corporation (SRC) program sponsored by DARPA and NSF under grant CCF-2316233. The content is solely the responsibility of the authors and does not necessarily represent the official views of the funding entities.\n\n## References\n\n* [1] Miltiadis Allamanis. The adverse effects of code duplication in machine learning models of code. In _Proceedings of the 2019 ACM SIGPLAN International Symposium on New Ideas, New Paradigms, and Reflections on Programming and Software_, pages 143-153, 2019.\n* [2] Aleksandar Bojchevski, Johannes Klicpera, Bryan Perozzi, Amol Kapoor, Martin Blais, Benedek Rozemberczki, Michal Lukasik, and Stephan Gunnemann. Scaling graph neural networks with approximate pagerank. In _Proceedings of the 26th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining_, pages 2464-2473, 2020.\n* [3] Ines Chami, Sami Abu-El-Haija, Bryan Perozzi, Christopher Re, and Kevin Murphy. Machine learning on graphs: A model and comprehensive taxonomy. _Journal of Machine Learning Research_, 23(89):1-64, 2022.\n* [4] Jianfei Chen, Jun Zhu, and Le Song. Stochastic training of graph convolutional networks with variance reduction. In _International Conference on Machine Learning_, pages 942-950. PMLR, 2018.\n* [5] Jie Chen, Tengfei Ma, and Cao Xiao. Fastgcn: Fast learning with graph convolutional networks via importance sampling. In _International Conference on Learning Representations_, 2018.\n* [6] Tianqi Chen, Lianmin Zheng, Eddie Yan, Ziheng Jiang, Thierry Moreau, Luis Ceze, Carlos Guestrin, and Arvind Krishnamurthy. Learning to Optimize Tensor Programs. In _Proceedings of the 32nd International Conference on Neural Information Processing Systems_, NeurIPS'18, page 3393-3404, Red Hook, NY, USA, 2018. Curran Associates Inc.\n* [7] Wei-Lin Chiang, Xuanqing Liu, Si Si, Yang Li, Samy Bengio, and Cho-Jui Hsieh. Cluster-gcn: An efficient algorithm for training deep and large graph convolutional networks. In _Proceedings of the 25th ACM SIGKDD international conference on knowledge discovery & data mining_, pages 257-266, 2019.\n* [8] Vijay Prakash Dwivedi, Chaitanya K Joshi, Thomas Laurent, Yoshua Bengio, and Xavier Bresson. Benchmarking graph neural networks. _arXiv preprint arXiv:2003.00982_, 2020.\n* [9] Matthias Fey, Jan E Lenssen, Frank Weichert, and Jure Leskovec. Gnnautoscale: Scalable and expressive graph neural networks via historical embeddings. In _International Conference on Machine Learning_, pages 3294-3304. PMLR, 2021.\n\n* Freitas et al. [2021] Scott Freitas, Yuxiao Dong, Joshua Neil, and Duen Horng Chau. A large-scale database for graph representation learning. In _Thirty-fifth Conference on Neural Information Processing Systems Datasets and Benchmarks Track (Round 1)_, 2021.\n* Hamilton et al. [2017] Will Hamilton, Zhitao Ying, and Jure Leskovec. Inductive representation learning on large graphs. _Advances in neural information processing systems_, 30, 2017.\n* Hu et al. [2020] Weihua Hu, Matthias Fey, Marinka Zitnik, Yuxiao Dong, Hongyu Ren, Bowen Liu, Michele Catasta, and Jure Leskovec. Open graph benchmark: Datasets for machine learning on graphs. _Advances in neural information processing systems_, 33:22118-22133, 2020.\n* Huang et al. [2018] Wenbing Huang, Tong Zhang, Yu Rong, and Junzhou Huang. Adaptive sampling towards fast graph representation learning. _Advances in neural information processing systems_, 31, 2018.\n* Jia et al. [2020] Zhihao Jia, Sina Lin, Mingyu Gao, Matei Zaharia, and Alex Aiken. Improving the accuracy, scalability, and performance of graph neural networks with roc. _Proceedings of Machine Learning and Systems_, 2:187-198, 2020.\n* Jouppi et al. [2020] Norman P. Jouppi, Doe Hyun Yoon, George Kurian, Sheng Li, Nishant Patil, James Laudon, Cliff Young, and David Patterson. A Domain-Specific Supercomputer for Training Deep Neural Networks. _Commun. ACM_, 63(7):67-78, June 2020. ISSN 0001-0782. doi: 10.1145/3360307.\n* Karypis and Kumar [1997] George Karypis and Vipin Kumar. Metis: A software package for partitioning unstructured graphs, partitioning meshes, and computing fill-reducing orderings of sparse matrices. 1997.\n* Kaufman et al. [2021] Samuel J. Kaufman, Phitchaya Mangpo Phothilimthana, Yanqi Zhou, Charith Mendis, Sudip Roy, Amit Sabne, and Mike Burrows. A Learned Performance Model for Tensor Processing Units. In _Proceedings of Machine Learning for Systems_, 2021.\n* Kingma and Ba [2014] Diederik P Kingma and Jimmy Ba. Adam: A method for stochastic optimization. _arXiv preprint arXiv:1412.6980_, 2014.\n* Kipf and Welling [2016] Thomas N Kipf and Max Welling. Semi-supervised classification with graph convolutional networks. 2016.\n* Loshchilov and Hutter [2017] Ilya Loshchilov and Frank Hutter. Decoupled weight decay regularization. _arXiv preprint arXiv:1711.05101_, 2017.\n* Markowitz et al. [2021] Elan Sopher Markowitz, Keshav Balasubramanian, Mehrnoosh Mirtaheri, Sami Abu-El-Haija, Bryan Perozzi, Greg Ver Steeg, and Aram Galstyan. Graph traversal with tensor functionals: A meta-algorithm for scalable learning. In _International Conference on Learning Representations_, 2021.\n* Paszke et al. [2017] Adam Paszke, Sam Gross, Soumith Chintala, Gregory Chanan, Edward Yang, Zachary DeVito, Zeming Lin, Alban Desmaison, Luca Antiga, and Adam Lerer. Automatic differentiation in pytorch. 2017.\n* Pham et al. [2017] Trang Pham, Truyen Tran, Hoa Dam, and Svetha Venkatesh. Graph classification via deep learning with virtual nodes. _arXiv preprint arXiv:1708.04357_, 2017.\n* Phothilimthana et al. [2023] Phitchaya Mangpo Phothilimthana, Sami Abu-El-Haija, Kaidi Cao, Bahare Fatemi, Charith Mendis, and Bryan Perozzi. Tpugraphs: A performance prediction dataset on large tensor computational graphs. _arXiv preprint arXiv:2308.13490_, 2023.\n* Rampasek et al. [2022] Ladislav Rampasek, Mikhail Galkin, Vijay Prakash Dwivedi, Anh Tuan Luu, Guy Wolf, and Dominique Beaini. Recipe for a general, powerful, scalable graph transformer. _arXiv preprint arXiv:2205.12454_, 2022.\n* Shi et al. [2022] Yu Shi, Shuxin Zheng, Guolin Ke, Yifei Shen, Jiacheng You, Jiyan He, Shengjie Luo, Chang Liu, Di He, and Tie-Yan Liu. Benchmarking graphormer on large-scale molecular modeling datasets. _arXiv preprint arXiv:2203.04810_, 2022.\n\n* Szklarczyk et al. [2019] Damian Szklarczyk, Annika L Gable, David Lyon, Alexander Junge, Stefan Wyder, Jaime Huerta-Cepas, Milan Simonovic, Nadezhda T Doncheva, John H Morris, Peer Bork, et al. String v11: protein-protein association networks with increased coverage, supporting functional discovery in genome-wide experimental datasets. _Nucleic acids research_, 47(D1):D607-D613, 2019.\n* [28] TensorFlow. XLA: Optimizing Compiler for TensorFlow. [https://www.tensorflow.org/xla](https://www.tensorflow.org/xla). [Online; accessed 19-September-2019].\n* Wan et al. [2021] Cheng Wan, Youjie Li, Cameron R Wolfe, Anastasios Kyrillidis, Nam Sung Kim, and Yingyan Lin. Pipegcn: Efficient full-graph training of graph convolutional networks with pipelined feature communication. In _International Conference on Learning Representations_, 2021.\n* Wan et al. [2022] Cheng Wan, Youjie Li, Ang Li, Nam Sung Kim, and Yingyan Lin. Bns-gcn: Efficient full-graph training of graph convolutional networks with partition-parallelism and random boundary node sampling. _Proceedings of Machine Learning and Systems_, 4:673-693, 2022.\n* Wei et al. [2020] Colin Wei, Sham Kakade, and Tengyu Ma. The implicit and explicit regularization effects of dropout. In _International conference on machine learning_, pages 10181-10192. PMLR, 2020.\n* Wu et al. [2018] Zhenqin Wu, Bharath Ramsundar, Evan N Feinberg, Joseph Gomes, Caleb Geniesse, Aneesh S Pappu, Karl Leswing, and Vijay Pande. Moleculenet: a benchmark for molecular machine learning. _Chemical science_, 9(2):513-530, 2018.\n* Xie et al. [2014] Cong Xie, Ling Yan, Wu-Jun Li, and Zhihua Zhang. Distributed power-law graph computing: Theoretical and empirical analysis. _Advances in neural information processing systems_, 27, 2014.\n* Xu et al. [2018] Keyulu Xu, Weihua Hu, Jure Leskovec, and Stefanie Jegelka. How powerful are graph neural networks? In _International Conference on Learning Representations_, 2018.\n* Ying et al. [2021] Chengxuan Ying, Tianle Cai, Shengjie Luo, Shuxin Zheng, Guolin Ke, Di He, Yanming Shen, and Tie-Yan Liu. Do transformers really perform badly for graph representation? _Advances in Neural Information Processing Systems_, 34:28877-28888, 2021.\n* You et al. [2020] Jiaxuan You, Zhitao Ying, and Jure Leskovec. Design space for graph neural networks. _Advances in Neural Information Processing Systems_, 33:17009-17021, 2020.\n* Yu et al. [2022] Haiyang Yu, Limei Wang, Bokun Wang, Meng Liu, Tianbao Yang, and Shuiwang Ji. Graphfm: Improving large-scale gnn training via feature momentum. In _International Conference on Machine Learning_, pages 25684-25701. PMLR, 2022.\n* Zeng et al. [2019] Hanqing Zeng, Hongkuan Zhou, Ajitesh Srivastava, Rajgopal Kannan, and Viktor Prasanna. Graphsaint: Graph sampling based inductive learning method. In _International Conference on Learning Representations_, 2019.\n* Zhang et al. [2017] Chenzi Zhang, Fan Wei, Qin Liu, Zhihao Gavin Tang, and Zhenguo Li. Graph edge partitioning via neighborhood heuristic. In _Proceedings of the 23rd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining_, pages 605-614, 2017.\n* Zhang et al. [2022] Hengrui Zhang, Zhongming Yu, Guohao Dai, Guyue Huang, Yufei Ding, Yuan Xie, and Yu Wang. Understanding gnn computational graph: A coordinated computation, io, and memory perspective. _Proceedings of Machine Learning and Systems_, 4:467-484, 2022.\n* Zheng et al. [2021] Lianmin Zheng, Ruochen Liu, Junru Shao, Tianqi Chen, Joseph E. Gonzalez, Ion Stoica, and Ameer Haj Ali. Tenset: A large-scale program performance dataset for learned tensor compilers. In _Thirty-fifth Conference on Neural Information Processing Systems Datasets and Benchmarks Track (Round 1)_, 2021. URL [https://openreview.net/forum?id=aIfp8kLuvc9](https://openreview.net/forum?id=aIfp8kLuvc9).\n* Zitnik et al. [2019] Marinka Zitnik, Rok Sosic, Marcus W Feldman, and Jure Leskovec. Evolution of resilience in protein interactomes across the tree of life. _Proceedings of the National Academy of Sciences_, 116(10):4426-4433, 2019.\n* Zou et al. [2019] Difan Zou, Ziniu Hu, Yewen Wang, Song Jiang, Yizhou Sun, and Quanquan Gu. Layer-dependent importance sampling for training deep and large graph convolutional networks. _Advances in neural information processing systems_, 32, 2019.\n\nProofs and Derivations\n\n**Theorem A.1**.: _Under proper a condition that \\(W\\cdot\\tilde{\\mathbf{h}}_{j}^{(i)}\\approx 0\\) and \\(W\\cdot\\mathbf{h}_{j}^{(i)}\\approx 0\\), where \\(W\\) is the first linear transformation in \\(F^{\\prime}\\), SED with a keep ratio \\(p\\) ensures to reduce bias term introduced by historical embeddings by a factor of \\(p\\), while introducing another regularization term._\n\nProof.: Let \\(\\delta^{(i)}\\triangleq\\mathbf{h}_{s}^{(i)}\\bigoplus\\tilde{\\mathbf{h}}_{j}^{(i)}- \\bigoplus\\mathbf{h}_{j}^{(i)}\\) be the perturbation on the graph embedding. We use ET to denote using the embedding table without applying SED. For GST+E, we have\n\n\\[\\delta_{j}^{(i)^{\\text{ET}}}=\\begin{cases}0&\\text{with prob. }\\frac{S^{(i)}}{J^{(i)}}\\\\ \\tilde{\\mathbf{h}}_{j}^{(i)}-\\mathbf{h}_{j}^{(i)}&\\text{with prob. }\\frac{J^{(i)}-S^{(i)}}{J^{(i)}} \\end{cases}\\]\n\nThe randomness above comes from the fact that each segment \\(\\mathcal{G}_{j}^{(i)}\\) is selected for backpropagation with probability \\(\\frac{S^{(i)}}{J^{(i)}}\\).\n\nFor SED, there are two folds of randomness when training on graph \\(\\mathcal{G}^{(i)}\\): randomly selecting \\(S^{(i)}\\) segments to train and randomly select stale embedding \\(\\tilde{\\mathbf{h}}_{j}^{(i)}\\) to drop. Thus we can rewrite \\(\\delta_{j}^{(i)}\\) as\n\n\\[\\delta_{j}^{(i)^{\\text{SED}}}=\\begin{cases}\\frac{(1-p)(J^{(i)}-S^{(i)})}{S^{( i)}}\\mathbf{h}_{j}^{(i)}&\\text{with prob. }\\frac{S^{(i)}}{J^{(i)}}\\\\ -\\mathbf{h}_{j}^{(i)}&\\text{with prob. }\\frac{(1-p)(J^{(i)}-S^{(i)})}{J^{(i)}}\\\\ \\tilde{\\mathbf{h}}_{j}^{(i)}-\\mathbf{h}_{j}^{(i)}&\\text{with prob. }\\frac{p(J^{(i)}-S^{(i)})}{J^{(i)}} \\end{cases}\\]\n\nWe apply Taylor expansion around \\(\\delta_{j}^{(i)}=0\\) on the final loss to analyze the effect of this perturbation. In Section A.2 of Wei et al. [31], when \\(W\\cdot\\tilde{\\mathbf{h}}_{j}^{(i)}\\approx 0\\) and \\(W\\cdot\\mathbf{h}_{j}^{(i)}\\approx 0\\), the perturbation of \\(\\delta^{(i)}\\) to the loss function might not be too large, so it supports the use of Taylor Expansion.\n\n\\[\\mathcal{L}(F^{\\prime}(\\mathbf{h}_{s}^{(i)}\\bigoplus\\tilde{\\mathbf{h}}_{j }^{(i)}))-\\mathcal{L}(F^{\\prime}(\\bigoplus\\mathbf{h}_{j}^{(i)}))\\] \\[= \\mathcal{L}(F^{\\prime}(\\bigoplus(\\mathbf{h}_{j}^{(i)}+\\delta_{j}^{(i) })))-\\mathcal{L}(F^{\\prime}(\\bigoplus\\mathbf{h}_{j}^{(i)}))\\] \\[\\approx \\sum_{j}D_{\\mathbf{h}_{j}^{(i)}}(\\mathcal{L}\\circ F^{\\prime})[\\mathbf{h}_ {j}^{(i)}]\\delta_{j}^{(i)}+\\frac{1}{2}{\\delta_{j}^{(i)}}^{\\top}(D_{\\mathbf{h}_{j} ^{(i)}}^{2}(\\mathcal{L}\\circ F^{\\prime})[\\mathbf{h}_{j}^{(i)}])\\delta_{j}^{(i)}\\]\n\nNote that we randomly select segments with index \\(s\\) during training, we can then derive an approximation of the expected difference during training as\n\n\\[\\mathbb{E}_{s}\\mathcal{L}(F^{\\prime}(\\mathbf{h}_{s}^{(i)}\\bigoplus \\tilde{\\mathbf{h}}_{j}^{(i)}))-\\mathcal{L}(F^{\\prime}(\\bigoplus\\mathbf{h}_{j}^{(i)})) \\tag{3}\\] \\[\\approx \\sum_{j}\\mathbb{E}_{\\delta_{j}^{(i)}}\\underbrace{D_{\\mathbf{h}_{j}^ {(i)}}(\\mathcal{L}\\circ F^{\\prime})[\\mathbf{h}_{j}^{(i)}]\\delta_{j}^{(i)}}_{B}+ \\underbrace{1}_{2}{\\delta_{j}^{(i)}}^{\\top}(D_{\\mathbf{h}_{j}^{(i)}}^{2}(\\mathcal{ L}\\circ F^{\\prime})[\\mathbf{h}_{j}^{(i)}])\\delta_{j}^{(i)}}_{R}\\]\n\nWe can then compare the effect of SED by substituting the two versions of \\(\\delta_{j}^{(i)}\\) into Eq. 3.\n\nSo for the first term, we have\n\n\\[\\mathbb{E}_{\\delta_{j}^{(i)}\\text{ET}}[B] =\\langle D_{\\mathbf{h}_{j}^{(i)}}(\\mathcal{L}\\circ F^{\\prime})[\\mathbf{h} _{j}^{(i)}],\\mathbb{E}\\delta_{j}^{(i)}\\rangle\\] \\[=\\langle D_{\\mathbf{h}_{j}^{(i)}}(\\mathcal{L}\\circ F^{\\prime})[\\mathbf{h} _{j}^{(i)}],\\frac{J^{(i)}-S^{(i)}}{J^{(i)}}\\mathbb{E}(\\tilde{\\mathbf{h}}_{j}^{(i)}- \\mathbf{h}_{j}^{(i)})\\rangle\\] \\[\\mathbb{E}_{\\delta_{j}^{(i)}\\text{sExp}}[B] =\\langle D_{\\mathbf{h}_{j}^{(i)}}(\\mathcal{L}\\circ F^{\\prime})[\\mathbf{h} _{j}^{(i)}],\\mathbb{E}\\delta_{j}^{(i)}\\rangle\\] \\[=\\langle D_{\\mathbf{h}_{j}^{(i)}}(\\mathcal{L}\\circ F^{\\prime})[\\mathbf{h} _{j}^{(i)}],\\frac{J^{(i)}-S^{(i)}}{J^{(i)}}\\mathbb{E}(\\tilde{\\mathbf{h}}_{j}^{(i)}- \\mathbf{h}_{j}^{(i)})*p\\rangle\\]whereas for the second term, we have\n\n\\[\\mathbb{E}_{\\delta_{j}^{(i)\\text{F}}}[R]= \\langle D_{\\mathbf{h}_{j}^{(i)}}^{2}(\\mathcal{L}\\circ F^{\\prime})[\\mathbf{h }_{j}^{(i)}],\\frac{\\mathbb{E}\\delta_{j}^{(i)}\\delta_{j}^{(i)^{\\top}}}{2}\\rangle\\] \\[= \\langle D_{\\mathbf{h}_{j}^{(i)}}^{2}(\\mathcal{L}\\circ F^{\\prime})[\\mathbf{ h}_{j}^{(i)}],\\frac{J^{(i)}-S^{(i)}}{2J^{(i)}}(\\tilde{\\mathbf{h}}_{j}^{(i)}-\\mathbf{h}_{j}^{ (i)})^{\\bigodot 2}\\rangle\\] \\[\\mathbb{E}_{\\delta_{j}^{(i)\\text{BD}}}[R] =\\langle D_{\\mathbf{h}_{j}^{(i)}}^{2}(\\mathcal{L}\\circ F^{\\prime})[ \\mathbf{h}_{j}^{(i)}],\\frac{\\mathbb{E}\\delta_{j}^{(i)}\\delta_{j}^{(i)^{\\top}}}{2}\\rangle\\] \\[= \\langle D_{\\mathbf{h}_{j}^{(i)}}^{2}(\\mathcal{L}\\circ F^{\\prime})[\\mathbf{ h}_{j}^{(i)}],(\\frac{J^{(i)}-S^{(i)})p}{2J^{(i)}}(\\tilde{\\mathbf{h}}_{j}^{(i)}-\\mathbf{ h}_{j}^{(i)})^{\\bigodot 2}\\] \\[+\\frac{(J^{(i)}-S^{(i)})(1-p)(J^{(i)}-pJ^{(i)}+pS^{(i)})}{2J^{(i)}S ^{(i)}}\\mathbf{h}_{j}^{(i)^{\\bigodot 2}}\\rangle\\rangle\\]\n\nIt is easy to check that the statement satisfies given the value calculated. \n\n## Appendix B Implementation Details\n\n### Missing Algorithm\n\n```\n0: A preprocessed training graph dataset \\(\\mathcal{D}_{\\text{train}}=\\{(\\bigoplus\\mathcal{G}_{j}^{(i)},y^{(i)})\\}_{i=1}^{n}\\). A parameterized backbone \\(F\\) and a prediction head \\(F^{\\prime}\\). A historical segment embedding table \\(\\mathcal{T}\\).\n1:for\\(t=1\\) to \\(T_{0}\\)do\n2:\\(\\mathcal{B}\\leftarrow\\text{SampleMiniBatch}(\\mathcal{D}_{\\text{train}})\\)\n3:for\\((\\mathcal{G}^{(i)},y^{(i)})\\) in \\(\\mathcal{B}\\)do\n4:\\(\\{\\mathcal{G}_{s}^{(i)}\\}_{s\\in\\mathcal{S}^{(i)}}\\leftarrow\\text{SampleGraphSegments}( \\mathcal{G}^{(i)})\\)\n5:\\(\\tilde{\\mathbf{h}}_{j}^{(i)}\\leftarrow\\mathcal{T}.\\text{LookUp}\\left((i,j)\\right)\\)for\\(j\\notin\\mathcal{S}^{(i)}\\)\n6:\\(\\mathbf{h}_{s}^{(i)}\\gets F(\\mathcal{G}_{s}^{(i)})\\)for\\(s\\in\\mathcal{S}^{(i)}\\)\n7:\\(\\mathcal{T}.\\text{InsertOrUpdate}((i,s),\\mathbf{h}_{s}^{(i)})\\)\n8:endfor\n9: SGD on \\(\\frac{1}{|\\mathcal{B}|}\\sum_{i}\\mathcal{L}\\left(F^{\\prime}(\\eta_{s}^{(i)} \\cdot\\mathbf{h}_{s}^{(i)}\\bigoplus(\\eta_{j}^{(i)}\\cdot\\tilde{\\mathbf{h}}_{j}^{(i)}),y ^{(i)}\\right)\\) {SED with \\(\\eta\\) defined in Equation 1}\n10:endfor\n11:# Prediction Head Finetuning\n12:\\(\\mathcal{T}.\\text{InsertOrUpdate}((i,j),F(\\mathcal{G}_{j}^{(i)}))\\) for every \\(\\mathcal{G}_{j}^{(i)}\\)\n13:for\\(t=T_{0}\\) to \\(T_{1}\\)do\n14:for\\(\\mathcal{G}^{(i)}\\) in \\(\\text{SampleMiniBatch}(\\mathcal{D}_{\\text{train}})\\)do\n15:\\(\\tilde{\\mathbf{h}}_{j}^{(i)}\\leftarrow\\mathcal{T}.\\text{LookUp}\\left((i,j)\\right)\\)for\\(j\\leq J^{(i)}\\)\n16:endfor\n17: SGD with loss \\(\\leftarrow\\frac{1}{|\\mathcal{B}|}\\sum_{i}\\mathcal{L}\\left(F^{\\prime}( \\bigoplus\\tilde{\\mathbf{h}}_{j}^{(i)})\\right)\\)\n18:endfor\n```\n\n**Algorithm 2** Pipeline of GST+EFD\n\nWe follow GraphGym [36] to represent design spaces of GNN as (message passing layer type, number of pre-process layers, number of message passing layers, number of post-process layers, activation, aggregation). Our code is implemented in PyTorch [22].\n\n\\begin{table}\n\\begin{tabular}{c c c c c c c} \\hline \\hline  & Avg. \\# nodes & Min. \\# nodes & Max. \\# nodes & Avg. \\# edges & Min. \\# edges & Max. \\# edges \\\\ \\hline MalNet-Tiny & 1,410 & 5 & 4,994 & 2,860 & 4 & 20,096 \\\\ MalNet-Large & 47,838 & 3,374 & 541,571 & 225,474 & 20,597 & 3,278,318 \\\\ TpuGraphs & 38,444 & 299 & 615,019 & 62,475 & 380 & 1,058,278 \\\\ \\hline \\hline \\end{tabular}\n\\end{table}\nTable 4: Overview of the graph datasets used in this study.\n\n**Implementation details for MalNet-Large.** We consider three model variations for the MalNet-Large dataset. Please refer to their hyperparameters in Table 5. We use Adam optimizer [18] with the base learning rate of 0.01 for GCN and SAGE. For GraphGPS, we use AdamW optimizer [20] with the cosine scheduler and the base learning rate of 0.0005. We use L2 regularization with a weight decay of 1e-4. We train for 600 epochs until convergence. For Prediction Head Finetuning, we finetune for another 100 epochs. We limit the maximum segment size to 5,000 nodes, and use a keep probability \\(p=0.5\\) if not otherwise specified. We train with CrossEntropy loss.\n\n**Implementation details for MalNet-Tiny.** We use the same model architectures/training schedules as in the MalNet-Large dataset. The only difference is that as graphs in MalNet-Tiny have no more than 5000 nodes, so we limit maximum segment size to 500 here.\n\n**Implementation details for TpuGraphs.** We only consider SAGE with configurations (SAGEConv, 0, 4, 3, 128, ReLU, sum) for the TpuGraphs dataset. We use Adam optimizer with the base learning rate of 0.0001. We train for 200,000 iterations until convergence. We by default limit the maximum segment size to 8,192 nodes, and use a keep probability \\(p=0.5\\) if not otherwise specified. Since we care more about relative ranking than the absolute runtime, we use PairwiseHinge loss within a batch during training:\n\n\\[\\mathcal{L}(\\widehat{y}_{1},\\widehat{y}_{2})=\\sum_{i}\\sum_{j}\\mathbb{I}[y_{i} >y_{j}]\\cdot\\max(0,1-(\\widehat{y}_{i}-\\widehat{y}_{j}))\\]\n\n## Appendix C Additional Results\n\n**Convergence analysis.** To study the effect on convergence for the proposed framework GST and technique SED, we visualize training/test curve per epoch on TpuGraphs dataset in Figure 5, MalNet-Tiny dataset in Figure 6. We show that the convergence speed of various methods studied are quite similar. In addition, the convergence speed in terms of iterations is similar between Full Graph Training and the proposed GST. Due to certain implementation overheads, e.g., on-the-fly graph segment extraction, embedding table query, we didn't observe speed up in terms of wall-clock time for our current implementation yet. Nevertheless, the implementation can be optimized further to reduce the overhead, we leave it for future work.\n\n**Ablation study on partition algorithms.** We incorporated an ablation study focusing on various graph partition algorithms. The test accuracy of GST+EFD using the SAGE backbone for 5 iterations is depicted in Table 6. Our findings indicate that the Random Edge-Cut algorithm delivers subpar results due to its inability to maintain the integrity of the subgraph structure. Some attentive readers might hypothesize that ignoring edges between segments could lead to a decrease in graph property prediction accuracy, leading us to further investigate Vertex-Cut partition algorithms. In theory, Vertex-Cut techniques, which distribute edges across different machines and duplicate nodes as necessary, are likely to result in less information loss compared to Edge-Cut methods. Our empirical findings show that all partition algorithms that manage to retain the local structure have quite similar performance levels. This suggests that edges connecting different segments do not have a significant impact on the final prediction accuracy.\n\n## Appendix D Broader Impact\n\nThe paper presents Graph Segment Training (GST), a framework for predicting properties of large graphs using a divide-and-conquer approach. This method addresses the challenge of memory\n\n\\begin{table}\n\\begin{tabular}{l|l l l} \\hline \\hline model & GCN & SAGE & GraphGPS \\\\ \\hline message passing layer type & GCNConv & SAGEConv & GatedGCN+Performer \\\\ pre-process layer num. & 1 & 1 & 0 \\\\ message passing layer num. & 2 & 2 & 5 \\\\ post-process layer num. & 1 & 1 & 3 \\\\ hidden dimension & 300 & 300 & 64 \\\\ activation & PReLU & PReLU & ReLU \\\\ aggregation & mean & mean & mean \\\\ \\hline \\hline \\end{tabular}\n\\end{table}\nTable 5: Detailed GNN/Graph Transformer designs used in MalNet-Tiny and MalNet-Large.\n\nlimitation during training and has the potential to bring significant impact across various domains, with notable efficiency and accuracy improvements. GST can be particularly beneficial to industries that need to manage and interpret large-scale graph data. In telecommunications, it can help optimize network infrastructure, while in cybersecurity, it could improve anomaly detection in network traffic. Furthermore, enabling the capability in large network analysis can contribute to more resilient infrastructure, enhancing the quality of life in many communities. As with any advancement in AI, there's a risk that the benefits of this technology will be unevenly distributed, potentially increasing economic disparity. Companies with access to large amounts of data and the computational resources to analyze it might reap disproportionate benefits, which could further exacerbate the digital divide.\n\n## Appendix E Limitations\n\nWhile Graph Segment Training (GST) represents a significant advancement in graph property prediction, there are several potential limitations to this approach, as highlighted below:\n\n**Segmentation Limitations**: The efficacy of the Graph Segment Training (GST) approach is influenced by the proficiency of the graph partitioning procedure. While our empirical findings indicate that various partitioning algorithms that maintain locality tend to produce satisfactory outcomes, partitions created through random edge-cut methods haven't demonstrated the same level of success.\n\nFigure 5: Accuracy curve on TpuGraphs dataset.\n\nFigure 6: Accuracy curve on MalNet-Tiny dataset.\n\n**Historical Embedding Table**: The use of a historical embedding table to obtain embeddings for non-sampled segments introduces additional implementation complexity and potential for errors. If not managed properly, it could lead to inefficient memory usage or even slower the training process.\n\n\\begin{table}\n\\begin{tabular}{l l|c c} \\hline \\hline  & & MalNet-Tiny & MalNet-Large \\\\ \\hline Edge-Cut & Random & 85.43\\(\\pm\\)0.98 & 74.02\\(\\pm\\)2.23 \\\\ Edge-Cut & Louvain & 88.95\\(\\pm\\)0.67 & 89.16\\(\\pm\\)0.85 \\\\ Edge-Cut & METIS & 89.24\\(\\pm\\)0.53 & 89.78\\(\\pm\\)0.68 \\\\ Vertex-Cut & Random & 88.12\\(\\pm\\)1.17 & 87.69\\(\\pm\\)1.51 \\\\ Vertex-Cut & BBH [33] & 88.79\\(\\pm\\)0.74 & 89.28\\(\\pm\\)0.93 \\\\ Vertex-Cut & NE [39] & 89.16\\(\\pm\\)0.70 & 89.49\\(\\pm\\)0.87 \\\\ \\hline \\hline \\end{tabular}\n\\end{table}\nTable 6: Test accuracy when combined with different partition algorithms.", "response": "## Review\n\n### Summary\n\nThis paper addresses the problem of large graph property prediction through a novel Graph Segment Training (GST) approach. The method segments large graphs to optimize memory usage and computational efficiency while maintaining predictive accuracy. By utilizing a historical embedding table and techniques like stale embedding dropout, the authors demonstrate a clear advancement in handling large-scale graph learning tasks. Empirical evaluations across multiple datasets substantiate the effectiveness of the proposed framework, showcasing its practical applicability and memory efficiency. However, the paper could benefit from a more thorough discussion on limitations and future research directions.\n\n### Strengths\n\n- Targets an important problem of property prediction on large graphs with high application value.\n- The GST framework employs a divide-and-conquer strategy, enhancing both memory efficiency and computational speed.\n- Comprehensive theoretical analysis and practical experimentation validate the proposed methods.\n- Innovative techniques like stale embedding dropout contribute to reducing bias in predictions.\n- Well-organized and clearly presented, making it accessible for readers.\n\n### Weaknesses\n\n- Lacks a detailed discussion on how to handle inter-segmentation learning for link predictions between different graph segments.\n- Insufficient comparison with existing baseline methods for large graph learning, limiting the evaluation of GST's effectiveness.\n- Could improve on discussing limitations and potential future directions of the proposed framework.\n\n### Questions\n\n- How does the proposed method adapt to graph properties that span across different segments?\n- What happens to connecting edges between segments during partitioning, and how might this impact performance?\n- Could the method's efficiency be compared with other established techniques for large graph property prediction?\n- How sensitive is the proposed framework to hyperparameter changes?\n\n### Soundness\n\n**Score:** 3\n\n**Description:** 3 = good. The proposed method is technically sound, with solid theoretical backing and empirical support, though some aspects could benefit from further clarification.\n\n### Presentation\n\n**Score:** 3\n\n**Description:** 3 = good. The paper is well-structured and easy to follow, although minor improvements in clarity regarding certain concepts would enhance understanding.\n\n### Contribution\n\n**Score:** 3\n\n**Description:** 3 = good. The contributions are significant, particularly in addressing large graph learning challenges, though the paper could expand on its societal implications.\n\n### Rating\n\n**Score:** 6\n\n**Description:** 6 = Weak Accept: The paper is technically solid with moderate-to-high impact, though it requires some improvements regarding the breadth of evaluation and discussions on limitations.\n\n### Paper Decision\n\n**Decision:** Accept (poster)\n\n**Reasons:** The paper presents an innovative approach to large graph property prediction, demonstrating both theoretical and empirical merit. While it has some minor weaknesses in experimental comparisons and discussions on limitations, the overall contributions and clarity of presentation warrant acceptance. The work is original, addresses a significant problem, and shows potential for impact in the field, making it a strong candidate for acceptance.\n"}
{"query": "You are a highly experienced, conscientious, and fair academic reviewer, please help me review this paper. The review should be organized into nine sections: \n1. Summary: A summary of the paper in 100-150 words.\n2. Strengths/Weaknesses/Questions: The Strengths/Weaknesses/Questions of paper, which should be listed in bullet points, with each point supported by specific examples from the article where possible.\n3. Soundness/Contribution/Presentation: Rate the paper's Soundness/Contribution/Presentation, and match this score to the corresponding description from the list below and provide the result. The possible scores and their descriptions are: \n    1 poor\n    2 fair\n    3 good\n    4 excellent\n4. Rating: Give this paper an appropriate rating, match this rating to the corresponding description from the list below and provide the result. The possible Ratings and their descriptions are: \n    1 strong reject\n    2 reject, significant issues present\n    3 reject, not good enough\n    4 possibly reject, but has redeeming facets\n    5 marginally below the acceptance threshold\n    6 marginally above the acceptance threshold\n    7 accept, but needs minor improvements \n    8 accept, good paper\n    9 strong accept, excellent work\n    10 strong accept, should be highlighted at the conference   \n5. Paper Decision: It must include the Decision itself(Accept or Reject) and the reasons for this decision, based on the criteria of originality, methodological soundness, significance of results, and clarity and logic of presentation.\n\nHere is the template for a review format, you must follow this format to output your review result:\n**Summary:**\nSummary content\n\n**Strengths:**\n- Strength 1\n- Strength 2\n- ...\n\n**Weaknesses:**\n- Weakness 1\n- Weakness 2\n- ...\n\n**Questions:**\n- Question 1\n- Question 2\n- ...\n\n**Soundness:**\nSoundness result\n\n**Presentation:**\nPresentation result\n\n**Contribution:**\nContribution result\n\n**Rating:**\nRating result\n\n**Paper Decision:**\n- Decision: Accept/Reject\n- Reasons: reasons content\n\n\nPlease ensure your feedback is objective and constructive. The paper is as follows:\n\nPerfograph: A Numerical Aware Program Graph Representation for Performance Optimization and Program Analysis\n\nAli TehraniJamsaz\\({}^{1}\\)1, Quazi Ishtiaque Mahmud\\({}^{1}\\)1, Le Chen1, Nesreen K. Ahmed2, Ali Jannesari1\n\n\\({}^{1}\\) _Iowa State University, Ames, Iowa, USA_\n\n{tehrani, mahmund, lechen, jannesari}@iastate.edu\n\n\\({}^{2}\\) _Intel Labs, Santa Clara, CA, USA_\n\nnesreen.k.ahmed@intel.com\n\nEqual contribution.\n\nFootnote 1: footnotemark:\n\n###### Abstract\n\nThe remarkable growth and significant success of machine learning have expanded its applications into programming languages and program analysis. However, a key challenge in adopting the latest machine learning methods is the representation of programming languages, which directly impacts the ability of machine learning methods to reason about programs. The absence of numerical awareness, aggregate data structure information, and improper way of presenting variables in previous representation works have limited their performances. To overcome the limitations and challenges of current program representations, we propose a graph-based program representation called Perfograph. Perfograph can capture numerical information and the aggregate data structure by introducing new nodes and edges. Furthermore, we propose an adapted embedding method to incorporate numerical awareness. These enhancements make Perfograph a highly flexible and scalable representation that effectively captures programs' intricate dependencies and semantics. Consequently, it serves as a powerful tool for various applications such as program analysis, performance optimization, and parallelism discovery. Our experimental results demonstrate that Perfograph outperforms existing representations and sets new state-of-the-art results by reducing the error rate by 7.4% (AMD dataset) and 10% (NVIDIA dataset) in the well-known Device Mapping challenge. It also sets new state-of-the-art results in various performance optimization tasks like Parallelism Discovery and NUMA and Prefetchers Configuration prediction.\n\n## 1 Introduction\n\nIn recent years, the remarkable success of machine learning has led to transformative advancements across numerous fields, including compiler optimization and program analysis. The applications include compiler heuristics prediction, optimization decisions, parallelism detection, etc. [4, 15]. The training process generally involves feeding program data as input and transforming it into a representation suitable for machine learning models. The selection of program representation is crucial, as it can significantly impact the performance of the machine learning model [12]. With the development of graph neural networks (GNNs), an increasing number of graph representations of programs have been incorporated into GNN models for program analysis [3, 28, 10]. One of the pioneering efforts in developing a comprehensive graph representation for programs is ProGraML[14]. ProGraML incorporates control, data, and call dependencies as integral components of a program's representation. In contrast to prior sequential learning systems for code, ProGraMLclosely resembles the intermediate representations used by compilers, and the propagation of information through these graphs mimics the behavior of typical iterative data-flow analyses. Despite the success that ProGraML has achieved, there are shortcomings in this current state-of-the-art program representation, especially for performance-oriented downstream tasks. These limitations stem from neglecting numerical values available at compile time and the inadequate representation of aggregate data types.\n\nIn this paper, we present Perfograph to address the limitations of the current state-of-the-art program representation. Additionally, we propose a novel way to embed numbers in programs in an elegant way so that our DL model will not face unknown numbers during inference time. Our experiments demonstrate that Perfograph sets new state-of-the-art results in numerous downstream tasks. For example, in the Device Mapping downstream task, Perfograph yield error rates as low as 6% and 10% depending on the target hardware. Moreover, Perfograph even outperforms the tools and models specially designed for specific tasks such as parallelism discovery.\n\nOverall, the main contributions of this paper are:\n\n* An enhanced compiler and language-agnostic program representation based on ProGraML that represents programs as graphs.\n* The proposed representation supports aggregate data types and provides numerical awareness, making it highly effective for performance optimization tasks.\n* Evaluation of the proposed representation on common downstream tasks and exceeding the performance of ProGraML.\n* Quantification of the proposed approach on a new set of downstream tasks such as parallelism discovery and configuration of NUMA systems.\n\nThe rest of the paper is structured as follows: Section 2 presents the related works. In section 3, we provide a motivational example, showing the limitations of ProGraML, the state-of-the-art program representation. This section is followed by section 4 where we present our proposed representation Perfograph along with the novel way of embedding numerical values. In section 5, experimental results on downstream tasks are provided, and finally, Section 6 concludes the paper and discusses some future works.\n\n## 2 Related Works\n\nMachine learning has brought significant advancements in many fields, and program analysis and software engineering are no exceptions. However, Machine Learning (ML) and Deep Learning (DL) models can not directly process raw source code to reason about programs. Therefore, researchers have explored different approaches to represent applications in a format suitable for DL models. Generally, there are three types of commonly used program presentations: sequence of tokens, Abstract Syntax Tree (AST), and Intermediate Representation (IR).\n\n**Sequence of tokens:** The initial attempts [16; 21; 27] represented source code as a sequence of tokens, such as identifiers, variable names, or operators. This approach intuitively treats programming languages similarly to natural languages. It allows for the utilization of advanced natural language process (NLP) techniques, such as large language models [19; 22; 23]. However, this token-based representation overlooks the inherent dependency information within the program's structure. It fails to capture the unique relationships and dependencies between different elements of the code, which can limit its effectiveness in tasks such as compiler optimization and code optimization.\n\n**AST:** An AST represents the structure of a program by capturing its hierarchical organization. It is constructed based on the syntactic rules of the programming language and provides a high-level abstraction of the code. Previous works have leveraged ASTs as inputs to tree-based models for various code analysis tasks like software defect prediction [17] and code semantic study [9]. Moreover, there have been efforts to augment ASTs into graphs that incorporate program analysis flows such as control flow and data flow. These AST-based graph representations capture more comprehensive code dependency information and have shown superior results compared to traditional approaches in previous works [2; 3].\n\n**IR:** IR is an intermediate step between the source code and the machine code generated by a compiler. Previous work [39] has utilized IR to train an encoding infrastructure for representing programs as a distributed embedding in continuous space. It augments the Symbolic encodings with the flow of information to capture the syntax as well as the semantics of the input programs. However, it generates embedding at the program or function level and also requires a data-flow analysis type for generating the embedding. In contrast, our approach derives embedding from the representation and works at the more fine-grained instruction level. More recent works [6; 14; 8] have leveraged IR-based graph representation to better capture essential program information, such as control flow, data flow, and dependencies. However, despite their success, IR-based graph representations have certain limitations. For example, these representations may not be numeric-aware or may lack the ability to adequately represent aggregate data types. In this work, we propose Perfograph, a graph representation based on IR, to address these limitations.\n\n## 3 Motivation\n\nAs stated in the related work section, program representations based on the intermediate representation are very effective in enabling DL models to automate the process of various optimizations. One such representation is ProGraML, whose performance surpasses other code representations, making it state-of-the-art in various optimizations and downstream tasks. However, despite its potential, it suffers from several limitations. To name a few: it is incapable of properly carrying out information regarding read and write operations to the memory location, has no support for aggregate data types, and discards numerical values. Listing 1 shows a code snippet where a 3-dimensional array is defined. Figure 1 shows the ProGraML representation of this code snippet. For illustration purposes, instruction nodes and control flow edges are shown in blue, whereas red represents variable, constant nodes, and data flow edges. Green edges show the call graph. As it can be seen, ProGraML fails to represent some critical information. For instance, code float arr[2][3][4] is converted to LLVM IR [2 x [3 x [4 x float]]]*, which is used to construct a node in ProGraML. It eliminates the aggregate data structure information, like the array's dimension. Leaving it up to the DL model to infer the meaning behind the numbers in [2 x [3 x [4 x float]]]*. Moreover, in this representation, only the type of numbers (e.g., int8,float) are considered, and the actual values of the numbers are not given attention. The absence of numerical awareness limits the performance of ProGraML in applications where numerical values play an important role. A numerically aware representation can help understand and optimize operations involving numeric data types, constants, and expressions. There are also some anomalies in the way temporary variables are depicted. For example, in 1, we see the fourth alloca node allocates memory for a variable, and two store instructions are applied on two separate nodes representing the variable. Thus, the information about the first store instruction is not carried out properly when the second store instruction happens. In the following section, we will see how Perfograph effectively addresses many limitations in the current state-of-the-art program representation. Perfograph uses ProGraML representation as its initial graph and reconstructs the graphs by addressing the aforementioned limitations.\n\n## 4 Perfograph: A fined-grained numerical aware graph representation\n\nPerfograph is graph representation based on LLVM IR. It is built on top of ProGraML; however, it does not suffer from the limitations that the ProGraML has, helping DL models to reason over the complex structure of the programs and enabling them to make more accurate optimization decisions,\n\nFigure 1: ProGraML representation of Listing 1.\n\nespecially in terms of performance optimization. Figure 2 shows how various enhancements and improvements are applied to construct a more precise representation. Consider a simple code example of defining a variable and increasing it by one {int i = 0; i++;}. Figure 1(a) shows the ProGraML representation of this code example. It has two store nodes, as one is responsible for storing 0 and the other one storing the incremented value of i. In the following subsection, we will explain how Perfograph is constructed by addressing the limitations shown in Figure 1(a).\n\n### Representing Local Identifiers and store instruction\n\n**Local Identifiers:** Local identifiers' names are preceded by % in LLVM Intermediate representation. Memory allocation on the stack is done by alloca instruction. One of the limitations of the current state-of-the-art program representation, ProGraML, is that it is unable to carry out information regarding the operations that happen to a memory location. For instance, in Figure 1(a), the two store nodes represent storing values of 0 and 1 to variable i. However, as shown, each store instruction node is connected to a separate variable node, making it difficult for the graph neural network to reason over the operations that happen to a memory location. For the embedding vector of the second store node in 1(a) to represent the fact that some information regarding the variable i has changed, one has to increase the number of GNN layers to 3 to support up to 3 hops when propagating the messages in GNN. This can potentially limit the ability of the GNN model if there are a greater number of hops between the two store nodes shown in Figure 1(a). To address this limitation, instead of having more than one variable node (oval-shape nodes) per identifier, Perfograph only considers one variable node in its graph representation. Any load or store instruction will refer to the same variable node. These changes are shown in Figure 1(b). We see that the store nodes in Figure 1(b) access the same memory location, thus representing the fact that those store instructions are modifying the same memory location.\n\nStore instruction: LLVM uses store instruction to write into memory. store instruction has two arguments: a value to store and the address to which it will store the value. ProGraML differentiates between these two arguments by adding a position feature to the edges as shown in Figure 1(a). However, since the store instruction modifies the contents at the corresponding memory address, we posit that it is better to reflect the fact the content of the identifier has changed. To present this information, Perfograph adds an extra edge from the store node to node representing the identifier whose value is modified by the store instruction. Figure 1(c) shows these changes in the graph constructed by Perfograph.\n\n**Numbers:** Numbers can be a significant factor in optimization decisions. For example, they can show the loop bound, and different optimizations can be considered depending on the loop bound. Perfograph, unlike ProGraML, not only considers the type of numbers such as 132, 164, float but also the actual values of the numbers. As illustrated in Figure 1(d), numerical constant nodes have the actual value of the number in their feature set in addition to the type of the number. Even though numerical constant nodes have the value of the number as one of their features, there is a need to embed the numbers in a way that unknown numbers will not be seen in the inference. Unlike other tokens, numbers are harder to embed as an infinite amount of numbers exists, and to handle all ranges of numbers, we need to have a very large vocabulary set.\n\nFigure 2: Perfograph addresses the existing limitations in program representation.\n\n### Numerical Awareness\n\nHowever, with a very large vocabulary size, the DL models may still encounter numbers in the inference phase that they have not seen in the training phase. We propose a novel way of embedding numbers called Digit Embedding. Figure 3 shows our approach. To embed a number, we first break down the number to its digits; then, we consider a position for each one of the digits. The goal is to let DL models realize the place value of each digit. Then, each digit and its corresponding position are embedded and summed together. Therefore, we will have an embedding representing the information about the digits and their positions. For instance, in Figure 3, we embed each digit and its corresponding position with an output dimension of 3. Since the number has four digits, the results would be a vector/tensor of size \\(4\\times 3\\). To make sure the Digit Embedding of numbers has the same length across numbers with varying sizes of digits, we apply an aggregation function over the embedding dimension. Since the output embedding dimension is three in this example, we would have one vector of length three representing the number after aggregation. The aggregation function can be of any type (Max, Mean, etc.).\n\n### Aggregate Data Types\n\nAggregate data types, such as arrays and vectors, are an essential part of applications. They play an important role in many applications, such as matrix multiplications. Thus, presenting these data types helps the DL models better understand programs. Current LLVM IR-based program representations fail to present aggregate data types appropriately. For example, consider a three-dimensional integer array. In LLVM IR, this array is shown as 3 x [2 x [3 x i32]]*. As can be seen, the length of the arrays and their data types are inferable. However, without proper representation, the DL model's capacities will be spent on learning these deterministic facts (i.e., the length of the arrays and their type). Perfograph considers aggregate data types as a new node type in its representations. Figure 3(b) shows how aggregate data types are supported by Perfograph. Unlike other LLVM IR-based representations, Perfograph supports multi-dimensional arrays and vectors. Perfograph creates a chain of nodes to present the different dimensions of the arrays. In Figure 3(a), we see there is a node representing the three-dimensional array [3 x [2 x [3 x i32]]*. Perfograph breaks down the corresponding node into three (since it is a three-dimensional array) white nodes as shown in Figure 3(b). Then, each node has a context representing that specific dimension of the array. For example, the context for the third dimension is [3 x i32], whereas for the second dimension, the context is [2 x [3 x i32]]. For each aggregate type node, in addition to the context of the node, we specifically add the length of the array and its type as additional features. For aggregate data types whose lengths are not known during compile time, we follow the LLVM conventions by considering the length of those data types as vscale. These enhancements will help the DL models to reason over the dimensions and types of aggregate data types. As a result, Perfograph will ultimately enable the DL models to have more accurate predictions for applications that deal with arrays and vectors.\n\nFigure 4: Perfograph supports aggregate data types.\n\nFigure 3: Overview of the digit embedding.\n\nExperimental Results and Downstream Tasks\n\nIn this section, we evaluate Perfograph on six downstream tasks. For each downstream task, we will explain the task itself, the dataset, and the baselines.\n\n### Experimental Setup\n\nIn our experiments, we use DGL's [40] RGCN [37] implementation for Perfograph representation. The graphs from Perfograph are treated as heterogeneous and managed via the HeteroGraphConv module. We use a hardware setup of two 18-core Intel Skylake 6140 CPUs and two NVIDIA Tesla V100-32GB GPUs. The embedding space for numbers is generated by extracting digits and positions from a numeric token of an IR statement, then passed to a PyTorch [31] embedding layer for digit and position embeddings. These are combined for the final numeric token embedding. Non-numeric tokens directly go through the PyTorch embedding layer. Each Perfograph heterogeneous node converts to a 120-dimensional vector via this embedding. We use the Adam[26] Optimizer, relu[1] activation function, a learning rate of \\(0.01\\), and hidden_dim parameter of \\(60\\). Mean aggregation is applied to combine different node-type results before a linear classification layer, which outputs a probability distribution for each class. The class with the highest probability is the prediction.\n\n### Device Mapping\n\n**Problem Definition:** We apply Perfograph to the challenging heterogeneous device mapping [13] problem. In this task, there are a number of OpenCL kernels that we need to predict which accelerator (CPU or GPU) yields higher performance. We compare Perfograph against DeepTune [13], Inst2Vec [6], and ProGraML[14]. The results of the baselines are quoted from [14].\n\n**Dataset:** For this task, we use the dataset published in [13]. In this dataset, there are 256 OpenCL kernels available, and 680 LLVM IR instances are extracted from them. There are two types of GPUs: AMD and NVIDIA. For each of the GPUs, the runtimes of the kernels are recorded in the dataset. For AMD, 276 kernels show better performance in GPU, while 395 kernels show better performance in CPU. Whereas for NVIDIA, 385 kernels have better runtimes with GPU, and 286 kernels have better runtimes with CPU. We consider this as a binary CPU or GPU classification problem.\n\n**Results:** As the dataset is small, we use the same 10-fold validation (with 80% training, 10% validation, and 10% testing) like ProGraML[14] and chose the model with the highest validation accuracy. The hand-crafted features of [20] are also used as graph-level features in our model to enhance the performance following the approach in [14]. Table 1 and 2 show the final precision, call, f1-score, and accuracy for AMD and NVIDIA devices. Figure 5 compares Perfograph with state-of-the-art models on the Device Mapping dataset. We can see that Perfograph sets new state-of-the-art results by achieving the lowest error rate among the baselines both for AMD and NVIDIA, indicating the effectiveness of Perfograph.\n\n### Parallelism Discovery\n\n**Problem Definition:** In this problem, given a sequential loop, we try to predict whether a loop can be executed in parallel. We treat this problem as a binary classification problem with two classes: Parallel and Non-Parallel.\n\n\\begin{table}\n\\begin{tabular}{l c c c c} \\hline \\hline  & Precision & Recall & F1-score & Accuracy \\\\ \\hline CPU & 0.94 & 0.94 & 0.94 & 0.94 \\\\ GPU & 0.94 & 0.94 & 0.94 & 0.94 \\\\ \\hline \\hline \\end{tabular}\n\\end{table}\nTable 1: Perfograph results for AMD devices.\n\nFigure 5: Performance comparison the device mapping task with state-of-the-art models [lower is better].\n\n**Dataset:** The OMP_Serial dataset [11] is used for this task. It contains around 6k compilable source C files with Parallel and Non-Parallel loops. The training dataset contains around 30k IR files. The OMP_Serial dataset has three test subsets to compare the performance with three traditional parallelism assistant tools: Pluto (4032 IR files), AutoPar (3356 IR files), and DiscoPoP (1226 IR files).\n\n**Results:** We evaluate Perfograph on all three subsets and compare it with traditional rule-based tools: Pluto [7], AutoPar [33], DiscoPoP [29], and also Deep Learning based tools: Graph2Par [11], ProGraML. Table 3 shows the results. The results of Pluto and Graph2par are reported from [11]. As ProGraML does not have this downstream task in their paper, we used the ProGraML representation in our pipeline to generate the results. Results show that traditional rule-based tools have the highest precision but the lowest accuracy because those tools are overly conservative while predicting parallel loops. So, they miss out on a lot of parallelism opportunities. Perfograph achieves considerably good precision scores across all the test subsets. In terms of accuracy, Perfograph surpasses the current state-of-the-art approaches by 2% in the Pluto and AutoPar subset. In the DiscoPoP subset, it achieves an impressive 99% accuracy and surpasses ProGraML by 9%.\n\n### Parallel Pattern Detection\n\n**Problem Definition:** Parallel loops often follow some specific patterns. Identifying parallel patterns is important because it helps developers understand how to parallelize a specific program since each parallel pattern needs to be treated differently. As a result, we apply Perfograph to identify potential parallel patterns in sequentially written programs. Only the three most common parallel patterns are considered: Do-all (Private), Reduction, and Stencil [34]. Given a loop, the task is to predict the pattern.\n\n**Dataset:** For this experiment, we also use the OMP_Serial dataset [11]. This dataset contains source codes of different parallel patterns. These programs are collected from well-known benchmarks like NAS Parallel Benchmark [24], PolyBench [32], BOTS benchmark [18], and the Starbench benchmark [5]. Then, template programming packages like Jinja [35] are used to create synthetic programs from the templates collected from the mentioned benchmarks. The dataset contains 200 Do-all (Private), 200 Reduction, and 300 Stencil loops.\n\n**Results:** We used 80% of the dataset for training and 20% for testing. Table 4 represents our findings. The results of Pragformer [25] and Graph2par [11] are reported from [11]. We compare with these two approaches as they are specifically developed for solving this problem. For generating the results with ProGraML, we used the ProGraML representation in our pipeline. We can see Perfograph achieves an impressive 99% accuracy on the OMP_Serial Parallel Pattern dataset. It surpasses the state-of-the-art ProGraML model by 3%. This indicates the strength of Perfograph to capture the syntactic and structural patterns embedded into source programs. From Table 4, we can also see Perfograph has high precision for all three patterns and achieves a high precision score for Do-all and Stencil patterns while maintaining very good accuracy.\n\n\\begin{table}\n\\begin{tabular}{c c c c c c} \\hline \\hline Subset & Approach & Precision & Recall & F1-score & Accuracy \\\\ \\hline \\multirow{4}{*}{Pluto} & Pluto & 1 & 0.39 & 0.56 & 0.39 \\\\  & Graph2par & 0.88 & 0.93 & 0.91 & 0.86 \\\\  & ProGraML & 0.88 & 0.88 & 0.87 & 0.89 \\\\  & **Perfograph** & 0.91 & 0.90 & 0.89 & **0.91** \\\\ \\hline \\multirow{4}{*}{autoPar} & AutoPar & 1 & 0.14 & 0.25 & 0.38 \\\\  & Graph2par & 0.90 & 0.79 & 0.84 & 0.80 \\\\  & ProGraML & 0.92 & 0.69 & 0.67 & 0.84 \\\\  & **Perfograph** & 0.85 & 0.91 & 0.85 & **0.86** \\\\ \\hline \\multirow{4}{*}{DiscoPoP} & DiscoPoP & 1 & 0.54 & 0.70 & 0.63 \\\\  & Graph2par & 0.90 & 0.79 & 0.84 & 0.81 \\\\ \\cline{1-1}  & ProGraML & 0.92 & 0.94 & 0.92 & 0.91 \\\\ \\cline{1-1}  & **Perfograph** & 0.99 & 1 & 0.99 & **0.99** \\\\ \\hline \\hline \\end{tabular}\n\\end{table}\nTable 3: Performance comparison of Perfograph on the OMP_Serial dataset.\n\n### NUMA and Prefetchers Configuration Prediction\n\n**Problem Definition:** An appropriate configuration of Non-Uniform Memory Access (NUMA) and hardware prefetchers significantly impacts program performance. In this experiment, we define the task of NUMA and prefetcher selection as predicting the right configuration within a given tuning parameter search space. We evaluate the performance of both ProGarML and Perfograph for this task by converting each program in the dataset to ProGarML and Perfograph graphs following the approach in [38].\n\n**Dataset:** We use the dataset in [38], which includes a diverse set of intermediate representation files coupled with the optimal configuration [36]. The dataset incorporates various LLVM compiler optimization flags to produce different forms of the same program. There are 57 unique kernels (IR files) in this dataset, and around 1000 optimization flags are applied, resulting in 57000 IR files in total. Each IR file within the dataset is accompanied by its runtime on two architectures, Sandy Bridge and Skylake, across thirteen different NUMA and prefetcher configurations.\n\n**Results:** Following the approach in the study of TehraniJamsaz _et al._, we partition the dataset into ten folds for cross-validation. Figure 5(a) and 5(b) illustrate the performance results in terms of error rates. On average, Perfograph outperforms ProGarML by achieving 3.5% and 1.8% better error rates on average for the Sandy Bridge and Skylake architecture, respectively. These improvements demonstrate the effectiveness of Perfograph compared to the state-of-the-art ProGarML.\n\n### Thread Coarsening Factor (TCF) Prediction\n\n**Problem Definition:** Thread coarsening is an optimization technique for parallel programs by fusing the operation of two or more threads together. The number of threads that can be fused together is known as the Thread Coarsening Factor (TCF). For a given program, the task is to predict the coarsening factor value (1, 2, 4, 8, 16, 32) that leads to the best runtime. The running time with coarsening factor 1 is used as the baseline for calculating speedups. For this task, we compare Perfograph against DeepTune [13], Inst2Vec [6] and ProGarML [14]. The results of the baselines are quoted from [6]. Since ProGarML has not been evaluated on this task in the past, we apply ProGarML representation in our setup for comparison.\n\n\\begin{table}\n\\begin{tabular}{c c c c c c} \\hline \\multicolumn{5}{c}{OMP\\_Serial Dataset.} \\\\ \\hline Approach & \\multicolumn{2}{c}{Pattern} & Precision & Recall & F1-score & Accuracy \\\\ \\hline \\multirow{2}{*}{Pragformer} & Do-all & 0.86 & 0.85 & 0.86 & \\multirow{2}{*}{0.86} \\\\  & Reduction & 0.89 & 0.87 & 0.87 & \\multirow{2}{*}{0.86} \\\\  & Stencil & N/A & N/A & N/A & \\\\ \\hline \\multirow{3}{*}{Graph2Par} & Do-all & 0.88 & 0.87 & 0.87 & \\multirow{3}{*}{0.9} \\\\  & Reduction & 0.9 & 0.89 & 0.91 & \\\\  & Stencil & N/A & N/A & N/A & \\\\ \\hline \\multirow{3}{*}{ProGarML} & Do-all & 0.92 & 0.90 & 0.91 & \\multirow{3}{*}{0.96} \\\\  & Reduction & 0.92 & 0.92 & 0.92 & \\\\  & Stencil & 0.98 & 1 & 0.99 & \\\\ \\hline \\multirow{3}{*}{**Perfograph**} & Do-all & 1 & 0.97 & 0.99 & \\multirow{3}{*}{**0.99**} \\\\  & Reduction & 0.97 & 1 & 0.99 & \\\\ \\cline{1-1}  & Stencil & 1 & 1 & 1 & \\\\ \\hline \\end{tabular}\n\\end{table}\nTable 4: Performance comparison for the parallel pattern detection task with Perfograph on the\n\nFigure 6: Breakdown of the NUMA and prefetchers configuration prediction per fold [lower is better].\n\n**Dataset:** We use the dataset of Ben-Nun et al. [13]. The dataset contains only 17 OpenCL kernels. For each kernel, the dataset has the runtime information on four different GPUs for the different thread coarsening factor values. Hence, for each kernel, we have the runtime corresponding to each thread coarsening factor value on a specific GPU device.\n\n**Results:** we design the problem as a multi-class classification problem where, given a kernel, we try to predict which thread coarsening factor provides the highest performance. As the dataset is very small, we apply a 17-fold cross-validation approach. In each fold, we train our model on 16 data points, and the model is tested on the one unseen data point that is left out of the training set. Figure 7 shows the comparison of kernels with the correct Thread Coarsening Factor (TCF) found by ProGraML and Perfograph. Across the four platforms in total Perfograph is able to correctly predict the TCF for 17 cases, whereas ProGraML is able to find only 9 cases. In two of the platforms (AMD Radeon HD 5900 and NVIDIA GTX 480) where ProGraML failed to find any kernel with the correct TCF, Perfograph can find three kernels in both of the platforms with the correct TCF value. As shown in 5, even though Perfograph outperforms ProGraML on most computing platforms, it falls behind inst2vec. We posit the reason is that inst2vec has a pretraining phase where it is trained using skip-gram. On the other hand, 17 kernels are very small. Therefore, a DL-based model is not able to generalize enough. However, we can see that even on a smaller dataset, Perfograph achieved comparable speedups with respect to the current state-of-the-art models.\n\n### Algorithm Classification\n\n**Problem Definition:** Previous downstream tasks showed that in most of the cases, Perfograph outperforms the baselines. Those tasks were mostly performance-oriented. We go further by applying Perfograph on a different downstream task, which is algorithm classification. The task involves classifying a source code into 1 of 104 classes. In this task, we compare the results of Perfograph to those of inst2vec, ProGraML. The results of the baselines are quoted from [14].\n\n**Dataset:** We use the POJ-104 dataset [30] in a similar setup as [14] that contains around 240k IR files for training and 10k files for testing.\n\n**Results:** For this task, inst2vec has error rate of 5.17, whereas ProGraML has error rate of 3.38. Perfograph yields an error rate of 5.00, which is better than inst2vec and slightly behind ProGraML. One of the reasons is that ProGraML already has a very small error rate in this task, leaving a very small gap for improvement; however still Perfograph's result is very close to that of ProGraML. We could not reproduce the results in ProGraML paper in our setup. When we applied ProGraML in our setup, the error rate of ProGraML was 6.00. Moreover, we posit that for algorithm classification, numbers are not a significant factor. Therefore, numerical awareness can confuse the models a little bit. However, this experiment shows that Perfograph is very close to ProGraML's performance in this task and shows the applicability of Perfograph to a wider range of downstream tasks.\n\n### Ablation Study\n\nWe further analyzed how each one of the enhancements in Perfograph affects the end results. We performed an ablation study on the Device Mapping task and trained our GNN models on variations of Perfograph.\n\n\\begin{table}\n\\begin{tabular}{l c c c c} \\hline \\hline Computing Platform & DeepTune inst2vec & ProGraML & Perfograph \\\\ \\hline AMD Radeon HD 5900 & 1.1 & **1.37** & 1.15 & 1.19 \\\\ AMD Tahiti 7970 & 1.05 & 1.1 & 1.00 & **1.14** \\\\ NVIDIA GTX 480 & **1.1** & 1.07 & 0.98 & 1.03 \\\\ NVIDIA Tesla K20c & 0.99 & **1.06** & 1.03 & 1.01 \\\\ \\hline \\hline \\end{tabular}\n\\end{table}\nTable 5: Speedups achieved by coarsening threads\n\nFigure 7: Correct TCF found by ProGraML vs Perfograph [higher is better].\n\n**Results without aggregate data type nodes:** First, aggregate data type nodes are removed from Perfograph representation. Please note that in this setup, the Digit Embedding is still applied. Table 6 and 7 shows the results. It can be seen that when the representation does not support aggregate data type, the error rate increases to 13% in AMD and 15% in the NVIDIA dataset. This clearly indicates that having aggregate-type nodes in the representation helped the model to learn the code features more accurately.\n\n**Results without Digit Embedding:** Then Digit Embedding is removed from the pipeline for the second experiment and the aggregate data type nodes are kept. Table 6 and 7 shows the results. We can see that unlike having aggregate data type nodes, removing digit embedding does not hurt the error rate that much for the task of device mapping. However, we can still see a small increase (1.1%) in the error rate for the AMD dataset. For the NVIDIA dataset, the error rate increases from 10.0 to 10.6%.\n\nFinally, we can conclude that both components in our representation helped the model to learn the code features better to some extent. However, aggregate data type nodes in the embedding helped our model more than Digit Embedding for the task of device mapping. The reason can be that there are not many numbers in the dataset. However, in tasks where there are many numbers, Digit Embedding can play a significant role.\n\n## 6 Conclusion and Future Work\n\nIn this paper, we presented Perfograph, an LLVM IR-based graph representation of programs that supports aggregate data types such as arrays and vectors and is numerical aware. Moreover, it addresses several limitations of the previous IR-based graph representations. Perfograph is evaluated on various downstream tasks, and experimental results indicate that Perfograph is indeed effective, outperforming state-of-the-art in most of the downstream tasks. Perfograph numerical awareness capability is limited to the numerical values that are available at the compile time. For future work, we intend to augment our representation by adding support for dynamic information and checking the possibility of integrating hardware performance counters with our representation. Moreover, we plan to develop a pre-trained embedding model using our representation. Having a pre-trained model will help to solve the problem of limited training samples in some downstream tasks.\n\n## 7 Acknowledgement\n\nThis project was funded by NSF (#2211982) and Intel Labs. We would like to thank them for their generous support. Additionally, we extend our gratitude to the Research IT team* of Iowa\n\n\\begin{table}\n\\begin{tabular}{c c} \\hline \\hline Approach & Error (\\%) \\\\ \\hline DeepTune [13] & 28.1 \\\\ in32vec [6] & 19.7 \\\\ ProGAML [14] & 13.4 \\\\ Perfograph (without aggregate data type nodes) & 13.0 \\\\ Perfograph (without digit embedding) & 7.1 \\\\ Perfograph (aggregate data type nodes + digit embedding) & 6.0 \\\\ \\hline \\hline \\end{tabular}\n\\end{table}\nTable 6: Summarizing Perfograph results for AMD device.\n\n\\begin{table}\n\\begin{tabular}{c c} \\hline \\hline Approach & Error (\\%) \\\\ \\hline DeepTune [13] & 39.0 \\\\ inst2vec [6] & 21.5 \\\\ ProGramML [14] & 20.0 \\\\ Perfograph (without aggregate data type nodes) & 15.0 \\\\ Perfograph (without digit embedding) & 10.6 \\\\ Perfograph (aggregate data type nodes + digit embedding) & 10.0 \\\\ \\hline \\hline \\end{tabular}\n\\end{table}\nTable 7: Summarizing Perfograph results for NVIDIA device.\n\nState University for their continuous support in providing access to HPC clusters for conducting the experiments of this research project.\n\n## References\n\n* [1] A. F. Agarap. Deep learning using rectified linear units (relu). _arXiv preprint arXiv:1803.08375_, 2018.\n* [2] M. Allamanis. Graph neural networks in program analysis. _Graph Neural Networks: Foundations, Frontiers, and Applications_, pages 483-497, 2022.\n* [3] M. Allamanis, M. Brockschmidt, and M. Khandemi. Learning to represent programs with graphs. _arXiv preprint arXiv:1711.00740_, 2017.\n* [4] M. Allamanis, E. T. Barr, P. Devanbu, and C. Sutton. A survey of machine learning for big code and naturalness. _ACM Computing Surveys (CSUR)_, 51(4):1-37, 2018.\n* [5] M. Andersch, B. Juurlink, and C. Chi. A benchmark suite for evaluating parallel programming models. In _Proceedings of Workshop on Parallel Systems and Algorithms (PARS)_, volume 28, pages 1-6, 2013.\n* [6] T. Ben-Nun, A. S. Jakobovits, and T. Hoefler. Neural code comprehension: A learnable representation of code semantics. _Advances in Neural Information Processing Systems_, 31, 2018.\n* [7] U. Bondhugula, A. Hartono, J. Ramanujam, and P. Sadayappan. A practical automatic polyhedral parallelizer and locality optimizer. _SIGPLAN Not._, 43(6):101-113, jun 2008. ISSN 0362-1340. doi: 10.1145/1379022.1375595. URL [https://doi.org/10.1145/1379022.1375595](https://doi.org/10.1145/1379022.1375595).\n* [8] A. Brauckmann, A. Goens, S. Ertel, and J. Castrillon. Compiler-based graph representations for deep learning models of code. In _Proceedings of the 29th International Conference on Compiler Construction_, pages 201-211, 2020.\n* [9] L. Chen, W. Ye, and S. Zhang. Capturing source code semantics via tree-based convolution over api-enhanced ast. In _Proceedings of the 16th ACM International Conference on Computing Frontiers_, pages 174-182, 2019.\n* [10] L. Chen, Q. I. Mahmud, and A. Jannesari. Multi-view learning for parallelism discovery of sequential programs. In _2022 IEEE International Parallel and Distributed Processing Symposium Workshops (IPDPSW)_, pages 295-303. IEEE, 2022.\n* [11] L. Chen, Q. I. Mahmud, H. Phan, N. Ahmed, and A. Jannesari. Learning to parallelize with openmp by augmented heterogeneous ast representation. _Proceedings of Machine Learning and Systems_, 5, 2023.\n* [12] Z. Chen and M. Monperrus. A literature study of embeddings on source code. _arXiv preprint arXiv:1904.03061_, 2019.\n* [13] C. Cummins, P. Petoumenos, Z. Wang, and H. Leather. End-to-end deep learning of optimization heuristics. In _2017 26th International Conference on Parallel Architectures and Compilation Techniques (PACT)_, pages 219-232. IEEE, 2017.\n* [14] C. Cummins, Z. V. Fisches, T. Ben-Nun, T. Hoefler, and H. Leather. Programl: Graph-based deep learning for program optimization and analysis. _arXiv preprint arXiv:2003.10536_, 2020.\n* [15] C. Cummins, B. Wasti, J. Guo, B. Cui, J. Ansel, S. Gomez, S. Jain, J. Liu, O. Teytaud, B. Steiner, et al. Compilergym: Robust, performant compiler optimization environments for ai research. In _2022 IEEE/ACM International Symposium on Code Generation and Optimization (CGO)_, pages 92-105. IEEE, 2022.\n* [16] H. K. Dam, T. Tran, and T. Pham. A deep language model for software code. _arXiv preprint arXiv:1608.02715_, 2016.\n* [17] H. K. Dam, T. Pham, S. W. Ng, T. Tran, J. Grundy, A. Ghose, T. Kim, and C.-J. Kim. Lessons learned from using a deep tree-based model for software defect prediction in practice. In _2019 IEEE/ACM 16th International Conference on Mining Software Repositories (MSR)_, pages 46-57. IEEE, 2019.\n* [18] A. Duran, X. Teruel, R. Ferrer, X. Martorell, and E. Ayguade. Barcelona openmp tasks suite: A set of benchmarks targeting the exploitation of task parallelism in openmp. In _2009 international conference on parallel processing_, pages 124-131. IEEE, 2009.\n* [19] Z. Feng, D. Guo, D. Tang, N. Duan, X. Feng, M. Gong, L. Shou, B. Qin, T. Liu, D. Jiang, et al. Codebert: A pre-trained model for programming and natural languages. _arXiv preprint arXiv:2002.08155_, 2020.\n\n* [20] D. Grewe, Z. Wang, and M. F. O'Boyle. Portable mapping of data parallel programs to opencl for heterogeneous systems. In _Proceedings of the 2013 IEEE/ACM International Symposium on Code Generation and Optimization (CGO)_, pages 1-10. IEEE, 2013.\n* [21] X. Gu, H. Zhang, D. Zhang, and S. Kim. Deep api learning. In _Proceedings of the 2016 24th ACM SIGSOFT international symposium on foundations of software engineering_, pages 631-642, 2016.\n* [22] D. Guo, S. Ren, S. Lu, Z. Feng, D. Tang, S. Liu, L. Zhou, N. Duan, A. Svyatkovskiy, S. Fu, et al. Graphcodebert: Pre-training code representations with data flow. _arXiv preprint arXiv:2009.08366_, 2020.\n* [23] D. Guo, S. Lu, N. Duan, Y. Wang, M. Zhou, and J. Yin. Unixcoder: Unified cross-modal pre-training for code representation. _arXiv preprint arXiv:2203.03850_, 2022.\n* [24] H.-Q. Jin, M. Frumkin, and J. Yan. The openmp implementation of nas parallel benchmarks and its performance. 1999.\n* [25] T. Kadosh, N. Hasabnis, T. Mattson, Y. Pinter, G. Oren, et al. Pragformer: Data-driven parallel source code classification with transformers. 2023.\n* [26] D. P. Kingma and J. Ba. Adam: A method for stochastic optimization. _arXiv preprint arXiv:1412.6980_, 2014.\n* [27] D. Levy and L. Wolf. Learning to align the source code to the compiled object code. In _International Conference on Machine Learning_, pages 2043-2051. PMLR, 2017.\n* [28] Y. Li, C. Gu, T. Dullien, O. Vinyals, and P. Kohli. Graph matching networks for learning the similarity of graph structured objects. In _International conference on machine learning_, pages 3835-3845. PMLR, 2019.\n* [29] Z. Li, R. Atre, Z. U. Huda, A. Jannesari, and F. Wolf. Unveiling parallelization opportunities in sequential programs. _Journal of Systems and Software_, 117:282-295, July 2016. doi: 10.1016/j.jss.2016.03.045.\n* [30] L. Mou, G. Li, L. Zhang, T. Wang, and Z. Jin. Convolutional neural networks over tree structures for programming language processing. In _Proceedings of the AAAI conference on artificial intelligence_, volume 30, 2016.\n* [31] A. Paszke, S. Gross, F. Massa, A. Lerer, J. Bradbury, G. Chanan, T. Killeen, Z. Lin, N. Gimelshein, L. Antiga, A. Desmaison, A. Kopf, E. Yang, Z. DeVito, M. Raison, A. Tejani, S. Chilamkurthy, B. Steiner, L. Fang, J. Bai, and S. Chintala. Pytorch: An imperative style, high-performance deep learning library. In _Advances in Neural Information Processing Systems 32_, pages 8024-8035. Curran Associates, Inc., 2019. URL [http://papers.neurips.cc/paper/9015-pytorch-an-imperative-style-high-performance-deep-learning-library.pdf](http://papers.neurips.cc/paper/9015-pytorch-an-imperative-style-high-performance-deep-learning-library.pdf).\n* [32] L.-N. Pouchet and T. Yuki. Polybench: The polyhedral benchmark suite (version 4.2), 2017.\n* [33] D. Quinlan and C. Liao. The ROSE source-to-source compiler infrastructure. In _Cetus users and compiler infrastructure workshop, in conjunction with PACT_, volume 2011, page 1. Citeseer, 2011.\n* [34] J. Reinders, B. Ashbaugh, J. Brodman, M. Kinsner, J. Pennycook, X. Tian, J. Reinders, B. Ashbaugh, J. Brodman, M. Kinsner, et al. Common parallel patterns. _Data Parallel C++ Mastering DPC++ for Programming of Heterogeneous Systems using C++ and SYCL_, pages 323-352, 2021.\n* [35] A. Ronacher. Jinja2 documentation. _Welcome to Jinja2--Jinja2 Documentation (2.8-dev)_, 2008.\n* [36] I. Sanchez Barrera, D. Black-Schaffer, M. Casas, M. Moreto, A. Stupnikova, and M. Popov. Modeling and optimizing numa effects and prefetching with machine learning. In _Proceedings of the 34th ACM International Conference on Supercomputing_, pages 1-13, 2020.\n* [37] M. Schlichtkrull, T. N. Kipf, P. Bloem, R. Van Den Berg, I. Titov, and M. Welling. Modeling relational data with graph convolutional networks. In _The Semantic Web: 15th International Conference, ESWC 2018, Heraklion, Crete, Greece, June 3-7, 2018, Proceedings 15_, pages 593-607. Springer, 2018.\n* [38] A. TehraniJamsaz, M. Popov, A. Dutta, E. Saillard, and A. Jannesari. Learning intermediate representations using graph neural networks for numa and prefetchers optimization. In _2022 IEEE International Parallel and Distributed Processing Symposium (IPDPS)_, pages 1206-1216. IEEE, 2022.\n* [39] S. VenkataKeerthy, R. Aggarwal, S. Jain, M. S. Desarkar, R. Upadrasta, and Y. Srikant. Ir2vec: Llvm ir based scalable program embeddings. _ACM Transactions on Architecture and Code Optimization (TACO)_, 17(4):1-27, 2020.\n* [40] M. Y. Wang. Deep graph library: Towards efficient and scalable deep learning on graphs. In _ICLR workshop on representation learning on graphs and manifolds_, 2019.", "response": "## Review\n\n### Summary\n\nThis paper presents PerfoGraph, a novel graph-based representation for LLVM IR programs, building upon the established ProGraML framework. The proposed enhancements aim to improve performance optimization and program analysis applications through modifications in node and edge representations, which include collapsing nodes for reused local identifiers, employing decimal-based digit embedding for numerical constants, and breaking down composite data types into multiple nodes. The evaluation demonstrates that PerfoGraph outperforms ProGraML and traditional rule-based methods across six performance-oriented tasks, indicating the effectiveness of these modifications in reducing prediction errors and enhancing program graph representations.\n\n### Strengths\n\n- The paper is well-organized and clearly presents its three design contributions.\n- Effective graph modifications enhance the learnability of LLVM-IR graphs for GNNs.\n- The evaluation is comprehensive, covering multiple performance-oriented tasks, showing significant accuracy improvements over ProGraML.\n- The novel digit embedding approach effectively captures numerical awareness.\n- The paper identifies critical distinctions between composite data structures and regular data types.\n\n### Weaknesses\n\n- The improvements over ProGraML appear to be incremental rather than groundbreaking.\n- Ablation studies isolating the contributions of each modification are lacking, making it difficult to assess the individual impact of changes.\n- Comparison against LLMs and more recent methodologies in the field is insufficient.\n- The clarity of some sections, particularly regarding the representation of multi-dimensional arrays and the rationale for certain design choices, is inadequate.\n- Potential higher computational requirements compared to ProGraML may hinder practical deployment.\n\n### Questions\n\n- How does the digit embedding ensure that unknown numbers are handled during testing, and what specific information is encoded?\n- How does PerfoGraph's representation of composite data types extend beyond arrays and vectors?\n- What steps have been taken to ensure the correctness of ProGraML's implementation compared to PerfoGraph?\n- Could the authors provide visualizations of the digit embedding values for better understanding?\n- Will the authors consider moving ablation studies to the main text for clarity?\n\n### Soundness\n\n**Score:** 3\n\n**Description:** 3 = good, as the paper presents a technically solid implementation but lacks thorough ablation studies and comparisons with the latest methodologies.\n\n### Presentation\n\n**Score:** 4\n\n**Description:** 4 = excellent, due to the clear organization and detailed evaluations, though some sections could benefit from improved clarity.\n\n### Contribution\n\n**Score:** 3\n\n**Description:** 3 = good, as the paper makes significant contributions to the field with novel enhancements, although the improvements may not be fundamentally new.\n\n### Rating\n\n**Score:** 7\n\n**Description:** 7 = Accept, but needs minor improvements, as the paper is technically sound with moderate-to-high impact but requires further clarity and exploration of certain aspects.\n\n### Paper Decision\n\n**Decision:** Accept (poster)\n\n**Reasons:** The paper addresses important issues in program representation and demonstrates significant improvements over the existing ProGraML framework. While there are some areas for improvement, including the need for more comprehensive evaluations and clarity in certain sections, the overall contributions are valuable and warrant acceptance.\n"}
{"query": "You are a highly experienced, conscientious, and fair academic reviewer, please help me review this paper. The review should be organized into nine sections: \n1. Summary: A summary of the paper in 100-150 words.\n2. Strengths/Weaknesses/Questions: The Strengths/Weaknesses/Questions of paper, which should be listed in bullet points, with each point supported by specific examples from the article where possible.\n3. Soundness/Contribution/Presentation: Rate the paper's Soundness/Contribution/Presentation, and match this score to the corresponding description from the list below and provide the result. The possible scores and their descriptions are: \n    1 poor\n    2 fair\n    3 good\n    4 excellent\n4. Rating: Give this paper an appropriate rating, match this rating to the corresponding description from the list below and provide the result. The possible Ratings and their descriptions are: \n    1 strong reject\n    2 reject, significant issues present\n    3 reject, not good enough\n    4 possibly reject, but has redeeming facets\n    5 marginally below the acceptance threshold\n    6 marginally above the acceptance threshold\n    7 accept, but needs minor improvements \n    8 accept, good paper\n    9 strong accept, excellent work\n    10 strong accept, should be highlighted at the conference   \n5. Paper Decision: It must include the Decision itself(Accept or Reject) and the reasons for this decision, based on the criteria of originality, methodological soundness, significance of results, and clarity and logic of presentation.\n\nHere is the template for a review format, you must follow this format to output your review result:\n**Summary:**\nSummary content\n\n**Strengths:**\n- Strength 1\n- Strength 2\n- ...\n\n**Weaknesses:**\n- Weakness 1\n- Weakness 2\n- ...\n\n**Questions:**\n- Question 1\n- Question 2\n- ...\n\n**Soundness:**\nSoundness result\n\n**Presentation:**\nPresentation result\n\n**Contribution:**\nContribution result\n\n**Rating:**\nRating result\n\n**Paper Decision:**\n- Decision: Accept/Reject\n- Reasons: reasons content\n\n\nPlease ensure your feedback is objective and constructive. The paper is as follows:\n\n# Structured State Space Models for In-Context Reinforcement Learning\n\nChris Lu\n\nFLAIR, University of Oxford\n\nWork done during an internship at DeepMind. Contact: christopher.lu@exeter.ox.ac.uk\n\nYannick Schroecker\n\nDeepMind\n\nAlbert Gu\n\nDeepMind\n\nEmilio Parisotto\n\nDeepMind\n\nJakob Foerster\n\nFLAIR, University of Oxford\n\nSatinder Singh\n\nDeepMind\n\nFeryal Behbahani\n\nDeepMind\n\n###### Abstract\n\nStructured state space sequence (S4) models have recently achieved state-of-the-art performance on long-range sequence modeling tasks. These models also have fast inference speeds and parallelisable training, making them potentially useful in many reinforcement learning settings. We propose a modification to a variant of S4 that enables us to initialise and reset the hidden state in parallel, allowing us to tackle reinforcement learning tasks. We show that our modified architecture runs asymptotically faster than Transformers in sequence length and performs better than RNN's on a simple memory-based task. We evaluate our modified architecture on a set of partially-observable environments and find that, in practice, our model outperforms RNN's while also running over five times faster. Then, by leveraging the model's ability to handle long-range sequences, we achieve strong performance on a challenging meta-learning task in which the agent is given a randomly-sampled continuous control environment, combined with a randomly-sampled linear projection of the environment's observations and actions. Furthermore, we show the resulting model can adapt to out-of-distribution held-out tasks. Overall, the results presented in this paper show that structured state space models are fast and performant for in-context reinforcement learning tasks. We provide code at [https://github.com/luchris429/s5rl](https://github.com/luchris429/s5rl).\n\n## 1 Introduction\n\nStructured state space sequence (S4) models [12] and their variants such as S5 [38] have recently achieved impressive results in long-range sequence modelling tasks, far outperforming other popular sequence models such as the Transformer [42] and LSTM [16] on the Long-Range Arena benchmark [41]. Notably, S4 was the first architecture to achieve a non-trivial result on the difficult Path-X task, which requires the ability to handle extremely long-range dependencies of lengths \\(16k\\).\n\nFurthermore, S4 models display a number of desirable properties that are not directly tested by raw performance benchmarks. Unlike transformers, for which the per step runtime usually scales quadratically with the sequence length, S4 models have highly-scalable inference runtime performance, asymptotically using _constant_ memory and time per step with respect to the sequence length. While LSTMs and other RNNs also have this property, empirically, S4 models are far more performant while also being _parallelisable across the sequence dimension_ during training.\n\nWhile inference-time is normally not included when evaluating on sequence modelling benchmarks, it has a large impact on the scalability and wallclock-time for reinforcement learning (RL) becausethe agent uses inference to collect data from the environment. Thus, transformers usually have poor runtime performance in reinforcement learning [33]. While transformers have become the default architecture for many supervised sequence-modelling tasks [42], RNNs are still widely-used for memory-based RL tasks [29].\n\nThe ability to efficiently model contexts that are orders of magnitude larger may enable new possibilities in RL. This is particularly applicable in meta-reinforcement learning (Meta-RL), in which the agent is trained to adapt across _multiple_ environment episodes. One approach to Meta-RL, RL\\({}^{2}\\)[8; 43], uses sequence models to directly learn across these episodes, which can often result in trajectories that are thousands of steps long. Most instances of RL\\({}^{2}\\) approaches, however, are limited to narrow task distributions and short adaptation horizons because of their limited effective memory length and slow training speeds.\n\nUnfortunately, simply applying S4 models to reinforcement learning is challenging. This is because the most popular training paradigm in on-policy RL with multiple actors involves collecting fixed-length environment trajectories, which often cross episode boundaries. RNNs handle episode boundaries by resetting the hidden state at those transitions when performing backpropagation through time. Unlike RNNs, S4 models cannot simply reset their hidden states within the sequence because they train using a fixed convolution kernel instead of using backpropagation through time.\n\nA recent modification to S4, called Simplified Structured State Space Sequence Models (S5), replaces this convolution with a _parallel scan_ operation [38], which we describe in Section 2. In this paper, we propose a modification to S5 that enables resetting its hidden state within a trajectory during the training phase, which in turn allows practitioners to simply replace RNNs with S5 layers in existing frameworks. We then demonstrate S5's performance and runtime properties on the simple bsuite memory-length task [32], showing that S5 achieves a higher score than RNNs while also being nearly two times faster when using their provided baseline algorithm. We also re-implement and open source the recently-proposed Partially Observable Process Gym (POPGym) suite [27] in pure JAX, resulting in end-to-end evaluation speedups of over \\(30x\\). When evaluating our architecture on this suite, we show that S5 outperforms GRU's while also running over six times faster, achieving state-of-the-art results on the \"Repeat Hard\" task, which all other architectures previously struggled to solve. We further show that the modified S5 architecture can tackle a long-context partially-observed Meta-RL task with episode lengths of up to \\(6400\\). Finally, we evaluate S5 on a challenging Meta-RL task in which the environment samples a random DMControl environment [40] and a random linear projection of the state and action spaces at the beginning of each episode. We show that the S5 agent achieves higher returns than LSTMs in this setting. Furthermore, we demonstrate that the resulting S5 agent performs well even on random linear projections of the state and action spaces of out-of-distribution held-out tasks.\n\n## 2 Background\n\n### Structured State Space Sequence Models\n\nState Space Models (SSMs) have been widely used to model various phenomenon using first-order differential equations [14]. At each timestep \\(t\\), these models take an input signal \\(u(t)\\). This is used to update a latent state \\(x(t)\\) which in turn computes the signal \\(y(t)\\). Some of the more widely-used\n\n\\begin{table}\n\\begin{tabular}{l|l l l l l}  & Inference & Training & Parallel & Variable Lengths & bsuite Score \\\\ \\hline RNNs & **O(1)** & **O(\\(L\\))** & No & **Yes** & No \\\\ Transformers & O(\\(L^{2}\\)) & O(\\(L^{2}\\)) & **Yes** & **Yes** & **Yes** \\\\ S5 with \\(\\bullet\\) & **O(1)** & **O(\\(L\\))** & **Yes** & No & N/A \\\\ S5 with \\(\\oplus\\) & **O(1)** & **O(\\(L\\))** & **Yes** & **Yes** & **Yes** \\\\ \\end{tabular}\n\\end{table}\nTable 1: The different properties of the different architectures. The asymptotic runtimes are in terms of the sequence length \\(L\\) assume a constant hidden size. The bsuite scores correspond to whether or not they achieve a perfect score in the median runs on the bsuite memory length environment.\n\ncanonical SSMs are continuous-time linear SSMs, which are defined by the following equations:\n\n\\[\\dot{x}(t) =\\mathbf{A}x(t)+\\mathbf{B}u(t) \\tag{1}\\] \\[y(t) =\\mathbf{C}x(t)+\\mathbf{D}u(t)\\]\n\nwhere \\(\\mathbf{A},\\mathbf{B},\\mathbf{C}\\), and \\(\\mathbf{D}\\) are matrices of appropriate sizes. To model sequences with a fixed step size \\(\\Delta\\), one can _discretise_ the SSM using various techniques, such as the zero-order hold method, to obtain a simple linear recurrence:\n\n\\[x_{n} =\\mathbf{\\bar{A}}x_{n-1}+\\mathbf{\\bar{B}}u_{n} \\tag{2}\\] \\[y_{n} =\\mathbf{\\bar{C}}x_{n}+\\mathbf{\\bar{D}}u_{n}\\]\n\nwhere \\(\\mathbf{\\bar{A}},\\mathbf{\\bar{B}},\\mathbf{\\bar{C}}\\), and \\(\\mathbf{\\bar{D}}\\) can be calculated as functions of \\(\\mathbf{A},\\mathbf{B},\\mathbf{C},\\mathbf{D}\\), and \\(\\Delta\\).\n\nS4 [12] proposed the use of SSMs for modelling long sequences and various techniques to improve its stability, performance, and training speeds when combined with deep learning. For example, S4 models use a special matrix initialisation to better preserve sequence history called HiPPO [11].\n\nOne of the primary strengths of the S4 model is that it can be converted to both a recurrent model, which allows for fast and memory-efficient inference-time computation, and a convolutional model, which allows for efficient training that is _parallelisable across timesteps_[13].\n\nMore recently, Smith et al. [38] proposed multiple simplifications to S4, called S5. One of its contributions is the use of _parallel scans_ instead of convolution, which vastly simplifies S4's complexity and enables more flexible modifications. Parallel scans take advantage of the fact that the composition of _associative_ operations can be computed in any order. Recall that for an operation \\(\\bullet\\) to be associative, it must satisfy \\((x\\bullet y)\\bullet z=x\\bullet(y\\bullet z)\\).\n\nGiven an associative binary operator \\(\\bullet\\) and a sequence of length \\(N\\), parallel scan returns:\n\n\\[[e_{1},e_{1}\\bullet e_{2},\\cdots,e_{1}\\bullet e_{2}\\bullet\\cdots\\bullet e_{N}] \\tag{3}\\]\n\nFor example, when \\(\\bullet\\) is addition, the parallel scan calculates the prefix-sum, which returns the running total of an input sequence. Parallel scans can be computed in \\(O(\\log(N))\\) time when given a sequence of length \\(N\\), given \\(N\\) parallel processors.\n\nS5's parallel scan is applied to initial elements \\(e_{0:N}\\) defined as:\n\n\\[e_{k}=(e_{k,a},e_{k,b}):=(\\mathbf{\\bar{A}},\\mathbf{\\bar{B}}u_{k}) \\tag{4}\\]\n\nWhere \\(\\mathbf{\\bar{A}}\\), \\(\\mathbf{\\bar{B}}\\), and \\(u_{k}\\) are defined in Equation 2. S5's parallel operator is then defined as:\n\n\\[a_{i}\\bullet a_{j}=(a_{j,a}\\odot a_{i,a},a_{j,a}\\otimes a_{i,b}+a_{j,b}) \\tag{5}\\]\n\nwhere \\(\\odot\\) is matrix-matrix multiplication and \\(\\otimes\\) is matrix-vector multiplication. The parallel scan then generates the recurrence in the hidden state \\(x_{n}\\) defined in Equation 2.\n\n\\[e_{1} =(\\mathbf{\\bar{A}},\\mathbf{\\bar{B}}u_{1}) =(\\mathbf{\\bar{A}},x_{1}) \\tag{6}\\] \\[e_{1}\\bullet e_{2} =(\\mathbf{\\bar{A}}^{2},\\mathbf{\\bar{A}}x_{1}+\\mathbf{\\bar{B}}u_{ 2}) =(\\mathbf{\\bar{A}}^{2},x_{2})\\] (7) \\[e_{1}\\bullet e_{2}\\bullet e_{3} =(\\mathbf{\\bar{A}}^{2},\\mathbf{\\bar{A}}x_{2}+\\mathbf{\\bar{B}}u_{ 3}) =(\\mathbf{\\bar{A}}^{3},x_{3}) \\tag{8}\\]\n\nNote that the model assumes a hidden state initialised to \\(x_{0}=0\\) by initialising the scan with \\(e_{0}=(\\mathbf{I},0)\\).\n\n### Reinforcement Learning\n\nA Markov Decision Process (MDP) [39] is defined as a tuple \\(\\langle\\mathcal{S},\\mathcal{A},R,P,\\gamma\\rangle\\), which defines the environment. Here, \\(\\mathcal{S}\\) is the set of states, \\(\\mathcal{A}\\) the set of actions, \\(R\\) the reward function that maps from a given state and action to a real value \\(\\mathbb{R}\\), \\(P\\) defines the distribution of next-state transitions given a state and action, and \\(\\gamma\\) defines the discount factor. The agent's objective is to find a policy \\(\\pi\\) (a function which maps from a given state to a distribution over actions) which maximises the expected discounted sum of returns.\n\n\\[\\mathbb{E}[\\mathbb{R}^{\\gamma}|\\pi]=\\mathbb{E}_{s_{0}\\sim d,a_{0: \\infty}\\sim\\pi,s_{1:\\infty}\\sim P}\\Big{[}\\sum_{t=0}^{\\infty}\\gamma^{t}R(s_{t},a_{ t})\\Big{]}\\]\n\nIn a Partially-Observed Markov Decision Process (POMDP), the agent receives an observation \\(o_{t}\\sim O(s_{t})\\) instead of directly observing the state. Because of this, the optimal policy \\(\\pi^{*}\\) does not depend just on the current observation \\(o_{t}\\), but (in generality) also on all previous observations \\(o_{0:t}\\) and actions \\(a_{0:t}\\).\n\n## 3 Method\n\nWe first modify to S5 to handle variable-length sequences, which makes the architecture more suitable for tackling POMDPs. We then introduce a challenging new Meta-RL setting that tests for broad generalisation capabilities.\n\n### Resettable S5\n\nImplementations of on-policy policy gradient algorithms with parallel environments often collect fixed-length trajectory \"rollouts\" from the environment for training, despite the fact that the environment episode lengths are often far longer and vary significantly. Thus, the collected rollouts (1) often begin within an episode and (2) may contain episode boundaries. Note that there are other, more complicated, approaches to rollout collection that can be used to collect full episode trajectories [24].\n\nTo handle trajectory rollouts that begin in the middle of an episode, sequence models must be able to access the state of memory that was present prior to the rollout's collection. Usually, this is done by storing the RNN's hidden state at the beginning of each rollout to perform truncated backpropagation through time [44]. This is challenging to do with transformers because they do not normally have an explicit recurrent hidden state, but instead simply retain the entire history during inference time. This is similarly challenging for S4 models since they assume that all hidden states are initialised identically to perform a more efficient backwards pass.\n\nTo handle episode boundaries within the trajectory rollouts, memory-based models must be able to reset their hidden state, otherwise they would be accessing memory and context from other episodes. RNNs can trivially reset their hidden state when performing backpropagation through time, and transformers can mask out the invalid transitions. However, S4 models have no such mechanism to do this.\n\nTo resolve both of these issues, we modify S5's associative operator to include a reset flag that preserves the associative property, allowing S5 to efficiently train over sequences of varying lengths and hidden state initialisations. We create a new associative operator \\(\\oplus\\) that operates on elements \\(e_{k}\\) defined:\n\n\\[e_{k}=(e_{k,a},e_{k,b},e_{k,c}):=(\\mathbf{\\bar{A}},\\mathbf{\\bar{B}}u_{k},d_{k}) \\tag{9}\\]\n\nwhere \\(d_{k}\\in\\{0,1\\}\\) is the binary \"done\" signal for the given transition from the environment.\n\nWe define \\(\\oplus\\) to be:\n\n\\[a_{i}\\oplus a_{j}:=\\begin{cases}(a_{j,a}\\odot a_{i,a},a_{j,a}\\otimes a_{i,b}+a_ {j,b},a_{i,c})&\\text{if }a_{j,c}=0\\\\ (a_{j,a},a_{j,b},a_{j,c})&\\text{if }a_{j,c}=1\\end{cases}\\]\n\nWe prove that this operator is associative in Appendix A. We now show that the operator computes the desired value. Assuming \\(e_{n,c}=1\\) corresponds to a \"done\" transition while all other timesteps before it (\\(e_{0:n-1,c}=0\\)) and after it (\\(e_{n+1:L,c}=0\\)) do not, we see:\n\n\\[e_{0}\\oplus\\cdots\\oplus e_{n-1} =(\\mathbf{\\bar{A}}^{n-1},\\mathbf{\\bar{A}}x_{n-2}+\\mathbf{\\bar{B}} u_{n-1},0)\\] \\[=(\\mathbf{\\bar{A}}^{n-1},x_{n-1},0)\\] \\[e_{0}\\oplus\\cdots\\oplus e_{n} =(\\mathbf{\\bar{A}},\\mathbf{\\bar{B}}u_{n},1)\\] \\[=(\\mathbf{\\bar{A}},x_{n},1)\\] \\[e_{0}\\oplus\\cdots\\oplus e_{n+1} =(\\mathbf{\\bar{A}}^{2},\\mathbf{\\bar{A}}x_{n}+\\mathbf{\\bar{B}}u_{n+ 1},1)\\] \\[=(\\mathbf{\\bar{A}}^{2},x_{n+1},1)\\]\n\nNote that even if there are multiple \"resets\" within the rollout, the desired value will still be computed.\n\n### Multi-Environment Meta-Learning with Random Projections\n\nMost prior work in Meta-RL has only demonstrated the ability to _adapt_ to a small range of similar tasks [2]. Furthermore, the action and observation spaces usually remain identical across different tasks, which severely limits the diversity of meta-learning environments. To achieve more general meta-learning, ideally the agent should learn from tasks of varying complexity and dynamics. Inspired by Kirsch et al. [21], we propose taking _random linear projections_ of the observation space and action space to a fixed size, allowing us to use the same model for tasks of varying observation and action space sizes. Furthermore, randomised projections _vastly_ increase the number of tasks in the meta-learning space. We can then evaluate the ability of our model to _generalise_ to unseen held-out tasks.\n\nWe provide pseudocode for the environment implementation in Algorithm 1.\n\n```\n1:Initialization: \\(\\mathbf{\\bar{A}}\\), \\(\\mathbf{\\bar{B}}\\), \\(\\mathbf{\\bar{A}}^{n}\\), \\(\\mathbf{\\bar{B}}^{n}\\), \\(\\mathbf{\\bar{A}}^{n}\\), \\(\\mathbf{\\bar{B}}^{n}\\), \\(\\mathbf{\\bar{A}}^{n}\\), \\(\\mathbf{\\bar{B}}^{n}\\), \\(\\mathbf{\\bar{A}}^{n}\\), \\(\\mathbf{\\bar{B}}^{n}\\), \\(\\mathbf{\\bar{A}}^{n}\\), \\(\\mathbf{\\bar{B}}^{n}\\), \\(\\mathbf{\\bar{A}}^{n}\\), \\(\\mathbf{\\bar{B}}^{n}\\), \\(\\mathbf{\\bar{A}}^{n}\\), \\(\\mathbf{\\bar{B}}^{n}\\), \\(\\mathbf{\\bar{A}}^{n}\\), \\(\\mathbf{\\bar{B}}^{n}\\), \\(\\mathbf{\\bar{A}}^{n}\\), \\(\\mathbf{\\bar{B}}^{n}\\), \\(\\mathbf{\\bar{A}}^{n}\\), \\(\\mathbf{\\bar{B}}^{n}\\), \\(\\mathbf{\\bar{A}}^{n}\\), \\(\\mathbf{\\bar{B}}^{n}\\), \\(\\mathbf{\\bar{B}}^{n}\\), \\(\\mathbf{\\bar{A}}^{n}\\), \\(\\mathbf{\\bar{B}}^{n}\\), \\(\\mathbf{\\bar{A}}^{n}\\), \\(\\mathbf{\\bar{B}}^{n}\\), \\(\\mathbf{\\bar{B}}^{n}\\), \\(\\mathbf{\\bar{A}}^{n}\\), \\(\\mathbf{\\bar{B}}^{n}\\), \\(\\mathbf{\\bar{B}}^{n}\\), \\(\\mathbf{\\bar{A}}^{n}\\), \\(\\mathbf{\\bar{B}}^{n}\\), \\(\\mathbf{\\bar{B}}^{n}\\), \\(\\mathbf{\\bar{A}}^{n}\\), \\(\\mathbf{\\bar{B}}^{n}\\), \\(\\mathbf{\\bar{B}}^{n}\\), \\(\\mathbf{\\bar{A}}^{n}\\), \\(\\mathbf{\\bar{B}}^{n}\\), \\(\\mathbf{\\bar{B}}^{n}\\), \\(\\mathbf{\\bar{A}}^{n}\\), \\(\\mathbf{\\bar{B}}^{n}\\), \\(\\mathbf{\\bar{B}}^{n}\\), \\(\\mathbf{\\bar{A}}^{n}\\), \\(\\mathbf{\\bar{B}}^{n}\\), \\(\\mathbf{\\bar{B}}^{n}\\), \\(\\mathbf{\\bar{A}}^{n}\\), \\(\\mathbf{\\bar{B}}^{n}\\), \\(\\mathbf{\\bar{B}}^{n}\\), \\(\\mathbf{\\bar{A}}^{n}\\), \\(\\mathbf{\\bar{B}}^{n}\\), \\(\\mathbf{\\bar{B}}^{n}\\), \\(\\mathbf{\\bar{A}}^{n}\\), \\(\\mathbf{\\bar{B}}^{n}\\), \\(\\mathbf{\\bar{B}}^{n}\\), \\(\\mathbf{\\bar{A}}^{n}\\), \\(\\mathbf{\\bar{B}}^{n}\\), \\(\\mathbf{\\bar{B}}^{n}\\), \\(\\mathbf{\\bar{A}}^{n}\\), \\(\\mathbf{\\bar{B}}^{n}\\), \\(\\mathbf{\\bar{B}}^{n}\\), \\(\\mathbf{\\bar{A}}^{n}\\), \\(\\mathbf{\\bar{B}}^{n}\\), \\(\\mathbf{\\bar{B}}^{n}\\), \\(\\mathbf{\\bar{A}}^{n}\\), \\(\\mathbf{\\bar{B}}^{n}\\), \\(\\mathbf{\\bar{B}}^{n}\\), \\(\\mathbf{\\bar{A}}^{n}\\), \\(\\mathbf{\\bar{B}}^{n}\\), \\(\\mathbf{\\bar{B}}^{n}\\), \\(\\mathbf{\\bar{A}}^{n}\\), \\(\\mathbf{\\bar{B}}^{n}\\), \\(\\mathbf{\\bar{B}}^{n}\\), \\(\\mathbf{\\bar{A}}^{n}\\), \\(\\mathbf{\\bar{B}}^{n}\\), \\(\\mathbf{\\bar{B}}^{n}\\), \\(\\mathbf{\\bar{A}}^{n}\\), \\(\\mathbf{\\bar{B}}^{n}\\), \\(\\mathbf{\\bar{B}}^{n}\\), \\(\\mathbf{\\bar{A}}^{n}\\), \\(\\mathbf{\\bar{B}}^{n}\\), \\(\\mathbf{\\bar{B}}^{n}\\), \\(\\mathbf{\\bar{A}}^{n}\\), \\(\\mathbf{\\bar{B}}^{n}\\), \\(\\mathbf{\\bar{B}}^{n}\\), \\(\\mathbf{\\bar{A}}^{n}\\), \\(\\mathbf{\\bar{B}}^{n}\\), \\(\\mathbf{\\bar{B}}^{n}\\), \\(\\mathbf{\\bar{A}}^{n}\\), \\(\\mathbf{\\bar{B}}^{n}\\), \\(\\mathbf{\\bar{B}}^{n}\\), \\(\\mathbf{\\bar{A}}^{n}\\), \\(\\mathbf{\\bar{B}}^{n}\\), \\(\\mathbf{\\bar{B}}^{n}\\), \\(\\mathbf{\\bar{B}}^{n}\\), \\(\\mathbf{\\bar{A}}^{n}\\), \\(\\mathbf{\\bar{B}}^{n}\\), \\(\\mathbf{\\bar{B}}^{n}\\), \\(\\mathbf{\\bar{A}}^{n}\\), \\(\\mathbf{\\bar{B}}^{n}\\), \\(\\mathbf{\\bar{B}}^{n}\\), \\(\\mathbf{\\bar{A}}^{n}\\), \\(\\mathbf{\\bar{B}}^{n}\\), \\(\\mathbf{\\bar{\\(O(\\log(N))\\) in the backwards pass during training time (given enough processors), it is still bottlenecked by rollout collection from the environment, which takes \\(O(N)\\) time. Because of the poor runtime performance of transformers for long sequences, we did not collect results for them in the following experiments.\n\n### POPGym Environments\n\nWe evaluate our S5 architecture on environments from the Partially Observable Process Gym (POPGym) [27] suite, a set of simple environments designed to benchmark memory in deep RL. To increase experiment throughput on a limited compute budget, we carefully re-implemented environments from the POPGym suite entirely in JAX [3] by leveraging existing implementations of CartPole and Pendulum in Gymnax [22]. PyTorch does not support the associative scan operation, so we could directly use the S5 architecture in POPGym's RLLib benchmark.\n\nMorad et al. [27] evaluated several architectures and found the Gated Recurrent Unit (GRU) [5] to be the most performant. Thus, we compare our results to the GRU architecture proposed in the original POPGym benchmark. Note that POPGym's reported results use RLLib's [24] implementation of PPO, which makes several non-standard code-level implementation decisions. For example, it uses a dynamic KL-divergence coefficient on top of the clipped surrogate objective of PPO [37] - a feature that does not appear in most PPO implementations [9]. We instead use a recurrent PPO implementation that is more closely aligned with StableBaselines3 [35] and CleanRL's [18] recurrent PPO implementations. We include more discussion and the hyperparameters in Appendix B.\n\nFigure 1: Evaluating S5, LSTM, and Self-Attention across different Bsuite memory lengths in terms of (a) memory usage, (b) runtime, and (c) return. Error bars report the standard error of the mean across 5 seeds. Runs were performed on a single NVIDIA A100.\n\nFigure 2: (a) Results across implemented environments in POPGym’s suite. Scores are normalised by the max-MMER per-environment. The shaded region represents the standard error of the mean across eight seeds. (b) The runtime for a single seed averaged across the environments for each architecture. Note that our implementation is end-to-end compiled to run entirely on a single NVIDIA A40.\n\nWe show the results in Figure 2 and Appendix C. Notably, the S5 architecture performs well on the challenging \"Repeat Previous Hard\" task, far outperforming all architectures tested in Morad et al. [27]. Furthermore, the S5 architecture also runs over six times faster than the GRU architecture.\n\n### Randomly-Projected CartPole In-Context\n\nWe first demonstrate the long-horizon in-context learning capabilities of the modified S5 architecture by meta-learning across random projections of POPGym's Stateless CartPole (CartPole without velocity observations) task. More specifically, we perform the observation and action projections described in Section 3.2 and report the results in Figure 3.\n\nAgents are given \\(16\\) trials per episode on randomly-sampled linear projections of StatelessCartPole's observation and action spaces. We find that S5 with \\(\\oplus\\) outperforms GRU's while running twice as quickly. Furthermore, we show that performance improves across trials, demonstrating in-context learning and show that the modified S5 architecture, unlike the GRU, can continue to learn even past its training horizon, which was previously shown using Transformers in Adaptive Agent Team et al. [1].\n\n### Multi-Environment Meta-Reinforcement Learning\n\nWe run our S5 architecture, an LSTM baseline, and a memory-less fully-connected network in the environment described in Section 3.2. For these experiments, we use Muesli [15] as our policy optimisation algorithm. We randomly project the environment observations to a fixed size of \\(12\\) and randomly project from an action space of size \\(2\\) to the corresponding environment's action space. We selected all of the DMControl environments and tasks that had observation and action spaces of size equal to or less than those values and split them into train and test set environments.\n\nWe use the S5 architecture described in Smith et al. [38], which consists of multiple stacked layers of S5 blocks. For both S5 and LSTM architectures, we found that setting the trajectory length equal to the maximum episode length \\(1k\\) achieved the best performance in this setting. We provide a more detailed description of our\n\nFigure 4: The mean of the return across all of the training environments. The shaded regions represent the range of returns reported across the three seeds. The environment observations and actions are randomly projected as described in Algorithm 1.\n\nFigure 3: (a) Performance and (b) runtime on randomly-projected StatelessCartPole across 4 seeds. The shaded region represents the standard error. (c) Shows performance at the end of training across different trials. We evaluate on \\(32\\) trials even though we only train on \\(16\\). GRU’s appear to have overfit to the training horizon while S5 models continue to perform well. The error bars represent the standard error across 4 seeds. Runs were performed on a single NVIDIA A100.\n\narchitecture and hyperparameters in the supplementary material.\n\n**In-Distribution Training Results** We meta-train the model across the six DMControl environments in Figure 5 and show the mean performance across them in Figure 4. Note that while these environments are in the training distribution, they are still being evaluated on _unseen random linear projections_ of the state and action spaces and are _not given task labels_. Thus, the agent must infer from the reward and obfuscated dynamics which environment it is in. In this setting, S5 outperforms LSTMs in both sample efficiency and ultimate performance.\n\n## 5 Related Work\n\nS4 models have previously been shown to work in a number of previous settings, including audio generation [10] and video modeling [28]. Concurrent work investigated S4 models in other architecture to symmetric models [20], loss functions [17], target values [30], or drift functions [25]. In contrast, this work achieves generalisation by vastly increasing the task distribution _without limiting the expressivity of the underlying model_, which eliminates the need for hand-crafted restrictions.\n\nFigure 5: Results across the different environments of the training distribution. The shaded regions represent the range of returns reported across the three seeds. The environment observations and actions are randomly projected as described in Algorithm 1\n\nFigure 6: The mean of the return across all of the unseen held-out environments. The shaded regions represent the range of returns reported across the three seeds. The environment observations and actions are randomly projected as described in Algorithm 1.\n\nSome works perform long-horizon meta-reinforcement learning through the use of evolution strategies [17; 25]. This is because RNNs and Transformers have historically struggled to model very long sequences due to computational constraints and vanishing or exploding gradients [26]. However, evolution strategies are notoriously sample inefficient and computationally expensive.\n\nOther works have investigated different sequence model architectures for memory-based reinforcement learning. Ni et al. [29] showed that using well-tuned RNNs can be particularly effective compared to many more complicated methods in POMDPs. Parisotto et al. [34] investigated the use of transformer-based models for memory-based reinforcement learning environments.\n\nSequence models have also been used for a number of other tasks in RL. For example, Transformers have been used for offline reinforcement learning [4], multi-task behavioural cloning [36], and algorithm distillation [23]. Concurrent work used transformers to also demonstrate out-of-distribution generalisation in meta-reinforcement learning by leveraging a large task space [1].\n\n## 6 Conclusion and Limitations\n\nIn this paper, we investigated the performance of the recently-proposed S5 model architecture in reinforcement learning. S5 models are highly promising for reinforcement learning because of their strong performance in sequence modelling tasks and their fast and efficient runtime properties, with clear advantages over RNNs and Transformers. After identifying challenges in integrating S5 models into existing recurrent reinforcement learning implementations, we made a simple modification to the method that allowed us to reset the hidden state within a training sequence.\n\nWe then showed the desirable properties S5 models in the bsuite memory length task. We demonstrated that S5 is _asymptotically_ faster than Transformers in the sequence length. Furthermore, we also showed that S5 models run nearly twice as quickly as LSTMs with the same number of parameters while outperforming them. We further evaluated our S5 architecture on environments in the POPGym suite [27], where we match or outperform RNNs while also running nearly five times faster. We achieve strong results in the \"Repeat Previous Hard\" task, which previous models struggled to solve.\n\nFinally, we proposed a new meta-learning setting in which we meta-learn across random linear projections of the observation and action spaces of randomly sampled DMControl tasks. We show that S5 outperforms LSTMs in this setting. We then evaluate the models on held-out DMControl tasks and demonstrate out-of-distribution performance to unseen tasks through in-context adaptation.\n\nThere are several possible ways to further investigate S5 models for reinforcement learning in future work. For one, it may be possible to learn or distill [23] entire reinforcement learning algorithms within an S5 model, given its ability to scale to extremely long contexts. Another direction would be to investigate S5 models for continuous-time RL settings [7]. While \\(\\Delta\\), the discrete time between timesteps, is fixed for the original S4 model, S5 can in theory use a different \\(\\Delta\\) for each timestep.\n\nFigure 7: Results when evaluating on held-out DMControl tasks. The shaded regions represent the range of returns reported across the three seeds. The environment observations and actions are randomly projected as described in Algorithm 1\n\n**Limitations:** There are several notable limitations of this architecture and analysis. Firstly, implementing the associative scan operator is currently not possible using PyTorch, limiting us to using the JAX [3] framework. Furthermore, on tasks where short rollouts are sufficient to achieve good performance, S5 offers limited speedups, as rolling out across time is no longer a bottleneck. Finally, it was not possible to perform a fully comprehensive hyperparameter sweep in our results in Section 4.4 because the experiments used significant amounts of compute.\n\n## Acknowledgments and Disclosure of Funding\n\nWork funded by DeepMind. We would like to thank Antonio Orvieto, Robert Lange, Junhyok Oh, Greg Farquhar, Ted Moskovitz, and the rest of the Discovery Team at DeepMind for their helpful discussions throughout the course of the project.\n\n## References\n\n* [1] Adaptive Agent Team, Jakob Bauer, Kate Baumli, Satinder Baveja, Feryal Behbahani, Avishkar Bhoopchand, Nathalie Bradley-Schmieg, Michael Chang, Natalie Clay, Adrian Collister, Vibhavari Dasagi, Lucy Gonzalez, Karol Gregor, Edward Hughes, Sheleem Kashem, Maria Loks-Thompson, Hannah Openshaw, Jack Parker-Holder, Shreya Pathak, Nicolas Perez-Nieves, Nemanja Rakicevic, Tim Rocktaschel, Yannick Schroecker, Jakub Sygnowski, Karl Tuyls, Sarah York, Alexander Zacherl, and Lei Zhang. Human-timescale adaptation in an open-ended task space. _arXiv e-prints_, 2023.\n* [2] Jacob Beck, Risto Vuorio, Evan Zheran Liu, Zheng Xiong, Luisa Zintgraf, Chelsea Finn, and Shimon Whiteson. A survey of meta-reinforcement learning. _arXiv preprint arXiv:2301.08028_, 2023.\n* [3] James Bradbury, Roy Frostig, Peter Hawkins, Matthew James Johnson, Chris Leary, Dougal Maclaurin, George Necula, Adam Paszke, Jake VanderPlas, Skye Wanderman-Milne, and Qiao Zhang. JAX: composable transformations of Python+NumPy programs, 2018. URL [http://github.com/google/jax](http://github.com/google/jax).\n* [4] Lili Chen, Kevin Lu, Aravind Rajeswaran, Kimin Lee, Aditya Grover, Misha Laskin, Pieter Abbeel, Aravind Srinivas, and Igor Mordatch. Decision transformer: Reinforcement learning via sequence modeling. _Advances in neural information processing systems_, 34:15084-15097, 2021.\n* [5] Kyunghyun Cho, Bart Van Merrienboer, Caglar Gulcehre, Dzmitry Bahdanau, Fethi Bougares, Holger Schwenk, and Yoshua Bengio. Learning phrase representations using rnn encoder-decoder for statistical machine translation. _arXiv preprint arXiv:1406.1078_, 2014.\n* [6] Shmuel Bar David, Itamar Zimerman, Eliya Nachmani, and Lior Wolf. Decision s4: Efficient sequence-based rl via state spaces layers. In _The Eleventh International Conference on Learning Representations_, 2023. URL [https://openreview.net/forum?id=kqHkCVS7wbj](https://openreview.net/forum?id=kqHkCVS7wbj).\n* [7] Kenji Doya. Reinforcement learning in continuous time and space. _Neural computation_, 12(1):219-245, 2000.\n* [8] Yan Duan, John Schulman, Xi Chen, Peter L Bartlett, Ilya Sutskever, and Pieter Abbeel. Rl2: Fast reinforcement learning via slow reinforcement learning. _arXiv preprint arXiv:1611.02779_, 2016.\n* [9] Logan Engstrom, Andrew Ilyas, Shibani Santurkar, Dimitris Tsipras, Firdaus Janoos, Larry Rudolph, and Aleksander Madry. Implementation matters in deep policy gradients: A case study on ppo and trpo. In _International Conference on Learning Representations_, 2020.\n* [10] Karan Goel, Albert Gu, Chris Donahue, and Christopher Re. It's raw! audio generation with state-space models. _arXiv preprint arXiv:2202.09729_, 2022.\n* [11] Albert Gu, Tri Dao, Stefano Ermon, Atri Rudra, and Christopher Re. Hippo: Recurrent memory with optimal polynomial projections. _Advances in neural information processing systems_, 33:1474-1487, 2020.\n\n* Gu et al. [2021] Albert Gu, Karan Goel, and Christopher Re. Efficiently modeling long sequences with structured state spaces. _arXiv preprint arXiv:2111.00396_, 2021.\n* Gu et al. [2021] Albert Gu, Isys Johnson, Karan Goel, Khaled Saab, Tri Dao, Atri Rudra, and Christopher Re. Combining recurrent, convolutional, and continuous-time models with linear state space layers. _Advances in neural information processing systems_, 34:572-585, 2021.\n* Hamilton [1994] James D Hamilton. State-space models. _Handbook of econometrics_, 4:3039-3080, 1994.\n* Hessel et al. [2021] Matteo Hessel, Ivo Danihelka, Fabio Viola, Arthur Guez, Simon Schmitt, Laurent Sifre, Theophane Weber, David Silver, and Hado Van Hasselt. Muesli: Combining improvements in policy optimization. In _International Conference on Machine Learning_, pages 4214-4226. PMLR, 2021.\n* Hochreiter and Schmidhuber [1997] Sepp Hochreiter and Jurgen Schmidhuber. Long short-term memory. _Neural computation_, 9(8):1735-1780, 1997.\n* Houthooft et al. [2018] Rein Houthooft, Yuhua Chen, Phillip Isola, Bradly Stadie, Filip Wolski, OpenAI Jonathan Ho, and Pieter Abbeel. Evolved policy gradients. _Advances in Neural Information Processing Systems_, 31, 2018.\n* Huang et al. [2022] Shengyi Huang, Rousslan Fernand Julien Dossa, Chang Ye, Jeff Braga, Dipam Chakraborty, Kinal Mehta, and Joao G.M. Araujo. Cleanrl: High-quality single-file implementations of deep reinforcement learning algorithms. _Journal of Machine Learning Research_, 23(274):1-18, 2022. URL [http://jmlr.org/papers/v23/21-1342.html](http://jmlr.org/papers/v23/21-1342.html).\n* Kapturowski et al. [2018] Steven Kapturowski, Georg Ostrovski, John Quan, Remi Munos, and Will Dabney. Recurrent experience replay in distributed reinforcement learning. In _International conference on learning representations_, 2018.\n* Kirsch et al. [2022] Louis Kirsch, Sebastian Flennerhag, Hado van Hasselt, Abram Friesen, Junhyuk Oh, and Yutian Chen. Introducing symmetries to black box meta reinforcement learning. In _Proceedings of the AAAI Conference on Artificial Intelligence_, volume 36, pages 7202-7210, 2022.\n* Kirsch et al. [2022] Louis Kirsch, James Harrison, Jascha Sohl-Dickstein, and Luke Metz. General-purpose in-context learning by meta-learning transformers. _arXiv preprint arXiv:2212.04458_, 2022.\n* Lange [2022] Robert Tjarko Lange. gymax: A JAX-based reinforcement learning environment library, 2022. URL [http://github.com/RobertTLange/gymax](http://github.com/RobertTLange/gymax).\n* Laskin et al. [2022] Michael Laskin, Luyu Wang, Junhyuk Oh, Emilio Parisotto, Stephen Spencer, Richie Steigerwald, DJ Strouse, Steven Hansen, Angelos Filos, Ethan Brooks, et al. In-context reinforcement learning with algorithm distillation. _arXiv preprint arXiv:2210.14215_, 2022.\n* Liang et al. [2018] Eric Liang, Richard Liaw, Robert Nishihara, Philipp Moritz, Roy Fox, Ken Goldberg, Joseph Gonzalez, Michael Jordan, and Ion Stoica. Rllib: Abstractions for distributed reinforcement learning. In _International Conference on Machine Learning_, pages 3053-3062. PMLR, 2018.\n* Lu et al. [2022] Chris Lu, Jakub Grudzien Kuba, Alistair Letcher, Luke Metz, Christian Schroeder de Witt, and Jakob Foerster. Discovered policy optimisation. _arXiv preprint arXiv:2210.05639_, 2022.\n* Metz et al. [2021] Luke Metz, C Daniel Freeman, Samuel S Schoenholz, and Tal Kachman. Gradients are not all you need. _arXiv preprint arXiv:2111.05803_, 2021.\n* Morad et al. [2023] Steven Morad, Ryan Kortvelesy, Matteo Bettini, Stephan Liwicki, and Amanda Prorok. POP-Gym: Benchmarking partially observable reinforcement learning. In _The Eleventh International Conference on Learning Representations_, 2023. URL [https://openreview.net/forum?id=chDrutUTsoK](https://openreview.net/forum?id=chDrutUTsoK).\n* Nguyen et al. [2022] Eric Nguyen, Karan Goel, Albert Gu, Gordon Downs, Preey Shah, Tri Dao, Stephen Baccus, and Christopher Re. S4nd: Modeling images and videos as multidimensional signals with state spaces. In _Advances in Neural Information Processing Systems_, 2022.\n\n* Ni et al. [2022] Tianwei Ni, Benjamin Eysenbach, and Ruslan Salakhutdinov. Recurrent model-free rl can be a strong baseline for many pomdps. In _International Conference on Machine Learning_, pages 16691-16723. PMLR, 2022.\n* Oh et al. [2020] Junhyuk Oh, Matteo Hessel, Wojciech M Czarnecki, Zhongwen Xu, Hado P van Hasselt, Satinder Singh, and David Silver. Discovering reinforcement learning algorithms. _Advances in Neural Information Processing Systems_, 33:1060-1070, 2020.\n* O'Keefe and Dostrovsky [1971] John O'Keefe and Jonathan Dostrovsky. The hippocampus as a spatial map: Preliminary evidence from unit activity in the freely-moving rat. _Brain research_, 1971.\n* Osband et al. [2019] Ian Osband, Yotam Doron, Matteo Hessel, John Aslanides, Eren Sezener, Andre Saraiva, Katrina McKinney, Tor Lattimore, Csaba Szepesvari, Satinder Singh, et al. Behaviour suite for reinforcement learning. _arXiv preprint arXiv:1908.03568_, 2019.\n* Parisotto and Salakhutdinov [2021] Emilio Parisotto and Ruslan Salakhutdinov. Efficient transformers in reinforcement learning using actor-learner distillation. _arXiv preprint arXiv:2104.01655_, 2021.\n* Parisotto et al. [2020] Emilio Parisotto, Francis Song, Jack Rae, Razvan Pascanu, Caglar Gulcehre, Siddhant Jayakumar, Max Jaderberg, Raphael Lopez Kaufman, Aidan Clark, Seb Noury, et al. Stabilizing transformers for reinforcement learning. In _International conference on machine learning_, pages 7487-7498. PMLR, 2020.\n* Raffin et al. [2021] Antonin Raffin, Ashley Hill, Adam Gleave, Anssi Kanervisto, Maximilian Ernestus, and Noah Dormann. Stable-baselines3: Reliable reinforcement learning implementations. _Journal of Machine Learning Research_, 22(268):1-8, 2021. URL [http://jmlr.org/papers/v22/20-1364.html](http://jmlr.org/papers/v22/20-1364.html).\n* Reed et al. [2022] Scott Reed, Konrad Zolna, Emilio Parisotto, Sergio Gomez Colmenarejo, Alexander Novikov, Gabriel Barth-Maron, Mai Gimenez, Yury Sulsky, Jackie Kay, Jost Tobias Springenberg, et al. A generalist agent. _arXiv preprint arXiv:2205.06175_, 2022.\n* Schulman et al. [2017] John Schulman, F. Wolski, Prafulla Dhariwal, Alec Radford, and Oleg Klimov. Proximal policy optimization algorithms. _ArXiv_, abs/1707.06347, 2017.\n* Smith et al. [2022] Jimmy TH Smith, Andrew Warrington, and Scott W Linderman. Simplified state space layers for sequence modeling. _arXiv preprint arXiv:2208.04933_, 2022.\n* Sutton and Barto [2018] Richard S Sutton and Andrew G Barto. _Reinforcement learning: An introduction_. 2018.\n* Tassa et al. [2018] Yuval Tassa, Yotam Doron, Alistair Muldal, Tom Erez, Yazhe Li, Diego de Las Casas, David Budden, Abbas Abdolmaleki, Josh Merel, Andrew Lefrancq, et al. Deepmind control suite. _arXiv preprint arXiv:1801.00690_, 2018.\n* Tay et al. [2020] Yi Tay, Mostafa Dehghani, Samira Abnar, Yikang Shen, Dara Bahri, Philip Pham, Jinfeng Rao, Liu Yang, Sebastian Ruder, and Donald Metzler. Long range arena: A benchmark for efficient transformers. _arXiv preprint arXiv:2011.04006_, 2020.\n* Vaswani et al. [2017] Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Lukasz Kaiser, and Illia Polosukhin. Attention is all you need. _Advances in neural information processing systems_, 30, 2017.\n* Wang et al. [2016] Jane X Wang, Zeb Kurth-Nelson, Dhruva Tirumala, Hubert Soyer, Joel Z Leibo, Remi Munos, Charles Blundell, Dharshan Kumaran, and Matt Botvinick. Learning to reinforcement learn. _arXiv preprint arXiv:1611.05763_, 2016.\n* Williams and Zipser [1995] Ronald J Williams and David Zipser. Gradient-based learning algorithms for recurrent. _Backpropagation: Theory, architectures, and applications_, 433:17, 1995.\n\nProof of Associativity of Binary Operator\n\nRecall that \\(\\oplus\\) is defined as:\n\n\\[a_{i}\\oplus a_{j}:=\\begin{cases}(a_{j,a}\\odot a_{i,a},a_{j,a}\\otimes a_{i,b}+a_{j, b},a_{i,c})&\\text{if }a_{j,c}=0\\\\ (a_{j,a},a_{j,b},a_{j,c})&\\text{if }a_{j,c}=1\\end{cases}\\]\n\nThis is equivalent to the following:\n\n\\[a_{i}\\oplus a_{j}:=\\begin{cases}((a_{i}\\bullet a_{j})_{a},(a_{i}\\bullet a_{j}) _{b},a_{i,c})&\\text{if }a_{j,c}=0\\\\ a_{j}&\\text{if }a_{j,c}=1\\end{cases}\\]\n\nwhere \\(\\bullet\\) is S5's binary operator defined in Equation 5. Note that \\(\\bullet\\)'s associativity was proved in Smith et al. [38]. Using this, we can prove the associativity of \\(\\oplus\\).\n\nLet \\(x\\), \\(y\\), and \\(z\\) refer to three elements. We will prove that for all possible values of \\(x\\), \\(y\\), and \\(z\\), \\(\\oplus\\) retains associativity.\n\n**Case 1: \\(z_{c}=1\\)**\n\n\\[(x\\oplus y)\\oplus z =z \\tag{10}\\] \\[=y\\oplus z\\] (11) \\[=x\\oplus(y\\oplus z) \\tag{12}\\]\n\n**Case 2: \\(z_{c}=0\\)** and \\(y_{c}=1\\)\n\n\\[(x\\oplus y)\\oplus z =y\\oplus z \\tag{13}\\] \\[\\text{Note that }(y\\oplus z)_{c} =1\\text{ thus,}\\] (14) \\[=x\\oplus(y\\oplus z) \\tag{15}\\]\n\n**Case 3: \\(z_{c}=0\\)** and \\(y_{c}=0\\)\n\n\\[(x\\oplus y)\\oplus z =((x\\bullet y)_{a},(x\\bullet y)_{b},x_{c})\\oplus z \\tag{16}\\] \\[=(((x\\bullet y)\\bullet z)_{a},((x\\bullet y)\\bullet z)_{b},x_{c})\\] (17) \\[=((x\\bullet(y\\bullet z))_{a},(x\\bullet(y\\bullet z))_{b},x_{c})\\] (18) \\[=x\\oplus((y\\bullet z)_{a},(y\\bullet z)_{b},y_{c})\\] (19) \\[=x\\oplus(y\\oplus z) \\tag{20}\\]\n\n## Appendix B Hyperparameters\n\n\\begin{table}\n\\begin{tabular}{l|l} Parameter & Value \\\\ \\hline Adam Learning Rate & 3e-4 \\\\ Value Function Weight & 1.0 \\\\ Muesli Auxiliary Loss Weight & 3.0 \\\\ Muesli Model Unroll Length & 1.0 \\\\ Encoder Layer Sizes & [512, 512] \\\\ Number of Environments & 1024 \\\\ Discount \\(\\gamma\\) & 0.995 \\\\ Rollout Length & 1000 \\\\ Offline Data Fraction & 0.0 \\\\ Total Frames & 2e9 \\\\ LSTM Hidden Size & 512 \\\\ Projected Observation Size & 12 \\\\ Projected Action Size & 2 \\\\ S5 Layers & 10 \\\\ S5 Hidden Size & 256 \\\\ S5 Discretization & ZOH \\\\ S5 \\(\\Delta\\) min & 0.001 \\\\ S5 \\(\\Delta\\) max & 0.1 \\\\ \\end{tabular}\n\\end{table}\nTable 4: Hyperparameters for training Muesli on Multi-Environment Meta-RL. These experiments were run using 64 TPUv3’s.\n\n## Appendix C POPGym Discussion\n\nMorad et al. [27] used RLLib's [24] implementation of PPO, which differs significantly from standard implementations of PPO. It uses a dynamic KL-divergence coefficient on top of the clipped surrogate objective of PPO [37]. Furthermore, they use advanced orchestration to return full episode trajectories, rather than using the more commonly-studied \"stored state\" [19] strategy.\n\nInstead, we follow the design decisions outlined in the StableBaselines3 [35] and CleanRL's [18] recurrent PPO implementations. While this results in different results shown in Table 5 for the same architecture, it recovers similar performance across the environments. Notably, our S5 architecture far outperforms the best performing architecture in Morad et al. [27] in the \"RepeatPreviousHard\" environment.\n\nWe used the learning rate, number of environments, unroll length, timesteps, epochs, and minibatches, GAE \\(\\lambda\\), and model architectures from Morad et al. [27]. However, we used the standard clipping coefficient \\(\\epsilon\\) of \\(0.2\\) instead of \\(0.3\\) to account for the lack of a dynamic KL-divergence coefficient. Note that we also adjusted the S5 architecture to contain approximately the same number of _parameters_ as the GRU implementation instead of matching the size of the _hidden state_, which was done in Morad et al. [27].\n\nWe did not evaluate our architecture across the full POPGym suite. To enable more rapid experimentation, we implement our algorithms and environments end-to-end in JAX [3, 25]. While the original POPGym results took \\(2\\) hours per trial with a GRU with a Quadro RTX 8000 and 24 CPUs, we could run our experiments using only \\(3\\)_minutes_ per trial on an NVIDIA A40. Because of this, we selected environments from Morad et al. [27] to implement in JAX. We chose the CartPole, Pendulum, and Repeat environments because they are modified versions of existing environments in Lange [22]. We found that the \"Easy\" and \"Medium\" versions of these environments were not informative, as most models perform well on them and only report the \"Hard\" difficulty results.\n\nWe attach our code in the supplementary materials.\n\n\\begin{table}\n\\begin{tabular}{r|c c c c|c c}  & Stateless & Noisy Stateless & Stateless & Noisy Stateless & Repeat Previous \\\\  & CartPole Hard & CartPole Hard & Pendulum Hard & Pendulum Hard & Hard \\\\ \\hline S5 (ours) & \\(\\mathbf{1.0\\pm 0.0}\\) & \\(\\mathbf{0.28\\pm 0.0}\\) & \\(\\mathbf{0.79\\pm 0.01}\\) & \\(0.55\\pm 0.01\\) & \\(\\mathbf{0.91\\pm 0.01}\\) \\\\ GRU (ours) & \\(\\mathbf{1.0\\pm 0.0}\\) & \\(0.27\\pm 0.0\\) & \\(0.75\\pm 0.0\\) & \\(\\mathbf{0.61\\pm 0.01}\\) & \\(-0.46\\pm 0.01\\) \\\\ MLP (ours) & \\(0.26\\pm 0.0\\) & \\(0.22\\pm 0.0\\) & \\(0.41\\pm 0.02\\) & \\(0.34\\pm 0.01\\) & \\(-0.48\\pm 0.00\\) \\\\ \\hline GRU & \\(\\mathbf{1.000\\pm 0.000}\\) & \\(0.390\\pm 0.007\\) & \\(\\mathbf{0.828\\pm 0.001}\\) & \\(\\mathbf{0.657\\pm 0.002}\\) & \\(-0.428\\pm 0.002\\) \\\\ MLP & \\(0.265\\pm 0.002\\) & \\(0.229\\pm 0.002\\) & \\(0.477\\pm 0.030\\) & \\(0.351\\pm 0.012\\) & \\(-0.486\\pm 0.002\\) \\\\ IndRNN & \\(\\mathbf{1.000\\pm 0.000}\\) & \\(\\mathbf{0.404\\pm 0.005}\\) & \\(0.804\\pm 0.023\\) & \\(0.521\\pm 0.109\\) & \\(-0.384\\pm 0.013\\) \\\\ LMU & \\(0.987\\pm 0.007\\) & \\(0.352\\pm 0.019\\) & \\(0.806\\pm 0.006\\) & \\(0.563\\pm 0.014\\) & \\(\\mathbf{0.191\\pm 0.041}\\) \\\\ S4D & \\(0.127\\pm 0.026\\) & \\(0.207\\pm 0.007\\) & \\(0.303\\pm 0.014\\) & \\(0.289\\pm 0.011\\) & \\(-0.491\\pm 0.001\\) \\\\ FART & \\(\\mathbf{0.996\\pm 0.000}\\) & \\(0.366\\pm 0.002\\) & \\(0.698\\pm 0.077\\) & \\(0.553\\pm 0.007\\) & \\(-0.485\\pm 0.001\\) \\\\ \\end{tabular}\n\\end{table}\nTable 5: Results in POPGym’s suite. The reported number is the max-mean episodic reward (MMER) used in Morad et al. [27]. To calculate this, we take the mean episodic reward for each epoch, and then take the maximum over all epochs. For our results above, the mean and standard deviation across eight seeds are reported. The results below are selected architectures from Morad et al. [27], which also includes the best-performing one from each environment. They report the mean and standard deviation across three trials.", "response": "## Review\n\n### Summary\n\nThe paper presents a modified version of the S5 architecture, which allows for resetting the recurrent state, enabling it to serve as a drop-in replacement for RNNs and Transformers in reinforcement learning (RL) tasks. The authors demonstrate that their approach outperforms existing models in terms of performance and computational efficiency across various partially observed RL environments. Specifically, the resettable S5 model showcases superior long-range sequence modeling capabilities and improved inference speed. Additionally, the paper introduces a meta-learning task with random projections of state and action spaces, further validating the efficacy of S5 in in-context adaptation scenarios.\n\n### Strengths\n\n- The paper is the first to leverage S4-like models in RL, showcasing an important contribution.\n- A novel associative operator is proposed to handle episodic resets in the S5 architecture.\n- S5 outperforms RNNs and Transformers in terms of task performance and computational efficiency.\n- The introduction of a meta-RL environment with random projections serves as a useful benchmark for future research.\n- The experimental setup covers both POMDPs and meta-learning, indicating a broad applicability of the method.\n\n### Weaknesses\n\n- The evaluation does not sufficiently explore the potential of S5 for longer sequence lengths, which is a key benefit of state-space models.\n- The experimental design lacks variety and depth, particularly in the choice of tasks used for evaluation.\n- The main contribution regarding the reset mechanism appears to be a trivial modification rather than a novel advancement.\n- There is insufficient discussion of the limitations of the proposed architecture and its comparison to existing models.\n- The narrative around in-context RL is confusing, and the evaluation protocol lacks clarity.\n\n### Questions\n\n- What is the rationale behind the improved out-of-distribution (OOD) generalization of the S5 model compared to baselines?\n- Can the authors clarify the evaluation procedure for the meta-learning tasks, particularly regarding training and held-out tasks?\n- How do models perform beyond a sequence length of 512, especially in relation to S5's capabilities?\n- Why is the performance gain of S5 primarily observed on the 'Repeat Previous Hard' task, and what implications does this have for the benchmark's validity?\n- Is there a reason for the absence of transformer-based baselines in the evaluation?\n\n### Soundness\n\n**Score:** 2\n\n**Description:** 2 = fair; while the methodology is generally sound, the contributions and evaluations are somewhat limited and lack depth.\n\n### Presentation\n\n**Score:** 2\n\n**Description:** 2 = fair; the writing has clarity issues, particularly regarding the explanation of key concepts and evaluation procedures.\n\n### Contribution\n\n**Score:** 3\n\n**Description:** 3 = good; the paper makes a notable contribution to the field by introducing a new application of the S5 architecture in RL, despite some concerns about the novelty of the modification.\n\n### Rating\n\n**Score:** 6\n\n**Description:** 6 = accept with minor revisions; while the paper shows promise and addresses significant challenges in RL, it requires some clarifications and improvements in experimental design.\n\n### Paper Decision\n\n**Decision:** Accept (poster)\n\n**Reasons:** The paper presents an innovative approach to leveraging the S5 architecture for reinforcement learning, demonstrating solid technical foundations and promising results. However, it requires clarity in its presentation and a more thorough exploration of its limitations and evaluations. The strengths outweigh the weaknesses, leading to a recommendation for acceptance with minor revisions.\n"}
{"query": "You are a highly experienced, conscientious, and fair academic reviewer, please help me review this paper. The review should be organized into nine sections: \n1. Summary: A summary of the paper in 100-150 words.\n2. Strengths/Weaknesses/Questions: The Strengths/Weaknesses/Questions of paper, which should be listed in bullet points, with each point supported by specific examples from the article where possible.\n3. Soundness/Contribution/Presentation: Rate the paper's Soundness/Contribution/Presentation, and match this score to the corresponding description from the list below and provide the result. The possible scores and their descriptions are: \n    1 poor\n    2 fair\n    3 good\n    4 excellent\n4. Rating: Give this paper an appropriate rating, match this rating to the corresponding description from the list below and provide the result. The possible Ratings and their descriptions are: \n    1 strong reject\n    2 reject, significant issues present\n    3 reject, not good enough\n    4 possibly reject, but has redeeming facets\n    5 marginally below the acceptance threshold\n    6 marginally above the acceptance threshold\n    7 accept, but needs minor improvements \n    8 accept, good paper\n    9 strong accept, excellent work\n    10 strong accept, should be highlighted at the conference   \n5. Paper Decision: It must include the Decision itself(Accept or Reject) and the reasons for this decision, based on the criteria of originality, methodological soundness, significance of results, and clarity and logic of presentation.\n\nHere is the template for a review format, you must follow this format to output your review result:\n**Summary:**\nSummary content\n\n**Strengths:**\n- Strength 1\n- Strength 2\n- ...\n\n**Weaknesses:**\n- Weakness 1\n- Weakness 2\n- ...\n\n**Questions:**\n- Question 1\n- Question 2\n- ...\n\n**Soundness:**\nSoundness result\n\n**Presentation:**\nPresentation result\n\n**Contribution:**\nContribution result\n\n**Rating:**\nRating result\n\n**Paper Decision:**\n- Decision: Accept/Reject\n- Reasons: reasons content\n\n\nPlease ensure your feedback is objective and constructive. The paper is as follows:\n\n# Bayesian Metric Learning for Uncertainty Quantification in Image Retrieval\n\n Frederik Warburg\n\nTechnical University of Denmark\n\nfrwa@dtu.dk\n\nEqual contribution\n\n&Marco Miani\n\nTechnical University of Denmark\n\nmmia@dtu.dk\n\n&Silas Brack\n\nTechnical University of Denmark\n\nsilasbrack@gmail.com\n\n&Soren Hauberg\n\nTechnical University of Denmark\n\nsohau@dtu.dk\n\n###### Abstract\n\nWe propose a Bayesian encoder for metric learning. Rather than relying on neural amortization as done in prior works, we learn a distribution over the network weights with the Laplace Approximation. We first prove that the contrastive loss is a negative log-likelihood on the spherical space. We propose three methods that ensure a positive definite covariance matrix. Lastly, we present a novel decomposition of the Generalized Gauss-Newton approximation. Empirically, we show that our Laplacian Metric Learner (LAM) yields well-calibrated uncertainties, reliably detects out-of-distribution examples, and has state-of-the-art predictive performance.\n\n## 1 Introduction\n\nMetric learning seeks data representations where similar observations are near and dissimilar ones are far. This elegantly allows for building retrieval systems with simple nearest-neighbor search. Such systems easily cope with a large number of classes, and new classes can organically be added without retraining. While these retrieval systems show impressive performance, they quickly, and with no raised alarms, deteriorate with out-of-distribution data [38]. In particular, in safety-critical applications, the lack of uncertainty estimation is a concern as retrieval errors may propagate unnoticed through the system, resulting in erroneous and possibly dangerous decisions.\n\nWe present the Laplacian Metric Learner (LAM) to estimate reliable uncertainties of image embeddings as demonstrated in Fig. 1. We learn a distribution over the network weights (weight posterior) from\n\nFigure 1: **Reliable stochastic embeddings.** LAM (ours) estimates reliable uncertainties of latent image embeddings that reflect the amount of blur, noise, or occlusion in the input image.\n\nwhich we obtain a stochastic representation by embedding an image through sampled neural networks. This Bayesian formulation has multiple benefits, namely (1) robustness to out-of-distribution examples, (2) calibrated in-distribution uncertainties, and (3) a slight improvement in predictive performance.\n\nMore specifically, our method extends the Laplace Approximation [27] for metric learning. We present a probabilistic interpretation of the contrastive loss [15] which justifies that it can be interpreted as an unnormalized negative log-likelihood. We propose three solutions to ensure a positive semidefinite Hessian for the contrastive loss and present two approaches to compute the Generalized Gauss-Newton [12] approximation for \\(\\ell_{2}\\)-normalized networks. Finally, we boost our method with the online training procedure from [29] to achieve state-of-the-art performance.\n\nWe are not the first to consider uncertainty quantification in image retrieval. Seminal works [38; 33] have addressed the lack of uncertainties in retrieval with _amortized inference_[14], where a neural network predicts a stochastic embedding. The issues with this approach are that (1) it requires strong assumptions on the distribution of the embedding, (2) the networks are often brittle and difficult to optimize, and (3) out-of-distribution detection relies on the network's capacity to extrapolate uncertainties. As neural networks extrapolate poorly [50], the resulting _predicted_ uncertainties are unreliable for out-of-distribution data [10] and are thus, in practice, of limited value.\n\nIn contrast, our method does not assume any distribution on the stochastic embeddings, is simple to optimize, and does not rely on a neural network to extrapolate uncertainties. Instead, our weight posterior is derived from the curvature of the loss landscape and the uncertainties of the latent embeddings deduced (rather than learned) with sampling. Empirically, we show that this leads to reliable out-of-distribution performance and calibrated uncertainties in both controlled toy experiments and challenging real-world applications such as bird, face, and place recognition.\n\n## 2 Related Work\n\n**Metric learning** attempts to map data to an embedding space, where similar data are close together and dissimilar data are far apart. This is especially useful for retrieval tasks with many classes and few observations per class such as place recognition [46] and face recognition [37] or for tasks where classes are not well-defined, such as food tastes [48] or narratives in online discussions [4].\n\nThere exist many metric losses that optimize for a well-behaved embedding space. We refer to the excellent survey by [30] for an overview. We here focus on the _contrastive loss_[15]\n\n\\[\\mathcal{L}_{\\text{con}}(\\theta)=\\frac{1}{2}\\|f_{\\theta}(x_{a})-f_{\\theta}(x_ {p})\\|^{2}+\\frac{1}{2}\\max\\left(0,m-\\|f_{\\theta}(x_{a})-f_{\\theta}(x_{n})\\|^{2 }\\right), \\tag{1}\\]\n\nwhich has shown state-of-the-art performance [30] and is one of the most commonly used metric losses. Here, \\(f_{\\theta}\\) is a neural network parametrized by \\(\\theta\\) which maps from the observation space to the embedding space. The loss consists of two terms, one that attracts observations from the same class (_anchor_\\(x_{a}\\) and _positive_\\(x_{p}\\)), and one that repels observations from different classes (_anchor_\\(x_{a}\\) and _negative_\\(x_{n}\\)). The margin \\(m\\geq 0\\) ensures that negatives are repelled sufficiently far. We will later present a probabilistic extension of the contrastive loss that allows us to learn stochastic, rather than deterministic, features in the embedding space.\n\n**Uncertainty in deep learning** is studied across many domains to mitigate fatal accidents and allow for human intervention when neural networks make erroneous predictions. Current methods can be divided into methods that apply amortized optimization to train a neural network to predict the parameters of the output distribution, and methods that do not. The amortized methods, best known from the variational autoencoder (VAE) [19; 36], seem attractive at first as they can directly estimate the output distribution (without requiring sampling), but they suffer from mode collapse and are sensitive to out-of-distribution data due to the poor extrapolation capabilities of neural networks [31; 10]. 'Bayes by Backprop' [2] learns a distribution over parameters variationally but is often deemed too brittle for practical applications. Alternatives to amortized methods includes deep ensembles [23], stochastic weight averaging (SWAG) [28], Monte-Carlo dropout [13] and Laplace Approximation (LA) [24; 27] which all approximate the generally intractable weight posterior \\(p(\\theta|\\mathcal{D})\\) of a neural network. We propose to extend LA to metric learning.\n\n**Laplace approximations (LA)** can be applied for every loss function \\(\\mathcal{L}\\) that can be interpreted as an unnormalized log-posterior by performing a second-order Taylor expansion around a chosen weight vector \\(\\theta^{*}\\) such that\n\n\\[\\mathcal{L}(\\theta)\\approx\\mathcal{L}^{*}+(\\theta-\\theta^{*})^{\\top}\\nabla \\mathcal{L}^{*}+\\frac{1}{2}(\\theta-\\theta^{*})^{\\top}\\nabla^{2}\\mathcal{L}^{*}( \\theta-\\theta^{*}), \\tag{2}\\]\n\nwhere \\(\\mathcal{L}^{*}\\) is the loss evaluated in \\(\\theta^{*}\\). Imposing the unnormalized log-posterior to be a second-order polynomial is equivalent to assuming the posterior to be Gaussian. If \\(\\theta^{*}\\) is a MAP estimate, the first-order term vanishes, and the second-order term can be interpreted as a precision matrix, the inverse of the covariance. Assuming \\(\\theta^{*}\\) is a MAP estimate, this second-order term is negative semi-definite for common (convex) supervised losses, such as the mean-squared error and cross-entropy. Recently, Daxberger et al. [8] demonstrated that post-hoc LA is scalable and produces well-behaved uncertainties for classification and regression. The Laplacian Autoencoder (LAE) [29] improves on the post-hoc LA with an online Monte Carlo EM training procedure to learn a well-behaved posterior. It demonstrates state-of-the-art uncertainty quantification for unsupervised representation learning.\n\n**Uncertainty in metric learning** is not new [44], but the majority of recent methods apply amortized inference to predict distributions in the embedding space [47; 3; 5; 33; 38; 39; 41], making them sensitive to mode collapse and out-of-distribution data. Alternatives like deep ensembles [42] and Monte-Carlo dropout [43] suffer from increased training time, poor empirical performance, and limited Bayesian interpretation [8]. We explore LA in metric learning and attain state-of-the-art performance.\n\n## 3 Laplacian Metric Learning\n\nTo perform Bayesian retrieval, we estimate the weight posterior of the embedding network \\(f_{\\theta}\\) such that we can sample data embeddings to propagate uncertainty through the decision process. The embedding network is parametrized by \\(\\theta\\in\\Theta\\) and trained with the contrastive loss. The network maps an image \\(x\\in\\mathcal{X}:=\\mathbb{R}^{HWC}\\) to an embedding \\(z\\in\\mathcal{Z}\\), which is restricted to be on a \\(Z\\)-dimensional unit sphere \\(\\mathcal{Z}:=\\mathcal{S}^{Z}\\). This spherical normalization is often done in retrieval to obtain faster retrieval and a slight performance boost [1; 35]. Fig. 2 illustrate our Bayesian mapping from image to latent space.\n\n**Laplace Approximation (LA) recap.** We apply LA to obtain the posterior over the weights \\(\\theta\\). LA comes in two flavors: (1) _The post-hoc LA_ is found by first training a standard deterministic network through gradient steps with the contrastive loss to find the _maximum a posteriori_ (MAP) parameters \\(\\theta^{*}\\). Since we are in a local optimum, the first-order term in the second-order Taylor expansion (Eq. 2) vanishes, and we can define the parameter distribution as\n\n\\[p(\\theta|\\mathcal{D})=\\mathcal{N}\\left(\\theta\\Big{|}\\theta^{*}, \\left(\\nabla_{\\theta}^{2}\\mathcal{L}_{\\text{con}}\\left(\\theta^{*};\\mathcal{D} \\right)+\\sigma_{\\text{prior}}^{-2}\\mathbb{I}\\right)^{-1}\\right). \\tag{3}\\]\n\nThe advantage of post-hoc LA is that the training procedure does not change, and already trained neural networks can be made Bayesian. In practice, however, we empirically observe post-hoc LA to be unstable. We refer to the large standard deviation in Table 1, Table 2, and Table 3 over random seeds. The instability stems from curvature differences between the local minimas through stochastic optimization. (2) _Online LA_[29] improves on this instability by marginalizing the LA during training with Monte Carlo EM. This helps the training recover a solution \\(\\theta^{*}\\) where the Hessian reflects the loss landscape. Specifically, at each step \\(t\\) during training, we keep in memory a Gaussian distribution on the parameters \\(q^{t}(\\theta)=\\mathcal{N}(\\theta|\\theta_{t},H_{\\theta_{t}}^{-1})\\). The parameters are updated through an expected gradient step\n\n\\[\\theta_{t+1}=\\theta_{t}+\\lambda\\mathbb{E}_{\\theta\\sim q^{t}}[ \\nabla_{\\theta}\\mathcal{L}_{\\text{con}}(\\theta;\\mathcal{D})] \\tag{4}\\]\n\nand a discounted Laplace update\n\n\\[H_{\\theta_{t+1}}=(1-\\alpha)H_{\\theta_{t}}+\\nabla_{\\theta}^{2} \\mathcal{L}_{\\text{con}}(\\theta;\\mathcal{D}), \\tag{5}\\]\n\nwhere \\(\\alpha\\) describes an exponential moving average, similar to momentum-like training. The initialization follows the isotropic prior \\(q^{0}(\\theta)=\\mathcal{N}(\\theta|0,\\sigma_{\\text{prior}}^{2}\\mathbb{I})\\).\n\nFigure 2: **Model overview**. We learn a distribution over parameters, such that we embed an image through sampled encoders \\(f_{\\theta}\\) to points \\(z_{i}\\) (red dots) in a latent space \\(\\mathcal{Z}\\). We reduce these latent samples to a single measure of uncertainty by estimating the parameters of a von Mises-Fisher distribution.\n\n**The probabilistic contrastive likelihood.** LA expects the loss function to be a log-posterior, that is log-likelihood plus log-prior. A simple choice of the log-prior is \\(\\|\\theta\\|_{2}^{2}\\) (weight decay). Further, we find that the contrastive loss Eq. 1 has a probabilistic interpretation and is a negative log-likelihood on the spherical space. We define the attractive term \\(\\mathcal{P}^{\\rightarrow\\rightarrow}(z|x,\\theta)\\sim\\mathcal{N}^{S}(z|f_{\\theta }(x),\\kappa)\\) and the repelling term \\(\\mathcal{P}^{\\leftarrow\\rightarrow}(z|x,\\theta)\\sim\\mathcal{N}^{S}(z|-f_{ \\theta}(x),\\kappa)\\) as Von Mises-Fisher distributions on the latent space spherical space, where the concentration parameter \\(\\kappa\\geq 0\\). The product of these two likelihoods yields a contrastive likelihood, which is valid on the spherical space, and its negative log-likelihood is equivalent to the contrastive loss (proof and details in 1).\n\nThe intuition, helped by electrostatics, is that the log probability density of these Von Mises-Fisher distributions (i.e. spherical Gaussians) plays the role of a potential energy. The precision parameter \\(\\kappa\\) controls the concentration of the potential energy and, consequently, the strength of the associated force. If \\(\\kappa=0\\) the distribution is uniform, the potential is constant and the associated force is zero. On the spherical space, moreover, a repulsive term from some point is equivalent to an attractive term from the antipodal point, which in turn is mathematically equivalent (see 1) to a negative precision \\(\\kappa\\). This is the fundamental reason behind the non positive-definiteness of the hessian.\n\n**Hessian of the contrastive loss.** Both post-hoc and online LA require the Hessian of the contrastive loss \\(\\nabla_{\\theta}^{2}\\mathcal{L}_{\\text{con}}(\\theta;\\mathcal{D})\\). The Hessian is commonly approximated with the Generalized Gauss-Newton (GGN) approximation [12, 8, 6, 11]. The GGN decomposes the loss into \\(\\mathcal{L}=g\\circ f\\), where \\(g\\) is usually chosen as the loss function and \\(f\\) the model function, and only \\(f\\) is linearized [22].\n\nHowever, in our case, this decomposition is non-trivial. Recall that the last layer of our network is an \\(\\ell_{2}\\) normalization layer, which projects embeddings onto a hyper-sphere. This normalization layer can either be viewed as part of the model \\(f\\) (linearized normalization layer) or part of the loss \\(g\\) (non-linearized normalization layer). The former can be interpreted as using the _Euclidean_ distance and the latter as using the _Arccos_ distance for the contrastive loss (see 1 and 1). These two share the zero- and first-order terms for normalized embeddings but not the second-order derivatives due to the GGN linearization. The Euclidean interpretation leads to simpler derivatives and interpretations, and we will therefore use it for our derivations. We emphasize that the Arccos is theoretically a more accurate approximation, because the \\(\\ell_{2}\\)-layer is not linearized, and we provide derivations in 1.\n\nThe GGN matrix for contrastive loss with the _Euclidean_ interpretation is given by\n\n\\[\\nabla_{\\theta}^{2}\\mathcal{L}_{\\text{con}}(\\theta;\\mathcal{I}) =\\sum_{ij\\in\\mathcal{I}}H_{\\theta}^{ij}=\\sum_{ij\\in\\mathcal{I}_{p} }H_{\\theta}^{ij}+\\sum_{ij\\in\\mathcal{I}_{n}}H_{\\theta}^{ij} \\tag{6}\\] \\[\\stackrel{{\\text{GGN}}}{{\\approx}}\\sum_{ij\\in \\mathcal{I}_{p}}J_{\\theta}^{ij\\,\\top}\\underbrace{\\big{(}\\begin{smallmatrix}1& -1\\\\ -1&1\\end{smallmatrix}\\big{)}}_{:=H_{p}}J_{\\theta}^{ij}+\\sum_{ij\\in\\mathcal{I}_{n }}J_{\\theta}^{ij\\,\\top}\\underbrace{\\big{(}\\begin{smallmatrix}-1&1\\\\ -1&1\\end{smallmatrix}\\big{)}}_{:=H_{n}}J_{\\theta}^{ij},\\]\n\nwhere \\(J_{\\theta}^{ij}=\\big{(}J_{\\theta}f_{\\theta}(x_{i})^{\\top},J_{\\theta}f_{\\theta} (x_{j})^{\\top}\\big{)}^{\\top}\\), with \\(J_{\\theta}\\) is the Jacobian wrt. the parameters and \\(H_{p}\\) and \\(H_{n}\\) are the Hessians of the contrastive loss wrt. the model output for positive and negative pairs. The first sum runs over positive pairs and the second sum runs over negative pairs _within the margin_. Negative pairs outside the margin do not contribute to the Hessian, and can therefore be ignored to reduce the computational load (1).\n\nThe eigenvalues of the Hessian wrt. to the output are \\((0,2)\\) and \\((-2,0)\\) for the positive \\(H_{p}\\) and negative \\(H_{n}\\) terms, so we are not guaranteed to have a positive semidefinite Hessian, \\(H_{\\theta}\\). To avoid covariances with negative eigenvalues, we propose three solutions to ensure a positive semidefinite Hessian. Proofs are in 1.\n\n**Ensuring positive definiteness of the covariance matrix.** We do not want to be restricted in the choice of the prior except to have non-zero precision, so we must ensure that \\(\\nabla_{\\theta}^{2}\\mathcal{L}_{\\text{con}}(\\theta^{*};\\mathcal{D})\\) is positive semidefinite. Differently from the standard convex losses, this is not ensured by the GGN approximation [18]. Our main insight is that we can ensure a positive semidefinite Hessian \\(H_{\\theta}\\) by only manipulating the Hessians \\(H_{p}\\) and \\(H_{n}\\) in Eq. 6.\n\n_1. Positive: The repelling term is ignored, such that only positive pairs contribute to the Hessian._\n\n\\[H_{p} =\\begin{pmatrix}1&-1\\\\ -1&1\\end{pmatrix}, H_{n} =\\begin{pmatrix}0&0\\\\ 0&0\\end{pmatrix} \\tag{7}\\]_2. Fixed: The cross derivatives are ignored._\n\n\\[H_{p} =\\left(\\begin{array}{cc}1&0\\\\ 0&1\\end{array}\\right), H_{n} =\\left(\\begin{array}{cc}-1&0\\\\ 0&-1\\end{array}\\right) \\tag{8}\\]\n\n_3. Full: Positive semidefiniteness is ensured with ReLU, \\(\\max(0,\\nabla_{\\theta}^{2}\\mathcal{L}_{\\text{con}}(\\theta;\\mathcal{D}))\\), on the Hessian of the loss wrt. the parameters._\n\n\\[H_{p} =\\left(\\begin{array}{cc}1&-1\\\\ -1&1\\end{array}\\right), H_{n} =\\left(\\begin{array}{cc}-1&1\\\\ 1&-1\\end{array}\\right) \\tag{9}\\]\n\nThe _positive_ approximation is inspired by [38], which only uses positive pairs to train an uncertainty module. The gradient arrows in Fig. 2(a) illustrate that negative pairs are neglected when computing the Hessian of the contrastive loss. The _fixed_ approximation considers one data point at a time, assuming the other one is fixed. Thus, given a pair of data points, this can be interpreted as first moving one data point, and then the second (rather than both at the same time). We formalize this in 1. Fig. 2(b) illustrate this idea when all points except \\(a\\) are fixed. Lastly, we propose the _full_ Hessian of the contrastive loss (Fig. 2(c)) and ensure positive semidefiniteness by computing the ReLU of the diagonal Hessian. This approximation can equivalently be interpreted as a projection into the space of psd matrixes. In practice, the Hessian scales quadratically in memory wrt. the number of parameters. To mitigate this, we approximate this Hessian by its diagonal and only apply the LA on the last layer [25; 9]. We experimentally find that the _fixed_ approximation yields the best performance (Section 4).\n\n**Hard negative mining.** Most pairs, namely the negatives outside the margin, do not contribute to the Hessian, so it is wasteful to compute their Hessian. Therefore, we use hard negative mining [30] to only compute the Hessian of pairs that have non-zero Hessian, _i.e._ the negative sample lie within the margin (illustrated with the dotted line in Fig. 3).\n\n**Von Mises-Fisher distribution.** To obtain a single measure of uncertainty from our sampled image embeddings (red dots in Fig. 4), we fit a von Mises-Fisher distribution. The von Mises-Fisher distribution describes a normal distribution where all probability mass lies on a \\(Z\\)-dimensional hyper-sphere. It is parametrized with a directional mean \\(\\mu\\) and a scalar concentration parameter \\(\\kappa\\), which can be interpreted as the inverse of an isotropic covariance \\(\\kappa=1/\\sigma^{2}\\), _i.e._, small \\(\\kappa\\) means high uncertainty and large \\(\\kappa\\) means low uncertainty. There exist several methods to estimate \\(\\kappa\\). We opt for the simplest and most computationally efficient [40] (see 1).\n\n## 4 Experiments\n\nWe benchmark our method against strong probabilistic retrieval models. Probabilistic Face Embeddings (PFE) [38] and Hedge Image Embedding (HIB) [33] perform amortized inference and thus estimate the mean and variance of latent observation. We also compare against MC Dropout [13] and Deep Ensemble [23], two approximate Bayesian methods, which have successfully been applied in image retrieval [43; 42].\n\nWe compare the models' _predictive performance_ with the recall (recall@\\(k\\)) and mean average precision (mAP@\\(k\\)) among the \\(k\\) nearest neighbors [47; 30; 1]. We evaluate the models' abilities to _interpolate_ and _extrapolate_ uncertainties by measuring the Area Under the Sparsification Curve (AUSC), Expected Calibration Error (ECE) on in-distribution (ID) data, the Area Under Receiver Operator Curve (AUROC), and Area Under Precision-Recall Curve (AUPRC) on out-of-distribution (OoD) data. We provide more details on these metrics in 1.\n\nFootnote 1: See [https://github.com/FrederikWarburg/bayesian-metric-learning](https://github.com/FrederikWarburg/bayesian-metric-learning)\n\nWe extend StochMan [11] with the Hessian backpropagation for the contrastive loss, and the PyTorch [34] code is publicly available2. 1 details the experimental setup.\n\nFootnote 2: See [https://github.com/FrederikWarburg/bayesian-metric-learning](https://github.com/FrederikWarburg/bayesian-metric-learning)\n\nFigure 3: **Hessian approximations.** To ensure a positive semidefinite Hessian approximation we propose three approximations. In (a) only the positives \\(p\\) contribute to the Hessian as the negatives \\(n\\) are ignored. In (b) we consider one point at a time, e.g., only the anchor \\(a\\) contributes. In (c) we consider all interactions.\n\n**Experimental Summary.** We first summarize our experimental results. Across five datasets, three network architectures, and three different sizes of the latent space (ranging from \\(3\\) to \\(2048\\)), we find that LAM has well-calibrated uncertainties, reliably detects OoD examples, and achieves state-of-the-art predictive performance. Fig. 3(a) shows that the uncertainties from online LAM reliably identify OoD examples. Online LAM outperforms other Bayesian methods, such as post-hoc LAM and MC dropout, on this task, which in turn clearly improves upon amortized methods that rely on a neural network to extrapolate uncertainties. Fig. 3(b) shows that LAM consistently matches or outperforms existing image retrieval methods in terms of predictive performance. We find that the fixed Hessian approximation with the Arccos distance performs the best, especially on higher dimensional data.\n\n**Ablation: Positive definiteness covariance matrix.** We experimentally study which method to ensure a positive semidefinite Hessian has the best performance measured in both predictive performance (mAP@5) and uncertainty quantification (AUROC, AUSC). We found that all methods perform similarly on simple datasets and low dimensional hyper-spheres, but the fixed approximation with Arccos distance performs better on more challenging datasets and higher dimensional hyper-spheres. We present results on one of these more challenging datasets, namely the LFW [17] face recognition dataset with the CUB200 [45] bird dataset as an OoD dataset. We use a ResNet50 [16] with a GeM pooling layer [35] and a \\(2048\\) dimensional embedding and diagonal, last-layer LA [8].\n\nTable 1 shows the performance for post-hoc and online LA with fixed, positive, or full Hessian approximation using either Euclidean or Arccos distance. Across all metrics, the online LA with Arccos distance and the fixed Hessian approximation performs similarly or the best. We proceed to benchmark this method against several strong probabilistic baselines on closed-set retrieval and a more challenging open-set retrieval.\n\n**Closed-Set Retrieval.** OoD capabilities are critical for identifying distributional shifts, outliers, and irregular user inputs, which might hinder the propagation of erroneous decisions in an automated system. We evaluate OoD performance on the commonly used benchmarks [31], where we use (1)\n\n\\begin{table}\n\\begin{tabular}{l l c c c} \\hline \\hline  & & mAP@5 \\(\\uparrow\\) & AUROC \\(\\uparrow\\) & AUSC \\(\\uparrow\\) \\\\ \\hline \\multirow{4}{*}{_S_} & Euclidean fix & 0.70 \\(\\pm\\) 0.0 & 0.57 \\(\\pm\\) 0.25 & 0.44 \\(\\pm\\) 0.01 \\\\  & Euclidean pos & 0.70 \\(\\pm\\) 0.0 & 0.58 \\(\\pm\\) 0.23 & 0.45 \\(\\pm\\) 0.01 \\\\  & Euclidean full & 0.70 \\(\\pm\\) 0.0 & 0.56 \\(\\pm\\) 0.26 & 0.44 \\(\\pm\\) 0.01 \\\\  & Arccos fix & 0.69 \\(\\pm\\) 0.0 & 0.53 \\(\\pm\\) 0.20 & 0.46 \\(\\pm\\) 0.02 \\\\  & Arccos pos & 0.70 \\(\\pm\\) 0.0 & 0.29 \\(\\pm\\) 0.11 & 0.48 \\(\\pm\\) 0.01 \\\\  & Arccos full & 0.69 \\(\\pm\\) 0.0 & 0.55 \\(\\pm\\) 0.18 & 0.45 \\(\\pm\\) 0.01 \\\\ \\hline \\multirow{4}{*}{_S_} & Euclidean fix & 0.63 \\(\\pm\\) 0.01 & 0.77 \\(\\pm\\) 0.04 & 0.31 \\(\\pm\\) 0.02 \\\\  & Euclidean pos & 0.70 \\(\\pm\\) 0.0 & 0.38 \\(\\pm\\) 0.10 & 0.47 \\(\\pm\\) 0.01 \\\\  & Euclidean full & 0.67 \\(\\pm\\) 0.01 & 0.59 \\(\\pm\\) 0.04 & 0.42 \\(\\pm\\) 0.01 \\\\  & Arccos fix & **0.71 \\(\\pm\\) 0.0** & **0.78 \\(\\pm\\) 0.18** & 0.50 \\(\\pm\\) 0.03 \\\\  & Arccos pos & 0.70 \\(\\pm\\) 0.0 & 0.23 \\(\\pm\\) 0.03 & 0.46 \\(\\pm\\) 0.00 \\\\  & Arccos full & **0.71 \\(\\pm\\) 0.0** & 0.70 \\(\\pm\\) 0.12 & **0.51 \\(\\pm\\) 0.01** \\\\ \\hline \\hline \\end{tabular}\n\\end{table}\nTable 1: **Ablation on Hessian approximation and GGN decomposition.** Online LA with the fixed approximation and Arccos distance performs best. Error bars show one std. across five runs.\n\nFigure 4: **Summary of experimental results.** LAM consistently outperforms existing methods on OoD detection, as measured by AUROC, and matches or surpasses in predictive performance measured by mAP@\\(k\\). Error bars show one std across five runs.\n\nFigure 5: **Calibration Curves.**FashionMNIST [49] as ID and MNIST [26] as OoD, and (2) CIFAR10 [21] as ID and SVHN [32] as OoD. We use, respectively, a standard \\(2\\)- or \\(3\\)-layer relu convolutional network followed by a single linear layer on which we compute LA with a diagonal Hessian.\n\n_FashionMNIST (ID) vs MNIST (_OoD_). Table 2 shows that both FFE and post-hoc LAM have a similar predictive performance to the deterministic model. This is not surprising, as both methods are initialized with the deterministic parameters, and then uncertainties are learned (PFE) or deduced (post-hoc LAM) with frozen weights. The awareness of uncertainties during training, grants the online LAM slightly higher predictive performance.\n\nPFE uses amortized inference to predict variances. This works reasonably within distribution but does not work well for OoD detection. This is because a neural network is trusted to extrapolate far away from the data distribution. Table 2 shows that MC dropout, LAM (post-hoc), and LAM (online) assign high uncertainty to observations outside the training distribution. Fig. 5 shows that both post-hoc and online LAM are near perfectly calibrated, giving very low ECE measures (Table 2).\n\n_CIFAR10 (ID) vs SVHN (_OoD_) is a slightly harder setting. Table 2 yields similar conclusions as before; Bayesian approaches such as MC dropout, LAM (post-hoc), and LAM (online) better detect OoD examples than neural amortized methods such as PFE. Online LAM has a similar predictive performance to state-of-the-art while having better ID (lower ECE and higher AUSC) and OoD (higher AUROC and AUPRC) performance. Fig. 5 shows the calibration plot for CIFAR10, where online LAM has near-perfect calibration. The CIFAR10 ROC curves (Fig. 6) show that online LAM is better at distinguishing ID and OoD examples.\n\n**Open-Set Retrieval.** A key advantage of metric learning methods is that they easily cope with a large number of classes and new classes can be added seamlessly. We therefore evaluate LAM's performance on challenging open-set retrieval, where none of the classes in the test set are available during training. We first test with CUB200 [45] as ID and CAR196 [20] as OoD similarly to Warburg et al. [47], and second, test with LFW [17] as ID and CUB200 as OoD. We use a ResNet50 [16] with a GeM pooling layer [35] and a \\(2048\\) dimensional embedding and diagonal, last-layer LA [8].\n\n_CUB200 (ID) vs CARS196 (_OoD_). The CUB-200-2011 dataset [45] has \\(200\\) bird species captured from different perspectives and in different environments. We follow the zero-shot train/test split [30]. In this zero-shot setting, the trained models have not seen any of the bird species in the test set, and the learned features must generalize well across species. Table 3 shows that LAM matches or surpasses the predictive performance of all other methods. LAM (post-hoc) achieves state-of-the-art\n\n\\begin{table}\n\\begin{tabular}{l l|l l l|l l|l l} \\hline \\hline  & & & \\multicolumn{2}{c|}{Image retrieval} & \\multicolumn{2}{c|}{OoD} & \\multicolumn{2}{c}{Calibration} \\\\  & & mAP@1 \\(\\uparrow\\) & mAP@5 \\(\\uparrow\\) & mAP@10 \\(\\uparrow\\) & AUROC \\(\\uparrow\\) & AUPRC \\(\\uparrow\\) & AUSC \\(\\uparrow\\) & ECE \\(\\downarrow\\) \\\\ \\hline \\multirow{4}{*}{\\begin{tabular}{} \\end{tabular} } & Deterministic & \\(0.78\\pm 0.01\\) & \\(0.73\\pm 0.01\\) & \\(0.72\\pm 0.01\\) & — & — & — & — \\\\  & Deep Ensemble & 0.69 & 0.62 & 0.59 & 0.41 & 0.46 & 0.61 & 0.04 \\\\  & PFE & \\(0.78\\pm 0.00\\) & \\(0.74\\pm 0.00\\) & \\(0.72\\pm 0.00\\) & \\(0.53\\pm 0.03\\) & \\(0.46\\pm 0.01\\) & \\(0.65\\pm 0.01\\) & \\(0.26\\pm 0.02\\) \\\\  & HIB & \\(0.69\\pm 0.08\\) & \\(0.63\\pm 0.09\\) & \\(0.61\\pm 0.09\\) & \\(0.60\\pm 0.12\\) & \\(0.60\\pm 0.11\\) & \\(0.65\\pm 0.08\\) & \\(0.54\\pm 0.08\\) \\\\  & MC dropout & 0.76\\pm 0.03 & \\(0.71\\pm 0.03\\) & \\(0.70\\pm 0.03\\) & \\(0.93\\pm 0.03\\) & \\(0.93\\pm 0.03\\) & \\(0.84\\pm 0.06\\) & \\(0.03\\pm 0.04\\) \\\\  & LAM (post-hoc) & \\(0.78\\pm 0.00\\) & \\(0.74\\pm 0.00\\) & \\(0.72\\pm 0.00\\) & \\(0.96\\pm 0.02\\) & \\(0.96\\pm 0.02\\) & \\(0.86\\pm 0.01\\) & \\(0.03\\pm 0.00\\) \\\\  & LAM (online) & \\(\\mathbf{0.81\\pm 0.00}\\) & \\(\\mathbf{0.77\\pm 0.01}\\) & \\(\\mathbf{0.76\\pm 0.01}\\) & \\(\\mathbf{0.98\\pm 0.01}\\) & \\(\\mathbf{0.98\\pm 0.01}\\) & \\(\\mathbf{0.89\\pm 0.01}\\) & \\(\\mathbf{0.02\\pm 0.00}\\) \\\\ \\hline \\multirow{4}{*}{\n\\begin{tabular}{} \\end{tabular} } & Deterministic & \\(\\mathbf{0.66\\pm 0.00}\\) & \\(0.59\\pm 0.00\\) & \\(0.58\\pm 0.00\\) & — & — & — & — \\\\  & Deep Ensemble & **0.66** & **0.61** & **0.59** & 0.42 & 0.67 & 0.72 & 0.02 \\\\  & MC dropout & \\(0.46\\pm 0.01\\) & \\(0.37\\pm 0.01\\) & \\(0.34\\pm 0.01\\) & \\(0.60\\pm 0.03\\) & \\(0.76\\pm 0.02\\) & \\(0.61\\pm 0.01\\) & \\(0.05\\pm 0.00\\) \\\\  & HIB & \\(0.11\\pm 0.01\\) & \\(0.07\\pm 0.00\\) & \\(0.05\\pm 0.00\\) & \\(0.44\\pm 0.17\\) & \\(0.70\\pm 0.1\\) & \\(0.29\\pm 0.03\\) & \\(0.04\\pm 0.02\\) \\\\  & PFE & \\(\\mathbf{0.66\\pm 0.00}\\) & \\(0.60\\pm 0.00\\) & \\(0.58\\pm 0.00\\) & \\(0.21\\pm 0.02\\) & \\(0.56\\pm 0.01\\) & \\(0.56\\pm 0.01\\) & \\(0.11\\pm 0.01\\) \\\\  & LAM (post-hoc) & \\(\\mathbf{0.66\\pm 0.00}\\) & \\(0.60\\pm 0.00\\) & \\(0.58\\pm 0.00\\) & \\(0.50\\pm 0.11\\) & \\(0.69\\pm 0.07\\) & \\(0.81\\pm 0.01\\) & \\(0.23\\pm 0.01\\) \\\\  & LAM (online) & \\(\\mathbf{0.66\\pm 0.01}\\) & \\(0.60\\pm 0.00\\) & \\(0.57\\pm 0.01\\) & \\(\\mathbf{0.78\\pm 0.04}\\) & \\(\\mathbf{0.85\\pm 0.03}\\) & \\(\\mathbf{0.83\\pm 0.01}\\) & \\(\\mathbf{0.01\\pm 0.00}\\) \\\\ \\hline \\hline \\end{tabular}\n\\end{table}\nTable 2: **Closed-set results.** LAM matches or outperforms existing methods in terms of predictive performance. It produces reliable uncertainties ID and OoD on two standard datasets FashionMNIST and CIFAR10. Error bars show one std. across five runs.\n\nFigure 6: **Recever Operator Curves.** LAM assign high uncertainty to OoD observations.\n\npredictive performance, while LAM (online) matches the predictive performance of the deterministic trained model while achieving state-of-the-art AUROC and AUPRC for OoD detection.\n\n_LFW (ID) vs CUB200 (OoD)._ Face recognition is another challenging metric learning task with many applications in security and surveillance. The goal is to retrieve images of the same person as in the query image. Table 3 shows that online LAM outperforms existing methods both in terms of predictive performance and uncertainty quantification. Fig. 7 shows that PFE assigns higher uncertainty to images from the ID dataset (faces) than those from the OoD dataset (birds). In contrast, both online and post-hoc LAM better associate high variance to OoD examples, while PFE predicts high variance to ID examples. Furthermore, online LAM seems to assign the highest variance to images in which the background is complex and thus camouflages the birds.\n\n**Visual Place recognition** is important for the long-term operation of autonomous robots [7], where the goal is to retrieve images taken within a radius of \\(25\\) meters from a query image. The high number of unique places and varying visual appearance of each location - including weather, dynamic, structural, viewpoint, seasonal, and day/night changes - makes visual place recognition a challenging metric learning\n\nFigure 8: **Sparsification curve. Online and post-hoc LAM’s sparsification curves monotonically increase, showing that they reliably associate higher uncertainty to harder observations.**\n\nFigure 7: **Images with lowest and highest variance for PFE, post-hoc LAM, and online LAM across LFW (ID) and CUB200 (OoD) datasets. LAM associates high uncertainty to OoD examples, and vice versa for PFE. Shows the best-performing PFE and online LAM across five runs.**\n\n\\begin{table}\n\\begin{tabular}{l l|l|l l|l l|l} \\hline \\hline  & & & \\multicolumn{2}{c|}{Image retrieval} & \\multicolumn{2}{c|}{OoD} & \\multicolumn{1}{c}{ID} \\\\  & & & mAP@1 \\(\\uparrow\\) & mAP@5 \\(\\uparrow\\) & mAP@10 \\(\\uparrow\\) & AUROC \\(\\uparrow\\) & AUPRC \\(\\uparrow\\) & AUSC \\(\\uparrow\\) \\\\ \\hline \\multirow{8}{*}{OoD} & Deterministic & \\(0.62\\pm 0.01\\) & \\(0.48\\pm 0.01\\) & \\(0.42\\pm 0.01\\) & — & — & \\multirow{4}{*}{\\(0.55\\)} & \\multirow{4}{*}{\\(0.21\\)} \\\\  & Deep Ensemble & \\(0.21\\) & \\(0.11\\) & \\(0.07\\) & \\(0.47\\) & \\(0.55\\) & \\(0.21\\) \\\\  & PFE & \\(0.62\\pm 0.01\\) & \\(0.5\\pm 0.01\\) & \\(0.43\\pm 0.01\\) & \\(0.44\\pm 0.16\\) & \\(0.5\\pm 0.08\\) & \\(0.61\\pm 0.02\\) \\\\  & HIB & \\(0.33\\pm 0.04\\) & \\(0.19\\pm 0.02\\) & \\(0.14\\pm 0.02\\) & \\(0.54\\pm 0.12\\) & \\(0.61\\pm 0.1\\) & \\(0.31\\pm 0.07\\) \\\\  & MC dropout & \\(0.61\\pm 0.00\\) & \\(0.48\\pm 0.00\\) & \\(0.42\\pm 0.00\\) & \\(0.73\\pm 0.08\\) & \\(0.68\\pm 0.07\\) & \\(0.63\\pm 0.01\\) \\\\  & LAM (post-hoc) & \\(\\textbf{0.65}\\pm\\textbf{0.01}\\) & \\(\\textbf{0.52}\\pm\\textbf{0.01}\\) & \\(\\textbf{0.45}\\pm\\textbf{0.01}\\) & \\(0.56\\pm 0.16\\) & \\(0.61\\pm 0.11\\) & \\(\\textbf{0.66}\\pm\\textbf{0.03}\\) \\\\  & LAM (online) & \\(0.61\\pm 0.00\\) & \\(0.48\\pm 0.00\\) & \\(0.42\\pm 0.00\\) & \\(\\textbf{0.80}\\pm\\textbf{0.03}\\) & \\(\\textbf{0.75}\\pm\\textbf{0.03}\\) & \\(0.63\\pm 0.01\\) \\\\ \\hline \\multirow{8}{*}{OoD} & Deterministic & \\(0.44\\pm 0.00\\) & \\(0.68\\pm 0.00\\) & \\(0.65\\pm 0.00\\) & — & — & — \\\\  & Deep Ensemble & \\(0.36\\) & \\(0.57\\) & \\(0.54\\) & \\(0.52\\) & \\(0.64\\) & \\(0.33\\) \\\\  & PFE & \\(0.44\\pm 0.00\\) & \\(0.68\\pm 0.00\\) & \\(0.65\\pm 0.00\\) & \\(0.03\\pm 0.02\\) & \\(0.41\\pm 0.0\\) & \\(0.49\\pm 0.01\\) \\\\  & MC dropout & \\(0.42\\pm 0.00\\) & \\(0.65\\pm 0.01\\) & \\(0.63\\pm 0.01\\) & \\(0.03\\pm 0.01\\) & \\(0.41\\pm 0.0\\) & \\(0.46\\pm 0.01\\) \\\\  & LAM (post-hoc) & \\(0.44\\pm 0.01\\) & \\(0.68\\pm 0.01\\) & \\(0.65\\pm 0.00\\) & \\(0.65\\pm 0.14\\) & \\(0.72\\pm 0.11\\) & \\(0.45\\pm 0.03\\) \\\\  & LAM (online) & \\(\\textbf{0.46}\\pm\\textbf{0.00}\\) & \\(\\textbf{0.71}\\pm\\textbf{0.00}\\) & \\(\\textbf{0.69}\\pm\\textbf{0.00}\\) & \\(\\textbf{0.71}\\pm\\textbf{0.22}\\) & \\(\\textbf{0.78}\\pm\\textbf{0.17}\\) & \\(\\textbf{0.50}\\pm\\textbf{0.02}\\) \\\\ \\hline \\hline \\end{tabular}\n\\end{table}\nTable 3: **Open-set results. LAM matches or outperforms existing methods in terms of predictive performance and produces state-of-the-art uncertainty quantification for challenging zero-shot metric learning datasets LFW and CUB200. Error bars show one std. across five runs.**problem. Reliable uncertainties and reliable out-of-distribution behavior are important to avoid incorrect loop-closure, which can deteriorate the autonomous robots' location estimate. We evaluate on MSLS [46], which is the largest and most diverse place recognition dataset currently available comprised of \\(1.6M\\) images from \\(30\\) cities spanning six continents. We use the standard train/test split, training on \\(24\\) cities and testing on six other cities. We use the same model as in open-set retrieval.\n\nTable 5 shows that online LAM yields state-of-the-art uncertainties for visual place recognition measured with AUSC, while matching the predictive performance of the alternative probabilistic and deterministic methods on both the MSLS validation and the challenge set. Fig. 8 shows the sparsification curves on the challenge set. Both online and post-hoc LAM have monotonically increasing sparsification curves, implying that when we remove the most uncertain observations, the predictive performance increase. This illustrates that LAM produces reliable uncertainties for this challenging open-set retrieval task. Fig. 9 shows the queries associated with the highest and lowest uncertainty. LAM predicts high uncertainty to images with are blurry, captured facing into the pavement, or contain mostly vegetation. These images do not have features that are descriptive of a specific place, making them hard to geographically locate.\n\n**Ablations.** We conduct ablations on LFW for online and post-hoc LAM. Table 4 shows that the predictive performance does not depend on the margin, however, OOD performance decreases significantly for larger margins. We find that LAM is robust to the number of pairs sampled per batch. We find that a larger latent space results in better-calibrated uncertainties. Online LAM is rather robust to the choice of memory factor \\(\\alpha\\). However, choosing a memory factor in the 0.0001-0.001 range gives both good AUROC and AUSC. This memory factor \\(\\alpha\\) can be interpreted as a momentum-like parameter that relates to the learning rate and indicates how fast we should update the current Hessian [29].\n\n\\begin{table}\n\\begin{tabular}{l|l l l} \\multicolumn{4}{l}{**Margin**} \\\\  & mAP@5 & AUROC & AUSC \\\\ \\hline\n\n[MISSING_PAGE_POST]\n\n \\multicolumn{4}{l}{0.001} & 0.69 & 0.86 & 0.43 \\\\ \\multicolumn{4}{l}{0.001} & 0.69 & 0.86 & 0.43 \\\\ \\multicolumn{4}{l}{\\end{tabular}\n\\end{table}\nTable 4: **Ablation on LFW.**\n\n\\begin{table}\n\\begin{tabular}{l|l l l l|l l l l l|l} \\multicolumn{9}{l}{**Margin**} \\\\  & R@1↑ & R@5↑ & R@1↑ & M@5↑ & M@1↑ & AUSC↑ & R@1↑ & R@5↑ & M@1↑ & AUSC↑ \\\\ \\hline Deterministic & **0.77** & **0.88** & **0.90** & **0.61** & **0.56** & & & **0.58** & **0.74** & **0.78** & **0.45** & 0.43 & — \\\\ MC Dropout & 0.75 & 0.87 & 0.87 & 0.59 & 0.54 & **0.77** & 0.55 & 0.71 & 0.76 & 0.43 & 0.41 & 0.57 \\\\ PFE & **0.77** & **0.88** & **0.90** & **0.61** & **0.56** & & 0.73 & **0.58** & **0.74** & **0.78** & **0.45** & **0.44** & 0.57 \\\\ LAM (post-hoc) & 0.76 & 0.86 & 0.89 & 0.60 & 0.55 & 0.74 & **0.58** & **0.74** & **0.78** & **0.45** & **0.44** & 0.59 \\\\ LAM (online) & 0.76 & 0.87 & **0.90** & 0.60 & **0.56** & **0.77** & 0.57 & **0.74** & **0.78** & **0.45** & 0.43 & **0.63** \\\\ \\multicolumn{9}{l}{} \\\\ \\end{tabular}\n\\end{table}\nTable 5: **Results on MSLS.** LAM yields state-of-the-art uncertainties and matches the predictive performance of deterministic trained models.\n\nFigure 9: **Images with lowest and highest variance** for PFE, post-hoc LAM, and online LAM across MSLS validation set. LAM reliably associates high uncertainty to images that are blurry, are captured facing the pavement, or contain vegetation. These images do not contain features that are descriptive of a specific place, making them especially challenging to geographically locate.\n\nLastly, we perform cold-posterior tempering of post-hoc LA by scaling the log-likelihood hessian with a factor \\(\\beta>0\\), such that the precision in Eq. 3 became \\(\\beta\\cdot\\nabla_{\\theta}^{2}\\mathcal{L}_{\\text{con}}\\left(\\theta^{*};\\mathcal{D} \\right)+\\sigma_{\\text{prior}}^{-2}\\mathbb{I}\\). We do not see any benefits from changing to other values than one.\n\n**Limitations.** Similar to other Bayesian methods, LAM relies on \\(n\\) samples to obtain uncertainties. This makes inferences \\(n\\) times slower. Computing the Hessian at every step during online LAM also makes training time slower. To combat long training times and high memory usage, we use a last-layer LAM and thus only estimate and sample for a weight posterior of the last layer. The last-layer LAM training time is \\(3\\) hours for online LAM vs \\(2.3\\) hours for deterministic contrastive loss on LFW, and \\(30\\) minutes vs \\(15\\) minutes loss on CUB200 on an NVIDIA RTX A5000.\n\n## 5 Conclusion\n\nIn this paper, we have introduced a Bayesian encoder for metric learning, the Laplacian Metric Learner (LAM), which uses the Laplace approximation. We prove that the contrastive loss is unnormalized negative log-likelihood on the spherical space, and develop three Hessian approximations, which ensures a positive definite covariance matrix. We propose a novel decomposition of the Generalized Gauss-Newton approximation that improves Hessian approximations of \\(\\ell_{2}\\)-normalized networks. Empirically, we demonstrate that LAM consistently produces well-calibrated uncertainties, reliably detects out-of-distribution examples, and achieves state-of-the-art predictive performance on both closed-set and challenging open-set image retrieval tasks.\n\n**Acknowledgement.** This work was supported by research grants (15334, 42062) from VILLUM FONDEN. This project received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement 757360). The work was partly funded by the Novo Nordisk Foundation through the Center for Basic Machine Learning Research in Life Science (NNF200C0062606).\n\n## References\n\n* [1] Relja Arandjelovic, Petr Gronat, Akihiko Torii, Tomas Pajdla, and Josef Sivic. Netvlad: CNN architecture for weakly supervised place recognition. In _Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition_, pages 5297-5307, 2016.\n* [2] Charles Blundell, Julien Cornebise, Koray Kavukcuoglu, and Daan Wierstra. Weight uncertainty in neural networks, 2015.\n* [3] Jie Chang, Zhonghao Lan, Changmao Cheng, and Yichen Wei. Data uncertainty learning in face recognition. In _Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition_, pages 5710-5719, 2020.\n* [4] Peter Ebert Christensen, Frederik Warburg, Menglin Jia, and Serge Belongie. Searching for structure in unfalsifiable claims. _arXiv preprint arXiv:2209.00495_, 2022.\n* [5] Sanghyuk Chun, Seong Joon Oh, Rafael Sampaio De Rezende, Yannis Kalantidis, and Diane Larlus. Probabilistic embeddings for cross-modal retrieval. In _Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition_, pages 8415-8424, 2021.\n* [6] Felix Dangel, Frederik Kunstner, and Philipp Hennig. BackPACK: Packing more into backprop. In _International Conference on Learning Representations (ICLR)_, 2020.\n* [7] Andrew J. Davison, Ian D. Reid, Nicholas D. Molton, and Olivier Stasse. Monoslam: Real-time single camera slam. _IEEE Transactions on Pattern Analysis and Machine Intelligence_, 29(6):1052-1067, 2007. doi: 10.1109/TPAMI.2007.1049.\n* [8] Erik Daxberger, Agustinus Kristiadi, Alexander Immer, Runa Eschenhagen, Matthias Bauer, and Philipp Hennig. Laplace redux-effortless Bayesian deep learning. In _NeurIPS_, 2021.\n* [9] John Denker and Yann LeCun. Transforming neural-net output levels to probability distributions. _Advances in Neural Information Processing Systems_, 1990.\n\n* Detlefsen et al. [2019] Nicki S Detlefsen, Martin Jorgensen, and Soren Hauberg. Reliable training and estimation of variance networks. In _Advances in Neural Information Processing Systems_, 2019.\n* Detlefsen et al. [2021] Nicki S. Detlefsen, Alison Pouplin, Cilie W. Feldager, Cong Geng, Dimitris Kalatzis, Helene Hauschultz, Miguel Gonzalez Duque, Frederik Warburg, Marco Miani, and Soren Hauberg. StochMan, 2021. URL [https://github.com/MachineLearningLifeScience/stochman/](https://github.com/MachineLearningLifeScience/stochman/).\n* Donese and Hagan [1997] F Dan Foresee and Martin T Hagan. Gauss-Newton approximation to Bayesian learning. In _Proceedings of International Conference on Neural Networks (ICNN'97)_, volume 3, pages 1930-1935. IEEE, 1997.\n* Gal and Ghahramani [2016] Yarin Gal and Zoubin Ghahramani. Dropout as a Bayesian approximation: Representing model uncertainty in deep learning. In _international conference on machine learning_, pages 1050-1059. PMLR, 2016.\n* Gershman and Goodman [2014] Samuel Gershman and Noah Goodman. Amortized inference in probabilistic reasoning. In _Proceedings of the annual meeting of the cognitive science society_, volume 36, 2014.\n* Hadsell et al. [2006] Raia Hadsell, Sumit Chopra, and Yann LeCun. Dimensionality reduction by learning an invariant mapping. In _2006 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR'06)_, volume 2, pages 1735-1742. IEEE, 2006.\n* He et al. [2016] Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep residual learning for image recognition. In _Proceedings of the IEEE conference on computer vision and pattern recognition_, pages 770-778, 2016.\n* Huang et al. [2007] Gary B. Huang, Manu Ramesh, Tamara Berg, and Erik Learned-Miller. Labeled faces in the wild: A database for studying face recognition in unconstrained environments. Technical Report 07-49, University of Massachusetts, Amherst, October 2007.\n* Immer et al. [2021] Alexander Immer, Maciej Korzepa, and Matthias Bauer. Improving predictions of Bayesian neural nets via local linearization. In _International Conference on Artificial Intelligence and Statistics_, pages 703-711. PMLR, 2021.\n* Kingma and Welling [2013] Diederik P Kingma and Max Welling. Auto-encoding variational Bayes. _arXiv preprint arXiv:1312.6114_, 2013.\n* Krause et al. [2013] J. Krause, M. Stark, J. Deng, and L. Fei-Fei. 3d object representations for fine-grained categorization. In _2013 IEEE International Conference on Computer Vision Workshops_, pages 554-561, 2013.\n* Krizhevsky [2009] Alex Krizhevsky. Learning multiple layers of features from tiny images, 2009.\n* Kunstner et al. [2019] Frederik Kunstner, Philipp Hennig, and Lukas Balles. Limitations of the empirical fisher approximation for natural gradient descent. _Advances in neural information processing systems_, 32, 2019.\n* Lakshminarayanan et al. [2017] Balaji Lakshminarayanan, Alexander Pritzel, and Charles Blundell. Simple and scalable predictive uncertainty estimation using deep ensembles. _Advances in Neural Information Processing Systems_, 30, 2017.\n* Laplace [1774] Pierre Simon Laplace. Memoire sur la probabilite des causes par les evenements. _Memoire de l'Academie Royale des Sciences_, 1774.\n* LeCun et al. [1989] Yann LeCun, John Denker, and Sara Solla. Optimal brain damage. _Advances in Neural Information Processing Systems_, 1989.\n* LeCun et al. [1998] Yann LeCun, Leon Bottou, Yoshua Bengio, and Patrick Haffner. Gradient-based learning applied to document recognition. _Proceedings of the IEEE_, 86(11):2278-2324, 1998.\n* MacKay [1992] David J. C. MacKay. A practical Bayesian framework for backpropagation networks. _Neural Computation_, 4(3):448-472, 1992.\n\n* [28] Wesley J Maddox, Pavel Izmailov, Timur Garipov, Dmitry P Vetrov, and Andrew Gordon Wilson. A simple baseline for Bayesian uncertainty in deep learning. _Advances in Neural Information Processing Systems_, 32, 2019.\n* [29] Marco Miani, Frederik Warburg, Pablo Moreno-Munoz, Nicke Skafte Detlefsen, and Soren Hauberg. Laplacian autoencoders for learning stochastic representations. In _Advances in Neural Information Processing Systems_, 2022.\n* [30] Kevin Musgrave, Serge Belongie, and Ser-Nam Lim. A metric learning reality check. In _European Conference on Computer Vision_, pages 681-699. Springer, 2020.\n* [31] Eric Nalisnick, Akihiro Matsukawa, Yee Whye Teh, Dilan Gorur, and Balaji Lakshminarayanan. Do deep generative models know what they don't know? _arXiv preprint arXiv:1810.09136_, 2018.\n* [32] Yuval Netzer, Tao Wang, Adam Coates, Alessandro Bissacco, Bo Wu, and Andrew Y Ng. Reading digits in natural images with unsupervised feature learning, 2011.\n* [33] Seong Joon Oh, Kevin Murphy, Jiyan Pan, Joseph Roth, Florian Schroff, and Andrew Gallagher. Modeling uncertainty with hedged instance embedding. _arXiv preprint arXiv:1810.00319_, 2018.\n* [34] Adam Paszke, Sam Gross, Soumith Chintala, Gregory Chanan, Edward Yang, Zachary DeVito, Zeming Lin, Alban Desmaison, Luca Antiga, and Adam Lerer. Automatic differentiation in PyTorch. In _NIPS-W_, 2017.\n* [35] Filip Radenovic, Giorgos Tolias, and Ondrej Chum. Fine-tuning CNN image retrieval with no human annotation. _IEEE transactions on pattern analysis and machine intelligence_, 41(7):1655-1668, 2018.\n* [36] Danilo Jimenez Rezende, Shakir Mohamed, and Daan Wierstra. Stochastic backpropagation and approximate inference in deep generative models. In _International conference on machine learning_, pages 1278-1286. PMLR, 2014.\n* [37] Florian Schroff, Dmitry Kalenichenko, and James Philbin. Facenet: A unified embedding for face recognition and clustering. In _Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition_, pages 815-823, 2015.\n* [38] Yichun Shi and Anil K Jain. Probabilistic face embeddings. In _Proceedings of the IEEE/CVF International Conference on Computer Vision_, pages 6902-6911, 2019.\n* [39] Yale Song and Mohammad Soleymani. Polysemous visual-semantic embedding for cross-modal retrieval. In _Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition_, pages 1979-1988, 2019.\n* [40] Suvrit Sra. A short note on parameter approximation for von Mises-Fisher distributions: and a fast implementation of i s (x). _Computational Statistics_, 27(1):177-190, 2012.\n* [41] Jennifer J Sun, Jiaping Zhao, Liang-Chieh Chen, Florian Schroff, Hartwig Adam, and Ting Liu. View-invariant probabilistic embedding for human pose. In _European Conference on Computer Vision_, pages 53-70. Springer, 2020.\n* [42] Ahmed Taha, Yi-Ting Chen, Teruhisa Misu, Abhinav Shrivastava, and Larry Davis. Unsupervised data uncertainty learning in visual retrieval systems. _arXiv preprint arXiv:1902.02586_, 2019.\n* [43] Ahmed Taha, Yi-Ting Chen, Xitong Yang, Teruhisa Misu, and Larry Davis. Exploring uncertainty in conditional multi-modal retrieval systems. _arXiv preprint arXiv:1901.07702_, 2019.\n* [44] Luke Vilnis and Andrew McCallum. Word representations via Gaussian embedding. _arXiv preprint arXiv:1412.6623_, 2014.\n* [45] Catherine Wah, Steve Branson, Peter Welinder, Pietro Perona, and Serge Belongie. Caltech-UCSD birds 200, 2011.\n\n* [46] Frederik Warburg, Soren Hauberg, Manuel Lopez-Antequera, Pau Gargallo, Yubin Kuang, and Javier Civera. Mapillary street-level sequences: A dataset for lifelong place recognition. In _Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR)_, June 2020.\n* [47] Frederik Warburg, Martin Jorgensen, Javier Civera, and Soren Hauberg. Bayesian triplet loss: Uncertainty quantification in image retrieval. In _Proceedings of the IEEE/CVF International Conference on Computer Vision_, pages 12158-12168, 2021.\n* [48] Michael Wilber, Iljung S Kwak, David Kriegman, and Serge Belongie. Learning concept embeddings with combined human-machine expertise. In _Proceedings of the IEEE International Conference on Computer Vision_, pages 981-989, 2015.\n* [49] Han Xiao, Kashif Rasul, and Roland Vollgraf. Fashion-MNIST: a novel image dataset for benchmarking machine learning algorithms, 2017.\n* [50] Keyulu Xu, Mozhi Zhang, Jingling Li, Simon S. Du, Ken ichi Kawarabayashi, and Stefanie Jegelka. How neural networks extrapolate: From feedforward to graph neural networks, 2021.", "response": "## Review\n\n### Summary\n\nThis paper presents a Bayesian metric learning approach for image retrieval, utilizing the Laplace approximation to learn a distribution over weights. The authors establish that the contrastive loss can be interpreted as a likelihood when projected onto the spherical space. They propose methods to ensure the Hessian is positive definite and conduct extensive empirical evaluations demonstrating advantages in calibrated uncertainties and out-of-distribution detection. The paper's novelty lies in its probabilistic view of contrastive loss and its applicability across various problems, making it a valuable contribution to the machine learning community.\n\n### Strengths\n\n- Well organized paper with clear writing and presentation of ideas.\n- The Bayesian approach for uncertainty measurement in metric learning is innovative.\n- The method shows improved results, especially on out-of-distribution examples, with comprehensive ablation studies.\n- The authors provide several analogies for various concepts, enhancing reader understanding.\n- Extensive empirical evaluations demonstrate the proposed approach's superiority over existing methods.\n\n### Weaknesses\n\n- Experimental results demonstrate only limited improvement over existing methods.\n- The method is computationally expensive, which may hinder practical applicability.\n- The focus on a classical contrastive loss posits questions about its generalizability to other loss functions.\n- The paper lacks clarity in presenting unique contributions amidst prior works and could benefit from improved organization.\n- The Laplace approximation's assumptions may not hold for large datasets and modern neural networks.\n\n### Questions\n\n- Could you provide a systematic analysis of the computational efficiency?\n- How does the proposed approach perform test-time retrieval, and how does its efficiency compare to amortized approaches?\n- Could the authors clarify the assumptions made regarding stochastic embeddings?\n- What are the primary roadblocks in applying the Laplacian autoencoder to probabilistic retrieval?\n- Are there alternative methods for approximating the Hessian while maintaining scalability and positive definiteness?\n\n### Soundness\n\n**Score:** 10\n\n**Description:** 3 = good; the methodology is solid but relies on assumptions that may not hold in all cases.\n\n### Presentation\n\n**Score:** 9\n\n**Description:** 3 = good; while the paper is generally well organized, it could benefit from clearer presentation and reduced reliance on supplemental material.\n\n### Contribution\n\n**Score:** 11\n\n**Description:** 3 = good; the paper presents significant contributions but could better articulate its unique aspects relative to prior work.\n\n### Rating\n\n**Score:** 18\n\n**Description:** 7 = accept, but needs minor improvements; the paper is technically solid and impactful but requires clarification and enhanced organization.\n\n### Paper Decision\n\n**Decision:** Accept (poster)\n\n**Reasons:** The paper offers a novel Bayesian approach to metric learning with significant empirical results and contributions to uncertainty quantification. Despite some weaknesses in computational efficiency and clarity, the strengths and potential impact on the field outweigh these concerns, justifying an acceptance decision.\n"}
{"query": "You are a highly experienced, conscientious, and fair academic reviewer, please help me review this paper. The review should be organized into nine sections: \n1. Summary: A summary of the paper in 100-150 words.\n2. Strengths/Weaknesses/Questions: The Strengths/Weaknesses/Questions of paper, which should be listed in bullet points, with each point supported by specific examples from the article where possible.\n3. Soundness/Contribution/Presentation: Rate the paper's Soundness/Contribution/Presentation, and match this score to the corresponding description from the list below and provide the result. The possible scores and their descriptions are: \n    1 poor\n    2 fair\n    3 good\n    4 excellent\n4. Rating: Give this paper an appropriate rating, match this rating to the corresponding description from the list below and provide the result. The possible Ratings and their descriptions are: \n    1 strong reject\n    2 reject, significant issues present\n    3 reject, not good enough\n    4 possibly reject, but has redeeming facets\n    5 marginally below the acceptance threshold\n    6 marginally above the acceptance threshold\n    7 accept, but needs minor improvements \n    8 accept, good paper\n    9 strong accept, excellent work\n    10 strong accept, should be highlighted at the conference   \n5. Paper Decision: It must include the Decision itself(Accept or Reject) and the reasons for this decision, based on the criteria of originality, methodological soundness, significance of results, and clarity and logic of presentation.\n\nHere is the template for a review format, you must follow this format to output your review result:\n**Summary:**\nSummary content\n\n**Strengths:**\n- Strength 1\n- Strength 2\n- ...\n\n**Weaknesses:**\n- Weakness 1\n- Weakness 2\n- ...\n\n**Questions:**\n- Question 1\n- Question 2\n- ...\n\n**Soundness:**\nSoundness result\n\n**Presentation:**\nPresentation result\n\n**Contribution:**\nContribution result\n\n**Rating:**\nRating result\n\n**Paper Decision:**\n- Decision: Accept/Reject\n- Reasons: reasons content\n\n\nPlease ensure your feedback is objective and constructive. The paper is as follows:\n\n# Smoothed Analysis of Sequential Probability Assignment\n\nAlankrita Bhatt\n\nCalifornia Institute of Technology\n\nabhatt@caltech.edu\n\n&Nika Haghtalab\n\nUniversity of California, Berkeley\n\nnika@berkeley.edu\n\n&Abhishek Shetty\n\nUniversity of California, Berkeley\n\nshetty@berkeley.edu\n\n###### Abstract\n\nWe initiate the study of smoothed analysis for the sequential probability assignment problem with contexts. We study information-theoretically optimal minmax rates as well as a framework for algorithmic reduction involving the _maximum likelihood estimator_ oracle. Our approach establishes a general-purpose reduction from minimax rates for sequential probability assignment for smoothed adversaries to minimax rates for transductive learning. This leads to optimal (logarithmic) fast rates for parametric classes and classes with finite VC dimension. On the algorithmic front, we develop an algorithm that efficiently taps into the MLE oracle, for general classes of functions. We show that under general conditions this algorithmic approach yields sublinear regret.\n\n## 1 Introduction\n\nSequential probability assignment -- also known as online learning under the logarithmic loss -- is a fundamental problem with far-reaching impact on information theory, statistics, finance, optimization, and sequential decision making [13, 19, 14, 11, 12, 13, 15, 16, 17, 18, 19, 20]. In recent years, methods for incorporating contexts or side information into sequential probability assignment have gained much attention [13, 14, 15, 16, 17, 18, 19, 21], in part due to their newly forged connection to sequential decision making applications, the contextual bandit problem, and learning in Markov Decision Processes (MDPs) (see e.g. [14] and [16]). In this setting, a forecaster who has access to historical data \\(x_{1:t-1},y_{1:t-1}\\) consisting of contexts \\(x_{\\tau}\\) (e.g., day \\(\\tau\\)'s meteorological information) and the outcomes \\(y_{\\tau}\\in\\{0,1\\}\\) (e.g., whether \\(\\tau\\) was a rainy day) wishes to predict \\(y_{t}\\) given a new context \\(x_{t}\\). The forecaster uses a _probability assignment_\\(p_{t}\\) to estimate the probability of \\(y_{t}=1\\) outcome and incurs the logarithmic loss, i.e., \\(-\\log p_{t}(y_{t})\\), which rewards the forecasters for having assigned high probability to the realized outcome. The goal of the forecaster is to suffer low _regret_ against a chosen reference class of predictors.\n\nA large body of prior work on sequential probability assignment with contexts has focused on settings where contexts are presented i.i.d. from an unknown distribution (see [14, 15, 16, 17, 18] and the references within); this problem is also referred to as conditional density estimation. In these cases, sequential probability assignment is known to enjoy small regret for several reference classes such as Vapnik-Chervonenkis (VC) classes. On the other hand, attempts to consider context distributions that evolve unpredictably andadversarially have faced strong impossibility results even for simple reference classes of predictors. For example, for the reference class of simple one-dimensional thresholds assigning \\(p_{t}=\\theta_{0}\\mathds{1}\\{x_{t}\\leq a\\}+\\theta_{1}\\mathds{1}\\{x_{t}>a\\}\\) for \\(a\\in[0,1]\\), regret is bounded by \\(O(\\log T)\\) in the i.i.d. case (Fogel and Feder, 2017) but is lower bounded by \\(\\Omega(T)\\) when the sequence of contexts is chosen adversarially (folklore e.g. Littestone (1988)). In the face of the increasing need to adapt to evolving contexts in modern applications, these impossibility results indicate that new models of adversarial behavior must be considered for obtaining rigorous guarantees that guide the design of sequential probability assignment in practical applications.\n\nIn recent years, _smoothed analysis_ of adaptive adversaries (Haghtalab et al., 2021; Rakhlin et al., 2011) has emerged as a framework for going beyond the worst-case adversaries while making minimal assumptions about the adaptive process that generates a sequence. In this setting, contexts are chosen from an evolving sequence of so-called \\(\\sigma\\)-smooth distributions, whose density is bounded above by \\(1/\\sigma\\) times that of a base measure (such as the uniform distribution). Remarkably, these methods, established by Haghtalab et al. (2021) for \\(0\\)-\\(1\\) loss and extended to regression by Haghtalab et al. (2022), Block et al. (2022), have established performance guarantees for the sequential prediction problem that matches the optimal performance in the i.i.d setting. This raises the question as to whether the _sequential probability assignment_ problem may similarly enjoy improved minmax regret bounds for smoothed adaptive sequences.\n\nBeyond minmax rates, an important feature of probability assignment and, its analogue, density estimation is the availability of fundamental and natural estimation techniques such as _maximum likelihood estimation (MLE)_. For i.i.d. sequences, under general conditions, MLE is known to be optimal asymptotically and often serves as a starting point for designing more sophisticated estimators. Going beyond i.i.d. sequences, we ask whether MLE can be made to achieve good statistical behavior on adaptive sequences. More generally, algorithmic perspective is increasingly important for the sequential probability assignment problem and its applications to contextual bandits and reinforcement learning (where algorithm design is as fundamental a consideration as minmax rates (Agarwal et al., 2014; Simchi-Levi and Xu, 2022; Foster and Rakhlin, 2020; Langford et al., 2007; Foster et al., 2021)). In this space, _oracle-efficient_ sequential decision making algorithms that repurpose existing offline algorithmic routines have received special interest (Kalai and Vempala, 2005; Dudik et al., 2020; Wang et al., 2022; Kakade et al., 2007; Simchi-Levi and Xu, 2022). Here again, recent progress on smoothed analysis for sequential prediction with \\(0\\)-\\(1\\) loss and regression loss (Haghtalab et al., 2021; Block et al., 2022) has shown promise in bridging the computational and information-theoretical gaps between what is obtainable in the i.i.d. case and for smoothed adaptive sequences.\n\nIn this paper, we initiate the study of smoothed analysis for sequential probability assignment and seek to understand fundamental information-theoretic limits on the _minmax regret_ as well as design _natural and oracle-efficient algorithms_ for this problem. Additionally, we investigate whether in the smoothed analysis setting, maximum likelihood estimation can efficiently address sequential probability assignment while achieving small regret. To the best of our knowledge, our work is the first to consider oracle-efficient algorithms (and particularly the MLE) for the sequential probability assignment problem.\n\n### Main results\n\n**Reduction to transductive learning.** Our first main result is a reduction from regret bounds against a smoothed adversary to regret bounds against (a generalized version of) a transductive adversary. That is, we show that the minimax regret in the smoothed analysis setting is upper bounded by the minimax regret in the setting where a set of contexts is provided to the learner and the adversary is constrained to picking the contexts from this set. For \\(\\mathcal{F}\\), a class of hypotheses mapping contexts to \\([0,1]\\), let us define the minmax regret in the transductive case over \\(T\\) times steps when a context set of size \\(M\\) is provided to the learner to be \\(\\underline{\\mathcal{R}}_{T}^{M}(\\mathcal{F})\\). We establish in Theorem 3.1 that for all \\(\\sigma\\)-smooth sequences the minmax regret \\(\\mathcal{R}_{T}(\\mathcal{F},\\sigma)\\) satisfies, for any \\(k>1\\),\n\n\\[\\mathcal{R}_{T}(\\mathcal{F},\\sigma)\\leq\\underline{\\mathcal{R}}_{T}^{kT}( \\mathcal{F})+T^{2}(1-\\sigma)^{k}.\\]\n\nFurthermore, in Theorem 3.4 we upper bound \\(\\underline{\\mathcal{R}}_{T}^{kT}(\\mathcal{F})\\) by connecting the worst case adversarial regret in this setting to the scale-sensitive VC dimension of \\(\\mathcal{F}\\) which is a prototypical offline complexity of the class. Our results obtain a logarithmic dependence on \\(1/\\sigma\\). In particular, in Corollary 3.1.1, we show that for VC classes (and parametric classes) the regret is bounded by \\(\\mathcal{R}_{T}(\\mathcal{F},\\sigma)\\leq O\\left(d\\log\\left(\\frac{T}{\\sigma}\\right)\\right)\\), where \\(d\\) is the VC dimension of class \\(\\mathcal{F}\\).\n\nEfficient Reduction from Sequential Probability Assignment to MLE.Our second contribution is initiating the study of oracle-efficient algorithm design for sequential probability assignment. In particular, for small alphabet size, we design a natural algorithm (Algorithm 1) that efficiently uses an MLE oracle and achieves sublinear regret in the smoothed setting. Our Theorem 4.1 gives a general regret bound in terms of the statistical complexity of the class \\(\\mathcal{F}\\) and the smoothness parameter \\(\\sigma\\).\n\nFor VC classes, this achieves regret rate of \\(T^{4/5}\\sqrt{\\frac{d}{\\sigma}}\\). To the best of our knowledge, this is the first _oracle-efficient_ algorithm and analysis of the follow-the-perturbed-leader style algorithms for the logarithmic loss.\n\nProbability assignment for VC classes.For VC classes \\(\\mathcal{F}\\), we explicitly construct sequential probability assignments and establish their regret guarantees in the smoothed setting. That is, we construct a probability assignment based on a Bayesian mixture over \\(\\mathcal{F}\\) that satisfies \\(\\mathcal{R}_{T}(\\mathcal{F},\\sigma)\\leq Cd\\log\\left(\\frac{T}{\\sigma}\\right)\\) where \\(d\\) is the VC dimension of class \\(\\mathcal{F}\\). While this approach is not oracle-efficient, it indeed achieves regret bound with optimal dependence on \\(T\\) and \\(\\sigma\\). This motivates a natural direction for future work as to whether such mixture-based methods can be implemented oracle efficiently or if there is a tradeoff between the regret and the computational complexity of sequential probability assignment.\n\n## 2 Preliminaries\n\nLet \\(\\mathcal{X}\\) be a set of _contexts_ and \\(\\mathcal{Y}=\\{0,1\\}\\). Then, the problem being studied entails a sequential game where at each timestep \\(t\\), based on the history of contexts \\(x_{1:t}:=(x_{1},\\ldots,x_{t})\\) where \\(x_{i}\\in\\mathcal{X}\\) and associated bits \\(y_{1:t-1}\\in\\{0,1\\}^{t-1}\\), the player must assign a probability \\(q(\\cdot|x_{1:t},y_{1:t-1})\\) to what the upcoming bit \\(y_{t}\\) will be. Once the bit \\(y_{t}\\) is revealed (possibly in an adversarial fashion) the player incurs loss \\(-\\log q(y_{t}|x_{1:t},y_{1:t-1})\\) and the game proceeds to the next step. For a _hypothesis class_\\(\\mathcal{F}\\subset\\{\\mathcal{X}\\rightarrow[0,1]\\}\\), the associated regret for a probability assignment strategy \\(\\mathscr{Q}=\\{q(\\cdot|x_{1:t},y_{1:t-1})\\}_{t=1}^{n}\\) for a fixed \\(x_{1:T},y_{1:T}\\) is\n\n\\[\\mathcal{R}_{T}(\\mathcal{F},x_{1:T},y_{1:T},\\mathscr{Q})=\\sum_{t=1}^{T}\\log \\frac{1}{q(y_{t}|x_{1:t},y_{1:t-1})}-\\inf_{f\\in\\mathcal{F}}\\sum_{t=1}^{T}\\log \\frac{1}{p_{f}(y_{t}|x_{t})} \\tag{1}\\]\n\nwhere \\(p_{f}(1|x_{t})=f(x_{t})\\); i.e. the function \\(f\\) assigns probability \\(\\operatorname{Bern}(f(x_{t}))\\) to the upcoming bit given the context \\(x_{t}\\).\n\nOur statistical results apply to a general loss function \\(\\ell\\) and general actions of the learner \\(a_{t}\\). For a set of inputs \\(\\big{\\{}(x_{i},y_{i})\\big{\\}}_{i=1}^{T}\\) specified by the adversary and a set of actions \\(\\big{\\{}a_{i}\\big{\\}}_{i=1}^{T}\\) of the learner, regret is defined by\n\n\\[\\mathcal{R}_{T}(\\mathcal{F},x_{1:T},y_{1:T},a_{1:T})=\\sum_{t=1}^{T}\\ell(a_{t}, (x_{t},y_{t}))-\\inf_{f\\in\\mathcal{F}}\\sum_{t=1}^{T}\\ell(f(x_{t}),(x_{t},y_{t} )),\\]\n\nwhere for log-loss \\(a_{t}=q(\\cdot|x_{1:t},y_{1:t-1})\\); the action is a probability mass function (pmf) over \\(\\{0,1\\}\\).\n\nThe regret in (1) is often studied under various adversary models; i.e. various different probabilistic assumptions (or lack thereof) on the model generating \\(x_{t}\\) and \\(y_{t}\\). In this work, we consider worst-case \\(y_{t}\\) (in contrast to the _realizable_ setting where \\(Y_{t}\\sim\\operatorname{Bern}(f^{*}(x_{t}))\\) for a fixed unknown \\(f^{*}\\in\\mathcal{F}\\)) and \\(X_{t}\\sim\\mathcal{D}_{t}\\) where \\(\\mathcal{D}_{t}\\)s form an adaptive sequence of smooth distributions.\n\n**Definition 2.1** (Smooth distribution and adversary [16]).: Consider a fixed and known base distribution \\(\\mu\\) on \\(\\mathcal{X}\\) (such as the uniform distribution if \\(\\mathcal{X}\\) supports it). A distribution \\(\\mathcal{D}\\) on \\(\\mathcal{X}\\) is said to be \\(\\sigma\\)-smooth if for all measurable sets \\(A\\subseteq\\mathcal{X},\\mathcal{D}(A)\\leq\\frac{\\mu(A)}{\\sigma}\\). We denote the set of all \\(\\sigma\\)-smooth distributions by \\(\\Delta_{\\sigma}\\left(\\mu\\right)\\). An adversary, characterized by a joint distribution \\(\\mathscr{D}\\) with \\(X_{t}\\sim\\mathcal{D}_{t}\\) (where \\(\\mathcal{D}_{t}\\) may possibly depend on the history) is said to be a \\(\\sigma\\)-smooth adaptive adversary if \\(\\mathcal{D}_{t}\\in\\Delta_{\\sigma}\\left(\\mu\\right)\\) for all \\(t\\in\\{1,\\ldots,T\\}\\).\n\nThe minmax regret for \\(\\sigma\\)-smooth adaptive adversaries is then given by\n\n\\[\\mathcal{R}_{T}(\\mathcal{F},\\sigma)=\\inf_{\\mathscr{Q}}\\sup_{\\sigma\\text{- smoothed }\\mathscr{D}}\\mathbb{E}_{X_{1:T}\\sim\\mathscr{D}}\\left[\\max_{y_{1:T}}\\mathcal{R}_{T} (\\mathcal{F},X_{1:T},y_{1:T},\\mathscr{Q})\\right],\\]\n\nwhere \\(\\mathscr{Q}\\) is the set of all probability assignment strategies.\n\nWe are particularly interested in how geometric properties of the function class \\(\\mathcal{F}\\) affect \\(\\mathcal{R}_{T}(\\mathcal{F},\\sigma)\\). There are several notions of covering numbers and combinatorial dimensions that quantify the \"richness\" and complexity of a class, but the _scale-sensitive VC dimension_ will be of particular interest to us and is invoked in our results.\n\n**Definition 2.2** (Scale-sensitive VC dimension).: Let \\(\\mathcal{F}\\) be a function class. For any \\(\\alpha>0\\) and points \\(x_{1},\\ldots,x_{m}\\in\\mathcal{X}\\), we say that \\(\\mathcal{F}\\) shatters the set \\(x_{1},\\ldots,x_{m}\\) at scale \\(\\alpha\\) if there exist \\(s_{1}\\ldots s_{m}\\in\\mathbb{R}\\) such that for each \\(\\epsilon\\in\\left\\{-1,1\\right\\}^{n}\\) there exists a function \\(f\\in\\mathcal{F}\\) such that \\(\\epsilon_{i}\\left(f(x_{i})-s_{i}\\right)\\geq\\frac{\\alpha}{2}\\). The scale sensitive VC dimension at scale \\(\\alpha\\) of \\(\\mathcal{F}\\), denoted by \\(\\operatorname{VC}\\left(\\mathcal{F},\\alpha\\right)\\) is defined as the largest \\(m\\) such that there is a set of \\(m\\) points \\(x_{1},\\ldots,x_{m}\\in\\mathcal{X}\\) such that \\(\\mathcal{F}\\) shatters the set at scale \\(\\alpha\\). The (traditional) VC dimension of a binary class \\(\\mathcal{F}\\) is defined as \\(\\operatorname{VC}\\left(\\mathcal{F}\\right)=\\lim_{\\alpha\\to 0^{+}} \\operatorname{VC}\\left(\\mathcal{F},\\alpha\\right)\\).\n\nThroughout, we use the following result of Haghtalab et al. (2021) about \\(\\sigma\\)-smooth distributions. This results aids us in reduction from smoothed learning to transductive learning.\n\n**Theorem 2.1** (Coupling Lemma of Haghtalab et al. (2021)).: _Let \\(\\mathscr{D}_{\\sigma}\\) be an adaptive sequence of \\(t\\)\\(\\sigma\\)-smooth distributions on \\(\\mathcal{X}\\). There is a coupling \\(\\Pi\\) such that_\n\n\\[(X_{1},Z_{1,1},\\ldots,Z_{1,K},\\ldots,X_{t},Z_{t,1},\\ldots,Z_{t,K})\\sim\\Pi\\]\n\n1. \\(X_{1},\\ldots,X_{t}\\) _is distributed according_ \\(\\mathscr{D}_{\\sigma}\\)_,_\n2. _For every_ \\(j\\leq t\\)_,_ \\(\\{Z_{i,k}\\}_{i\\geq j,k\\in[K]}\\) _are uniformly and independently distributed on_ \\(\\mathcal{X}\\)_, conditioned on_ \\(X_{1},\\ldots,X_{j-1}\\)_._\n3. _For any_ \\(t\\)_, with probability at least_ \\(1-(1-\\sigma)^{K}\\)_,_ \\(X_{t}\\in\\{Z_{t,k}\\}_{k=1:K}\\)_._\n\n## 3 General reduction to transductive learning\n\nIn this section, we will consider the minimax regret for the smoothed online learning game with respect to the loss function1\\(\\ell\\) against a general class of functions \\(\\mathcal{F}\\). In Section 3.1, we will show that the minimax regret can be reduced to the minimax regret for a version of transductive learning with respect to the same loss function and class of functions. In Section 3.2, we give general upper bounds for the transductive setting. There is a subtle but important difference between reduction that directly involve regret compared to recent efforts (such as Haghtalab et al. (2021), Block et al. (2022), Haghtalab et al. (2022)) using reductions between proxies of regret, such as covering numbers and sequential complexities. This is particularly important for log loss since its complexity is not captured by covering numbers. We discuss this point further in Section 3.2.2.\n\nFootnote 1: The reduction works for general loss functions with the property that the worst-case regret for horizon \\(T\\) is bounded by \\(T\\), but one can think about \\(\\ell\\) being the log-loss throughout this section for concreteness.\n\n### Regret-to-regret Reduction\n\nWe work with a general loss function \\(\\ell\\) and general actions of the learner \\(a_{t}\\). We note that, we can write the minmax value of the smoothed setting in extensive form as\n\n\\[\\mathcal{R}_{T}(\\mathcal{F},\\sigma)=\\sup_{\\mathcal{D}_{1}\\in \\Delta_{\\sigma}(\\mu)}\\mathbb{E}_{X_{1}\\sim\\mathcal{D}_{1}}\\inf_{a_{1}}\\sup_{y _{1}}\\sup_{\\mathcal{D}_{2}\\in\\Delta_{\\sigma}(\\mu)}\\mathbb{E}_{X_{2}\\sim\\mathcal{ D}_{2}}\\inf_{a_{2}}\\sup_{y_{2}}\\ldots \\tag{2}\\] \\[\\ldots\\sup_{\\mathcal{D}_{T}\\in\\Delta_{\\sigma}(\\mu)}\\mathbb{E}_{X_ {T}\\sim\\mathcal{D}_{T}}\\inf_{a_{T}}\\sup_{y_{T}}\\mathcal{R}(\\mathcal{F},X_{1:T },y_{1:T},a_{1:T}).\\]\n\nIn order to bound this, we consider a generalization of the notion of online learning that is referred to as transductive learning. In this setting, at the start of the interaction the adversary chooses a set of contexts \\(X=\\left\\{X_{i}\\right\\}_{i=1}^{M}\\) for some \\(M\\geq T\\) and provides this to the player. The game proceeds as before with the adversary picking \\((x_{t},y_{t})\\) at time \\(t\\) and the learner picking an action \\(a_{t}\\) and suffering a loss \\(\\ell(a_{t},(x_{t},y_{t}))\\). However, the adversary is now constrained to pick \\(x_{t}\\in X\\) at all times \\(t\\). We can then define the minmax regret indexed by \\(X\\) as\n\n\\[\\underline{\\mathcal{R}}_{T}(\\mathcal{F},X):=\\left[\\max_{x_{1}\\in X}\\inf_{a_{1} }\\sup_{y_{1}}\\ldots\\max_{x_{T}\\in X}\\inf_{a_{T}}\\sup_{y_{T}}\\mathcal{R}( \\mathcal{F},x_{1:T},y_{1:T},a_{1:T})\\right].\\]\n\nFurthermore, define the worst-case transductive learning regret for sets of size \\(M\\) as \\(\\underline{\\mathcal{R}}_{T}^{M}(\\mathcal{F})=\\max_{X\\subseteq\\mathcal{X},|X|=M} \\underline{\\mathcal{R}}_{T}(\\mathcal{F},X)\\). In the following theorem, we show that the regret against \\(\\sigma\\)-smoothed adversaries is bounded by the regret in the transductive learning setting when the set of contexts is drawn from the base distribution \\(\\mu\\).\n\n**Theorem 3.1**.: _Let \\(\\mathcal{F}\\) be any class of functions from \\(\\mathcal{X}\\) to \\(\\mathbb{R}\\) and let \\(\\sigma\\in(0,1]\\). Then, for any \\(T\\) and \\(k\\), we have_\n\n\\[\\mathcal{R}_{T}(\\mathcal{F},\\sigma)\\leq\\underline{\\mathcal{R}}_{T}^{kT}( \\mathcal{F})+T^{2}(1-\\sigma)^{k}.\\]\n\nTheorem 3.1 shows that we can reduce the problem of evaluating the minimax regret for smoothed adversaries to evaluating the minimax regret for transductive learning. Note that the second term \\((1-\\sigma)^{k}\\leq e^{-k\\sigma}\\) and thus, in order to get bounds that are sublinear one needs to consider \\(k=c\\log T/\\sigma\\) for an appropriate absolute constant \\(c\\). As we will see in the next section, this leads to logarithmic dependence on \\(\\sigma^{-1}\\). Moreover, by Theorem 3.1 and since \\(\\mathbb{E}_{X\\sim\\mu^{T}}\\underline{\\mathcal{R}}_{T}(\\mathcal{F},X)\\leq\\mathcal{ R}_{T}(\\mathcal{F},\\sigma)\\), we can see that the transductive learning regret exactly captures the smoothed regret up to \\(\\operatorname{polylog}\\left(\\frac{T}{\\sigma}\\right)\\) factors.\n\n### Bounds for Transductive Learning\n\nIn this section, we discuss ways to upper bound transductive learning regret \\(\\underline{\\mathcal{R}}_{T}^{M}(\\mathcal{F})\\) so as to achieve bounds on \\(\\mathcal{R}_{T}(\\mathcal{F},\\sigma)\\) via Theorem 3.1.\n\n#### 3.2.1 Using Covering Numbers\n\nOne of the approaches common in online learning is to characterize the regret in terms of geometric properties (such as covering numbers) of the function class \\(\\mathcal{F}\\). The notion of covering required varies depending on the loss function and the stochastic properties of the data--typically completely adversarial problems require stronger notions of sequential coverings (Rakhlin et al., 2015a,b) while for stochastic problems usually weaker _offline coverings_ suffice. In our smoothed case, we show that the offline complexity notion of scale-sensitive VC dimension as defined in Definition 2.2 is adequate. Similar ideas were considered for the case of regression and convex Lipshitz losses in Haghtalab et al. (2022), Block et al. (2022).\n\nLet us first define the notion of approximation according to which a cover will be constructed; we will consider a pointwise approximation. This notion is similar to the notion of global sequential covering in Wu et al. (2022).\n\n**Definition 3.1**.: Let \\(\\mathcal{F}\\) be a function class. A set of functions \\(\\tilde{\\mathcal{F}}\\) is said to be a \\(\\epsilon\\)-covering of \\(\\mathcal{F}\\) if for any \\(f\\in\\mathcal{F}\\) there exists \\(g\\in\\tilde{\\mathcal{F}}\\) such that \\(\\sup_{x\\in\\mathcal{X}}|f(x)-g(x)|\\leq\\epsilon\\). We will use \\(\\mathcal{N}\\left(\\mathcal{F},\\epsilon\\right)\\) to denote the size of the minimal \\(\\epsilon\\)-covering of \\(\\mathcal{F}\\).\n\nNote that while the metric in Definition 3.1 is quite stringent, using this cover in the transductive learning case requires us to only consider function classes with _bounded domain size_. We capture this using the following theorem.\n\n**Theorem 3.2** (Upper bound on transductive learning).: _Let \\(\\mathcal{F}\\) be a function class and \\(\\epsilon>0\\). Then,_\n\n\\[\\underline{\\mathcal{R}}_{T}^{kT}(\\mathcal{F})\\leq\\inf_{\\epsilon}\\left\\{\\sup_{Z \\subset\\mathcal{X}\\setminus Z|=kT}\\log\\mathcal{N}\\left(\\mathcal{F}|_{Z}, \\epsilon\\right)+2\\epsilon T\\right\\},\\]\n\n_where \\(\\mathcal{F}|_{Z}\\) is the projection of hypothesis class \\(\\mathcal{F}\\) on the set \\(Z\\)._The proof of this theorem follows from relating transductive learning to the worst sequential prediction on a finite set of points using formalism presented in Wu et al. (2022). The proof of this theorem is deferred to the Appendix C.\n\nNext, we recall that the covering number \\(\\mathcal{N}(\\mathcal{F},\\epsilon)\\) is bounded as a function of the scale sensitive VC dimension of the class \\(\\mathcal{F}\\) and the number of points in the domain.\n\n**Theorem 3.3** (Rudelson and Vershynin (2006)).: _There exist universal constants \\(c,C\\) such that for all \\(\\alpha>0\\), any function class \\(\\mathcal{F}\\) defined on a finite set \\(\\mathcal{X}\\), and \\(\\epsilon>0\\), we have_\n\n\\[\\log\\mathcal{N}(\\mathcal{F},\\epsilon)\\leq C\\cdot\\mathrm{VC}(\\mathcal{F},c \\alpha\\epsilon)\\log^{1+\\alpha}\\left(\\frac{C|\\mathcal{X}|}{\\mathrm{VC}(\\mathcal{ F},c\\epsilon)\\epsilon}\\right).\\]\n\nFinally, putting together Theorem 3.1, Theorem 3.2 and Theorem 3.3 we get the following.\n\n**Theorem 3.4** (Minimax smoothed regret and scale-sensitive VC dimension).: \\[\\mathcal{R}_{T}(\\mathcal{F},\\sigma)\\leq\\inf_{k,\\alpha,\\epsilon>0}\\left\\{C \\cdot\\mathrm{VC}(\\mathcal{F},c\\alpha\\epsilon)\\log^{1+\\alpha}\\left(\\frac{CkT}{ \\mathrm{VC}(\\mathcal{F},c\\epsilon)\\epsilon}\\right)+2\\epsilon T+T^{2}\\left(1- \\sigma\\right)^{k}\\right\\}.\\]\n\nWe can instantiate the bound in Theorem 3.4 in terms of \\(T\\) and \\(\\sigma\\) for two particularly interesting cases: when \\(\\mathrm{VC}(\\mathcal{F},\\epsilon)\\) scales as \\(d\\log\\left(1/\\epsilon\\right)\\) (often referred to as parametric classes) and when \\(\\mathrm{VC}(\\mathcal{F},\\epsilon)\\) scales as \\(\\epsilon^{-p}\\) (often referred to as nonparametric classes). A canonical example of the former are _VC classes_; for a class with VC dimension \\(d\\), \\(\\mathrm{VC}(\\mathcal{F}^{\\mathrm{VC}},\\epsilon)=Cd\\log\\left(\\frac{1}{\\epsilon}\\right)\\) (see for example (Vershynin, 2018, Theorem 8.3.18)). A canonical example of the latter are functions of bounded variation, \\(\\mathcal{F}^{\\mathrm{BV}}\\) which have \\(\\mathrm{VC}(\\mathcal{F}^{\\mathrm{BV}},\\epsilon)=\\frac{C}{\\epsilon}\\) (see for example (Musayeva, 2020; Bartlett et al., 1997)). This class is known to have unbounded sequential covering numbers (Rakhlin et al., 2010) and therefore is not learnable with a worst-case adversary--this can be seen as a simple consequence of the fact that \\(\\mathcal{F}^{\\mathrm{BV}}\\) contains all one-dimensional thresholds.\n\n**Corollary 3.4.1** (Rates for parametric and nonparametric classes).: _If \\(\\mathrm{VC}(\\mathcal{F},\\epsilon)=d\\log\\left(1/\\epsilon\\right)\\), then for a large enough \\(T\\)_\n\n\\[\\mathcal{R}_{T}(\\mathcal{F},\\sigma)\\leq O\\left(d\\cdot\\mathrm{poly}\\log\\left( \\frac{T}{\\sigma}\\right)\\right).\\]\n\n_If \\(\\mathrm{VC}(\\mathcal{F},\\epsilon)=\\epsilon^{-p}\\), then_\n\n\\[\\mathcal{R}_{T}(\\mathcal{F},\\sigma)\\leq O\\left(T^{\\frac{p}{p+1}}\\cdot\\mathrm{ poly}\\log\\left(\\frac{T}{\\sigma}\\right)\\right).\\]\n\nIn particular, note that Corollary 3.4.1 shows that for VC classes \\(\\mathcal{R}_{T}(\\mathcal{F}^{\\mathrm{VC}},\\sigma)=\\widetilde{O}\\left(d\\cdot \\log\\left(\\frac{T}{\\sigma}\\right)\\right)\\) (tight, see also concurrent work (Wu et al., 2023) for a similar bound) and for functions of bounded variation \\(\\mathcal{R}_{T}(\\mathcal{F}^{\\mathrm{BV}},\\sigma)=\\widetilde{O}\\left(\\sqrt{T} \\log\\left(\\frac{1}{\\sigma}\\right)\\right)\\); note that the minmax rates for the worst-case adversary scale as \\(\\Omega(T)\\) for both these cases. We note that the above bound may be loose for general nonparametric classes, but should be improvable using a multiscale (chaining) version of Theorem 3.4 but we do not focus on this here.\n\nThough the above results give satisfactory bounds in the minimax sense for many classes \\(\\mathcal{F}\\) of interest, it is useful to consider explicit constructions of probability assignment rules. For the case of finite VC dimension, we give an explicit probability assignment rule by considering a discretization of the class and using a mixture probability assignment rule. In particular, this strategy (denoted by \\(\\mathcal{Q}^{\\mathrm{VC}}\\)) yields optimal regret \\(\\mathcal{R}_{T}(\\mathcal{F}^{\\mathrm{VC}},\\sigma,\\mathcal{Q}^{\\mathrm{VC}}) \\leq O\\left(d\\log\\left(\\frac{T}{\\sigma}\\right)\\right)\\). For the formal statements, proofs and detailed discussion see Appendix D.\n\n#### 3.2.2 Examples without Covering numbers\n\nThis reduction approach to characterizing minmax regret in the log-loss is interesting since all previous approaches have used covering numbers of some kind--either sequential covering numbersor stronger notions of global covering. However, in stark contrast to the 0/1 loss and several other loss functions (Rakhlin et al., 2015b), covering numbers cannot capture the minmax regret for the log-loss, at least in the adversarial case. Consider the following class of functions on context set\n\n\\[\\mathcal{X}\\,=\\,\\mathbb{B}_{2}\\text{ (where $\\mathbb{B}_{2}$ denotes the unit $\\ell_{2}$ Euclidean ball) }\\,\\mathcal{F}^{\\mathrm{Lin}}\\,:=\\,\\left\\{x\\mapsto\\frac{(x,w)+1}{2}\\middle|w \\in\\mathbb{B}_{2}\\right\\}.\\]\n\nFor this class, Rakhlin and Sridharan (2015) construct a follow-the-regularized leader (FTRL) based algorithm achieving regret \\(O(\\sqrt{T})\\). However, Bilodeau et al. (2020) show an upper bound on the regret in terms of sequential covering numbers which is not improvable in general--this shows that sequential covering numbers are not adequate to capture the minmax regret rates for the log-loss. Wu et al. (2022b) further consolidate this by considering the following class, closely resembling \\(\\mathcal{F}^{\\mathrm{Lin}}\\), \\(\\mathcal{F}^{\\mathrm{AbsLin}}:=\\left\\{x\\mapsto\\middle|\\left\\langle x,w\\right\\rangle \\middle|\\,\\middle|w\\in\\mathbb{B}_{2}\\right\\}.\\)(Wu et al., 2022b, Example 2, Theorem 6) establish that the minmax regret for \\(\\mathcal{F}^{\\mathrm{AbsLin}}\\) is \\(\\widetilde{\\Theta}(T^{2/3})\\), demonstrating the surprising fact that by a simple linear transformation of the hypothesis class (which does not change its covering number) one can obtain minmax rates that differ by a polynomial factor! On the other hand, our reduction-based approach bypasses the need for using any covering based arguments and therefore would lead to tight (at least up to poly \\(\\log(T/\\sigma)\\)) rates.\n\nWe remark that exact characterizations of the minmax regret with log-loss (often referred to as the minmax redundancy in the information theory literature) in the no-context (adversarial) case is most often calculated by studying the so-called _stochastic complexity_ of the class \\(\\mathcal{F}\\)(Rissanen, 1996). This can be extended to (worst-case) transductive learning with contexts \\(x_{1},\\ldots,x_{T}\\); in this case the minmax optimal regret for a fixed horizon is achieved by the normalized maximum likelihood (NML) probability assignment (Shtar'kov, 1987), and can be expressed as \\(\\underline{\\mathcal{R}}_{T}^{T}(\\mathcal{F})=\\max_{x_{1}:x}\\log\\left(\\sum_{y_{ 1}:r\\in\\{0,1\\}^{T}}\\max_{f\\in\\mathcal{F}}\\prod_{t=1}^{T}p_{f}(y_{t}|x_{t}) \\right).\\) This expression has been evaluated previously for online logistic regression (Jacquet et al., 2021) and more general hypothesis classes (Wu et al., 2022b). It is an intriguing question to understand what properties of \\(\\mathcal{F}\\) the stochastic complexity depends on, given that the above examples illustrates that covering numbers do not capture it. Our reduction provides a technique to use such fine-grained understanding of the regret to directly lift the bounds to the more general smoothed adversary setting.\n\n## 4 Oracle-Efficient Smoothed Sequential Probability Assignment\n\nIn the previous section, we consider a purely statistical perspective on the minimax value of the sequential probability assignment problem for smoothed adversaries. In this section, we will focus on an algorithmic perspective and design an algorithm that is efficiently implemented using calls to an MLE oracle. We will focus on the setting when the base measure is the uniform measure on the input space \\(\\mathcal{X}\\) and the label space \\(\\mathcal{Y}=\\{0,1\\}\\). In this setting, we are given access to an oracle \\(\\mathrm{OPT}\\) which given a data set \\(S=\\{x_{i},y_{i}\\}_{i=1}^{m}\\) outputs a hypothesis that minimizes the loss on \\(S\\). That is,\n\n\\[\\mathrm{OPT}\\left(S\\right)=\\operatorname*{argmin}_{h\\in\\mathcal{F}}\\frac{1}{m} \\sum_{i=1}^{m}\\ell\\left(h,(x_{i},y_{i})\\right).\\]\n\nIn the context of the logarithmic loss, this corresponds to maximum likelihood estimation. Most of the analysis holds for a general loss function \\(\\ell\\) (with regret scaling appropriate bounds on the values and derivatives), but for clarity one can think of \\(\\ell\\) as the log-loss. In particular, Algorithm 1 is written for the log-loss.\n\nThe main framework we work in is the follow-the-perturbed-leader (FTPL) framework. Here, our algorithm uses the oracle on a data set consisting of the historical samples and a set of hallucinated samples. The hallucinated samples are intended to \"stabilize\" the predictions of the algorithm. This gives us a probability assignment \\(\\mathscr{Q}^{\\mathrm{FTPL}}=\\{q^{\\mathrm{FTPL}}(\\cdot|x_{1:t},y_{1:t-1})\\}_{ t=1}^{T}\\).\n\nIn order to state the regret bound, we need the following notions. For any class \\(\\mathcal{F}\\), define the truncated class \\(\\mathcal{F}_{\\alpha}\\) as \\(\\mathcal{F}_{\\alpha}=\\left\\{f_{\\alpha}:f_{\\alpha}(x)=\\frac{f(x)+\\alpha}{1+2 \\alpha}\\text{ where }f\\in\\mathcal{F}\\right\\}.\\) We will also need the notion of Rademacher complexity \\(\\operatorname{Rad}\\left(\\mathcal{F},T\\right)=\\sup_{X\\subset\\mathcal{X}|X|=T} \\mathbb{E}_{\\varepsilon}\\left[\\sup_{f\\in\\mathcal{F}}\\frac{1}{T}\\sum_{x\\in X} \\epsilon_{x}f\\left(x\\right)\\right].\\)\n\n**Theorem 4.1** (Main Regret Bound).: _For any hypothesis class \\(\\mathcal{F}\\) and parameters \\(n,\\alpha\\), we have that the regret of Algorithm 1 for \\(\\sigma\\)-smoothed adversaries is bounded as_\n\n\\[\\mathcal{R}_{T}(\\mathcal{F},\\sigma,\\mathscr{Q}^{\\mathrm{FTPL}}) \\leq n\\log\\left(\\frac{1}{\\alpha}\\right)+\\alpha T+T\\sqrt{\\log\\left( \\frac{1}{\\alpha}\\right)\\cdot\\frac{1}{\\sigma n}}\\] \\[\\qquad\\qquad+T\\cdot\\inf_{m\\leq n}\\left\\{\\frac{1}{\\alpha}\\mathrm{ Rad}\\left(\\mathcal{F}_{\\alpha},n/m\\right)+\\frac{n\\left(1-\\sigma\\right)^{m}\\log\\left(1/ \\alpha\\right)}{m}+e^{-n/8}\\right\\}.\\]\n\nWe will instantiate this bound for case when the class \\(\\mathcal{F}\\) has bounded VC dimension. For such classes, it is known that the Rademacher complexity is bounded. We state this in the following corollary.\n\n**Corollary 4.1.1**.: _Let \\(\\mathcal{F}\\) be a hypothesis class such that the Rademacher complexity is bounded as \\(\\mathrm{Rad}\\left(\\mathcal{F}_{\\alpha},T\\right)=cT^{-\\omega}\\), then we have \\(\\mathcal{R}_{T}(\\mathcal{F},\\sigma,\\mathscr{Q}^{\\mathrm{FTPL}})\\leq T^{\\frac {2}{2+\\omega}}\\sqrt{\\frac{1}{\\sigma}}\\cdot\\mathrm{poly}\\log\\left(\\frac{Td}{ \\sigma}\\right)\\)._\n\nNote that, in particular, for VC classes we have \\(\\mathcal{R}_{T}\\left(\\mathcal{F}^{\\mathrm{VC}},\\sigma\\right)\\) scales as \\(T^{4/5}\\). Improving this to achieve the minimax rate discussed in Corollary 3.4.1 is an interesting open question.\n\n_Remark_.: A slightly improved regret scaling as \\(T^{3/4}\\) can be achieved by assuming access to an oracle that can optimize a mixed objective function involving the log-loss and signed sum of the functions in class. However, these oracles do not have the natural interpretation in terms of maximum likliehood estimation.\n\n### Analysis\n\nThe main challenge in designing algorithms in the follow-the-perturbed-leader framework is designing the distribution of the hallucinated samples so as to balance the tradeoff between the \"stability\" of the algorithm i.e. how little the algorithm changes its prediction from time step to time step, and the \"perturbation\" i.e. how much the addition of the hallucinated samples changes the prediction of the algorithm from the outputting the best hypothesis on the historical samples. This is captured by the following lemma.\n\n**Lemma 4.2** (Follow the Perturbed Leader bound).: _Let \\(\\ell\\) be a convex loss function, and let \\(\\mathcal{F}\\) be a hypothesis class. Let \\(\\mathcal{D}_{t}\\) denote the distribution of the adversary at time \\(t\\) and let \\(\\mathcal{Q}_{t}\\) denote the distribution of the hypothesis \\(h_{t}\\) output by Algorithm 1. Then, we have that the regret of Algorithm 1 (where we use \\(\\widetilde{s}_{t}=(\\widetilde{x}_{t},\\widetilde{y}_{t})\\) to denote a hallucinated data point)2 is bounded by_\n\nFootnote 2: With some abuse of notation, we consider \\(\\mathcal{D}_{t}\\) to be over \\(\\mathcal{X}\\times\\{0,1\\}\\).\n\n\\[\\sum_{i=1}^{T}\\underbrace{\\mathbb{E}}_{s_{t}\\sim\\mathcal{D}_{t}}\\left( \\underset{h_{t}\\sim\\mathcal{Q}_{t}}{\\mathbb{E}}[\\ell(h_{t},s_{t})]-\\underset{h _{t+1}\\sim\\mathcal{Q}_{t+1}}{\\mathbb{E}}[\\ell(h_{t+1},s_{t})]\\right)+\\underbrace {\\mathbb{E}\\left[\\sup_{h\\in\\mathcal{F}_{\\sigma}}\\sum_{i=1}^{N}\\ell(h,\\tilde{s}_{ t})-\\ell(h^{*},\\tilde{s}_{t})\\right]}_{\\text{Permutation}}+\\alpha T.\\]\n\n_where \\(h^{*}=\\operatorname*{argmin}_{h\\in\\mathcal{F}_{\\sigma}}\\sum_{i=1}^{T}\\ell(h,s _{t})\\)._We provide a proof in Appendix E for completeness. Given this decomposition of the regret, we need to handle both terms carefully. Just to appreciate the tradeoff, note that as we increase the number of hallucinated examples, the perturbation term generally increases, but the stability term generally decreases. First, let us focus on the stability term which is harder to deal with. The main approach we will use is a generalization of the decomposition of the stability term introduced in Haghtalab et al. (2022) even when the losses are unbounded, as is the case with the log-loss. The main idea is to decompose the stability term in terms of the distance between the distribution of the average prediction at the next time step and the distribution of the current time step, as captured by the \\(\\chi^{2}\\) distance and a term that captures how different the predictions of the algorithm are when the sample is resampled from the same distribution. The proof can be found in Appendix F. 3\n\nFootnote 3: For the particular use in our analysis a simpler version of the lemma similar to Haghtalab et al. (2022) suffices but we prove a general version since we believe such a version is useful in providing improved regret bounds for the problem.\n\n**Lemma 4.3** (\\(\\chi^{2}\\) + Generalization \\(\\Rightarrow\\) Stability).: _Let \\(\\mathcal{Q}_{t}\\) denote learner's distribution over \\(\\mathcal{F}\\) in at round \\(t\\), \\(\\mathcal{D}_{t}\\) be adversary's distribution at time \\(t\\) (given the history \\(s_{1},\\cdots,s_{t-1}\\)), \\(s_{t}\\sim\\mathcal{D}_{t}\\) be the realized adversarial instance at time \\(t\\), and \\(s^{\\prime}_{t}\\) be an independent copy \\(s^{\\prime}_{t}\\sim\\mathcal{D}_{t}\\). Let \\(R^{(t+1)}\\) refers to the randomness used by the algorithm in round \\(t+1\\). Then,_\n\n\\[\\mathop{\\mathbb{E}}_{s_{t}\\sim\\mathcal{D}_{t}} \\left(\\mathop{\\mathbb{E}}_{h_{t}\\sim\\mathcal{Q}_{t}}[\\ell(h_{t},s _{t})]-\\mathop{\\mathbb{E}}_{h_{t+1}\\sim\\mathcal{Q}_{t+1}}[\\ell(h_{t+1},s_{t})]\\right)\\] \\[\\leq\\sqrt{\\frac{1}{2}}\\chi^{2}(\\mathop{\\mathbb{E}}_{s_{t}\\sim \\mathcal{D}_{t}}[\\mathcal{Q}_{t+1}],\\mathcal{Q}_{t})\\cdot\\log\\left(\\frac{1}{ \\alpha}\\right)+\\mathop{\\mathbb{E}}_{s_{t},s^{\\prime}_{t}\\sim\\mathcal{D}_{t};R^ {(t+1)}}[\\ell(h_{t+1},s^{\\prime}_{t})-\\ell(h_{t+1},s_{t})].\\]\n\nGiven this lemma, we move on to bounding the \\(\\chi^{2}\\) divergence between the distribution of the average prediction at the next time step and the distribution of the current time step. This is done using the Ingster method to bound the divergence of mixtures. We include a proof in Appendix G for completeness.\n\n**Lemma 4.4** (Bound on \\(\\chi^{2}\\)).: \\(\\chi^{2}\\left(\\mathop{\\mathbb{E}}_{s_{t}\\sim\\mathcal{D}_{t}}[\\mathcal{Q}_{t+1} ],\\mathcal{Q}_{t}\\right)\\leq\\frac{2}{\\sigma n}\\)_._\n\nNext, we move on second term in Lemma 4.3. Note that this term involves the difference between the loss of the hypothesis output at time \\(t+1\\) evaluated on two independent points \\(s_{t}\\) and \\(s^{\\prime}_{t}\\) drawn from \\(\\mathcal{D}_{t}\\). The main idea to bound the term is to use a stronger version of the coupling lemma Theorem 2.1 which allows us to extract subsequences of points sampled according to smooth distributions from iid samples from the base measure. This allows us to relate the required generalization bound to the Rademacher complexity of the class composed with the loss. Using the truncation and the contraction principle, we get the desired bound. The proof can be found in Appendix H.\n\n**Lemma 4.5** (Generalization).: _Let \\(h_{t+1}\\) denote the hypothesis output by the Algorithm 1 at time \\(t+1\\). Then, for any \\(m\\leq n\\), we have_\n\n\\[\\mathop{\\mathbb{E}}_{s_{t},s^{\\prime}_{t}\\sim\\mathcal{D}_{t};R^{(t+1)}}\\left[ \\ell(h_{t+1},s^{\\prime}_{t})-\\ell(h_{t+1},s_{t})\\right]\\leq\\frac{1}{\\alpha} \\mathrm{RAD}\\left(\\mathcal{F}_{\\alpha},n/m\\right)+\\frac{n\\left(1-\\sigma\\right) ^{m}\\log\\left(1/\\alpha\\right)}{m}+e^{-n/8}.\\]\n\nThe final term that we want to bound is the perturbation term. In order to bound this note that we set our truncation parameter \\(\\alpha\\) such that the loss our the prediction made by our algorithm is bounded and consequently the perturbation term is bounded by \\(n\\log\\frac{1}{\\alpha}\\), see Appendix I. Theorem 4.1 follows by combining the above results.\n\n## 5 Conclusions and Open Problems\n\nIn this paper, we initiated the study of sequential probability assignment with smoothed adversaries. We characterize the minimax regret in terms of the minimax regret for transductive learning and use this to provide tight regret bounds, e.g., for VC classes. Furthermore, we initiate the study of oracle efficiency in this setting and show that sublinear regret can be achieve for general classes.\n\nOur work motivates several directions for future work. An interesting direction is whether the optimal \\(O(\\log(T))\\) regret is achievable for some classes, such as VC classes, using oracle-efficient algorithms. More generally, are there computational barriers to obtaining fast rates in prediction with log-loss in this setting?\n\n## Acknowledgments and Disclosure of Funding\n\nThis work was supported in part by the National Science Foundation under grant CCF-2145898, a C3.AI Digital Transformation Institute grant, and Berkeley AI Research Commons grants. This work was partially done while authors were visitors at the Simons Institute for the Theory of Computing.\n\n## References\n\n* Rissanen (1983) Jorma Rissanen. A universal data compression system. _IEEE Transactions on information theory_, 29(5):656-664, 1983.\n* Rissanen (1984) Jorma Rissanen. Universal coding, information, prediction, and estimation. _IEEE Transactions on Information theory_, 30(4):629-636, 1984.\n* Cover (1991) Thomas M Cover. Universal portfolios. _Mathematical finance_, 1(1):1-29, 1991.\n* Feder et al. (1992) Meir Feder, Neri Merhav, and Michael Gutman. Universal prediction of individual sequences. _IEEE transactions on Information Theory_, 38(4):1258-1270, 1992.\n* Xie and Barron (1997) Qun Xie and Andrew R Barron. Minimax redundancy for the class of memoryless sources. _IEEE Transactions on Information Theory_, 43(2):646-657, 1997.\n* Merhav and Feder (1998) Neri Merhav and Meir Feder. Universal prediction. _IEEE Transactions on Information Theory_, 44(6):2124-2147, 1998.\n* Xie and Barron (2000) Qun Xie and Andrew R Barron. Asymptotic minimax regret for data compression, gambling, and prediction. _IEEE Transactions on Information Theory_, 46(2):431-445, 2000.\n* Yang and Barron (1999) Yuhong Yang and Andrew Barron. Information-theoretic determination of minimax rates of convergence. _Annals of Statistics_, pages 1564-1599, 1999.\n* Jiao et al. (2013) Jiantao Jiao, Haim H Permuter, Lei Zhao, Young-Han Kim, and Tsachy Weissman. Universal estimation of directed information. _IEEE Transactions on Information Theory_, 59(10):6220-6242, 2013.\n* Orabona and Pal (2016) Francesco Orabona and David Pal. Coin betting and parameter-free online learning. _Advances in Neural Information Processing Systems_, 29, 2016.\n* Foster et al. (2018) Dylan J Foster, Satyen Kale, Haipeng Luo, Mehryar Mohri, and Karthik Sridharan. Logistic regression: The importance of being improper. In _Conference On Learning Theory_, pages 167-208. PMLR, 2018.\n* Rakhlin and Sridharan (2015) Alexander Rakhlin and Karthik Sridharan. Sequential probability assignment with binary alphabets and large classes of experts. _arXiv preprint arXiv:1501.07340_, 2015.\n* Fogel and Feder (2017) Yaniv Fogel and Meir Feder. On the problem of on-line learning with log-loss. In _2017 IEEE International Symposium on Information Theory (ISIT)_, pages 2995-2999. IEEE, 2017.\n* Fogel and Feder (2018) Yaniv Fogel and Meir Feder. Universal batch learning with log-loss. In _2018 IEEE International Symposium on Information Theory (ISIT)_, pages 21-25. IEEE, 2018.\n* Bhatt and Kim (2021) Alankrita Bhatt and Young-Han Kim. Sequential prediction under log-loss with side information. In _Algorithmic Learning Theory_, pages 340-344. PMLR, 2021.\n* Bilodeau et al. (2021) Blair Bilodeau, Dylan J Foster, and Daniel M Roy. Minimax rates for conditional density estimation via empirical entropy. _arXiv preprint arXiv:2109.10461_, 2021.\n* Wu et al. (2022a) Changlong Wu, Mohsen Heidari, Ananth Grama, and Wojciech Szpankowski. Expected worst case regret via stochastic sequential covering. _arXiv preprint arXiv:2209.04417_, 2022a.\n* Foster and Krishnamurthy (2021) Dylan J Foster and Akshay Krishnamurthy. Efficient first-order contextual bandits: Prediction, allocation, and triangular discrimination. _Advances in Neural Information Processing Systems_, 34:18907-18919, 2021.\n* Zhao et al. (2018)Dylan J Foster, Sham M Kakade, Jian Qian, and Alexander Rakhlin. The statistical complexity of interactive decision making. _arXiv preprint arXiv:2112.13487_, 2021a.\n* Littlestone [1988] Nick Littlestone. Learning quickly when irrelevant attributes abound: A new linear-threshold algorithm. _Machine learning_, 2:285-318, 1988.\n* Haghtalab et al. [2021] Nika Haghtalab, Tim Roughgarden, and Abhishek Shetty. Smoothed analysis with adaptive adversaries. In _2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS)_, pages 942-953. IEEE, 2021.\n* Rakhlin et al. [2011] Alexander Rakhlin, Karthik Sridharan, and Ambuj Tewari. Online learning: Stochastic, constrained, and smoothed adversaries. In _Advances in neural information processing systems_, pages 1764-1772, 2011.\n* Haghtalab et al. [2022] Nika Haghtalab, Yanjun Han, Abhishek Shetty, and Kunhe Yang. Oracle-efficient online learning for beyond worst-case adversaries. _Advances in Neural Information Processing Systems_, 2022.\n* Block et al. [2022] Adam Block, Yuval Dagan, Noah Golowich, and Alexander Rakhlin. Smoothed online learning is as easy as statistical learning. In _Conference on Learning Theory_, pages 1716-1786. PMLR, 2022.\n* Agarwal et al. [2014] Alekh Agarwal, Daniel Hsu, Satyen Kale, John Langford, Lihong Li, and Robert Schapire. Taming the monster: A fast and simple algorithm for contextual bandits. In _International Conference on Machine Learning_, pages 1638-1646. PMLR, 2014.\n* Simchi-Levi and Xu [2022] David Simchi-Levi and Yunzong Xu. Bypassing the monster: A faster and simpler optimal algorithm for contextual bandits under realizability. _Mathematics of Operations Research_, 47(3):1904-1931, 2022.\n* Foster and Rakhlin [2020] Dylan Foster and Alexander Rakhlin. Beyond ucb: Optimal and efficient contextual bandits with regression oracles. In _International Conference on Machine Learning_, pages 3199-3210. PMLR, 2020.\n* Langford et al. [2007] John Langford, Lihong Li, and Alexander Strehl. Vowpal wabbit open source project. 2007.\n* Foster et al. [2021b] Dylan J Foster, Alexander Rakhlin, David Simchi-Levi, and Yunzong Xu. Instance-dependent complexity of contextual bandits and reinforcement learning: A disagreement-based perspective. 2021b.\n* Kalai and Vempala [2005] Adam Kalai and Santosh Vempala. Efficient algorithms for online decision problems. _Journal of Computer and System Sciences_, 71(3):291-307, 2005.\n* Dudik et al. [2020] Miroslav Dudik, Nika Haghtalab, Haipeng Luo, Robert E Schapire, Vasilis Syrgkanis, and Jennifer Wortman Vaughan. Oracle-efficient online learning and auction design. _Journal of the ACM (JACM)_, 67(5):1-57, 2020.\n* Wang et al. [2022] Guanghui Wang, Zihao Hu, Vidya Muthukumar, and Jacob Abernethy. Adaptive oracle-efficient online learning. _Advances in Neural Information Processing Systems_, 2022.\n* Kakade et al. [2007] Sham M Kakade, Adam Tauman Kalai, and Katrina Ligett. Playing games with approximation algorithms. In _Proceedings of the thirty-ninth annual ACM symposium on Theory of computing_, pages 546-555, 2007.\n* Haghtalab et al. [2020] Nika Haghtalab, Tim Roughgarden, and Abhishek Shetty. Smoothed analysis of online and differentially private learning. In H. Larochelle, M. Ranzato, R. Hadsell, M.F. Balcan, and H. Lin, editors, _Advances in Neural Information Processing Systems_, volume 33, pages 9203-9215. Curran Associates, Inc., 2020.\n* Rakhlin et al. [2015a] Alexander Rakhlin, Karthik Sridharan, and Ambuj Tewari. Sequential complexities and uniform martingale laws of large numbers. _Probability Theory and Related Fields_, 161(1-2):111-153, 2015a.\n* Rakhlin et al. [2015b] Alexander Rakhlin, Karthik Sridharan, and Ambuj Tewari. Online learning via sequential complexities. _J. Mach. Learn. Res._, 16(1):155-186, 2015b.\n* Rakhlin et al. [2015c]Changlong Wu, Mohsen Heidari, Ananth Grama, and Wojciech Szpankowski. Precise regret bounds for log-loss via a truncated bayesian algorithm. _Advances in Neural Information Processing Systems_, 2022b.\n* Rudelson and Vershynin (2006) Mark Rudelson and Roman Vershynin. Combinatorics of random processes and sections of convex bodies. _Annals of Mathematics_, pages 603-648, 2006.\n* Vershynin (2018) Roman Vershynin. _High-dimensional probability: An introduction with applications in data science_. Cambridge university press, 2018.\n* Musayeva (2020) Khadija Musayeva. Sample complexity result for multi-category classifiers of bounded variation. _arXiv preprint arXiv:2003.09176_, 2020.\n* Bartlett et al. (1997) Peter L Bartlett, Sanjeev R Kulkarni, and S Eli Posner. Covering numbers for real-valued function classes. _IEEE transactions on information theory_, 43(5):1721-1724, 1997.\n* Rakhlin et al. (2010) Alexander Rakhlin, Karthik Sridharan, and Ambuj Tewari. Online learning: Random averages, combinatorial parameters, and learnability. _Advances in Neural Information Processing Systems_, 23, 2010.\n* Wu et al. (2023) Changlong Wu, Ananth Grama, and Wojciech Szpankowski. Online learning in dynamically changing environments. _arXiv preprint arXiv:2302.00103_, 2023.\n* Bilodeau et al. (2020) Blair Bilodeau, Dylan Foster, and Daniel Roy. Tight bounds on minimax regret under logarithmic loss via self-concordance. In _International Conference on Machine Learning_, pages 919-929. PMLR, 2020.\n* Rissanen (1996) Jorma J Rissanen. Fisher information and stochastic complexity. _IEEE transactions on information theory_, 42(1):40-47, 1996.\n* Shtar'kov (1987) Yurii Mikhailovich Shtar'kov. Universal sequential coding of single messages. _Problemy Peredachi Informatsii_, 23(3):3-17, 1987.\n* Jacquet et al. (2021) Philippe Jacquet, Gil Shamir, and Wojciech Szpankowski. Precise minimax regret for logistic regression with categorical feature values. In _Algorithmic Learning Theory_, pages 755-771. PMLR, 2021.\n* Krichevsky and Trofimov (1981) Raphail Krichevsky and Victor Trofimov. The performance of universal encoding. _IEEE Transactions on Information Theory_, 27(2):199-207, 1981.\n* Kamath et al. (2015) Sudeep Kamath, Alon Orlitsky, Dheeraj Pichapati, and Ananda Theertha Suresh. On learning distributions from their samples. In _Conference on Learning Theory_, pages 1066-1100. PMLR, 2015.\n* Spielman and Teng (2004) Daniel A Spielman and Shang-Hua Teng. Smoothed analysis of algorithms: Why the simplex algorithm usually takes polynomial time. _Journal of the ACM (JACM)_, 51(3):385-463, 2004.\n* Block and Simchowitz (2022) Adam Block and Max Simchowitz. Efficient and near-optimal smoothed online learning for generalized linear functions. _Advances in neural information processing systems_, 2022.\n* Kalai and Vempala (2002) Adam Tauman Kalai and Santosh Vempala. Efficient algorithms for universal portfolios. _Journal of Machine Learning Research_, pages 423-440, 2002.\n* Luo et al. (2018) Haipeng Luo, Chen-Yu Wei, and Kai Zheng. Efficient online portfolio with logarithmic regret. _Advances in neural information processing systems_, 31, 2018.\n* Zimmert et al. (2022) Julian Zimmert, Naman Agarwal, and Satyen Kale. Pushing the efficiency-regret pareto frontier for online learning of portfolios and quantum states. In _Conference on Learning Theory_, pages 182-226. PMLR, 2022.\n* Jezequel et al. (2022) Remi Jezequel, Dmitrii M Ostrovskii, and Pierre Gaillard. Efficient and near-optimal online portfolio selection. _arXiv preprint arXiv:2209.13932_, 2022.\n* Zimmert et al. (2022)Tim Van Erven, Dirk Van der Hoeven, Wojciech Kotlowski, and Wouter M Koolen. Open problem: Fast and optimal online portfolio selection. In _Conference on Learning Theory_, pages 3864-3869. PMLR, 2020.\n* Cover and Ordentlich (1996) Thomas M Cover and Erik Ordentlich. Universal portfolios with side information. _IEEE Transactions on Information Theory_, 42(2):348-363, 1996.\n* Cross and Barron (2003) Jason E Cross and Andrew R Barron. Efficient universal portfolios for past-dependent target classes. _Mathematical Finance: An International Journal of Mathematics, Statistics and Financial Economics_, 13(2):245-276, 2003.\n* Gyorfi et al. (2006) Laszlo Gyorfi, Gabor Lugosi, and Frederic Udina. Nonparametric kernel-based sequential investment strategies. _Mathematical Finance: An International Journal of Mathematics, Statistics and Financial Economics_, 16(2):337-357, 2006.\n* Bhatt et al. (2023) Alankrita Bhatt, J Jon Ryu, and Young-Han Kim. On universal portfolios with continuous side information. In _International Conference on Artificial Intelligence and Statistics_. PMLR, 2023.\n* Cesa-Bianchi and Lugosi (2006) Nicolo Cesa-Bianchi and Gabor Lugosi. _Prediction, learning, and games_. Cambridge university press, 2006.\n\nRelated work\n\nSequential probability assignment is a classic topic in information theory with extensive literature, see the survey by Merhav and Feder (1998) and the references within. In particular, the idea of probability assignments that are Bayesian mixtures over the reference class of distributions (Krichevsky and Trofimov, 1981) is of central importance--such mixture probability assignments arise as the optimal solution to several operational information theoretic and statistical problems (Kamath et al., 2015). It is also known that the Bayesian mixture approach often outperforms the \"plug-in\" approach of estimating a predictor from the reference class and then playing it (Merhav and Feder, 1998). A similar Bayesian mixture probability assignment in the contextual probability assignment problem was used by Bhatt and Kim (2021), where the covering over the VC function class was obtained in a _data-dependent manner_. This idea of using a mixture over an empirical covering along with a so-called \"add-\\(\\beta\\)\" probability assignment was then used by Bilodeau et al. (2021). Combining this with the key idea of discretizing the class of functions as per the Hellinger divergence induced metric, they obtained matching rates for several interesting classes in the _realizable_ case (i.e. \\(y_{t}\\big{|}x_{t}\\) generated according to a fixed unknown distribution in the reference class); see also Yang and Barron (1999) for more intuition behind usage of Hellinger coverings for stochastic data. Recent work of Wu et al. (2022, 2022) has also employed an empirical covering with an add-\\(\\beta\\) probability assignment for both stochastic and adversarial adversaries.\n\nA complementary approach, more common in the online learning literature is to study fundamental limits of sequential decision making problems non-constructively (i.e. providing bounds on the minmax regret without providing a probability assignment that achieves said regret). This sequential complexities based approach of Rakhlin et al. (2015, 2015) has been employed for the log-loss by Rakhlin and Sridharan (2015) and Bilodeau et al. (2020); however the latter suggests that sequential complexities might not fully capture the log-loss problem.\n\nSmoothed analysis, initiated by Spielman and Teng (2004) for the study of efficiency of algorithms such as the simplex method, has recently shown to be effective in circumventing both statistical and computational lower bounds in online learning for classification and regression Haghtalab et al. (2021); Rakhlin et al. (2011); Block et al. (2022); Haghtalab et al. (2020); Block and Simchowitz (2022). This line of work establishes that smoothed analysis is a viable line of attack to construct statistically and computationally efficient algorithms for sequential decision making problems and has developed tools such as the coupling lemma to understand smoothed sequential decision-making. But handling the unbounded and non-Lipschitz log-loss and MLE oracle is the main conceptual novelty introduced in this work.\n\nDue to the fundamental nature of the problem, the notion of computational efficiency for sequential probability assignment and the closely related problem of portfolio selection has been considered in the literature. Kalai and Vempala (2002) presents an efficient implementation of Cover's universal portfolio algorithm using techniques from Markov chain Monte Carlo. Recently, there has been a flurry of interest in using follow the regularized leader (FTRL) type techniques to achieve low regret and low complexity simultaneously (Luo et al., 2018; Zimmert et al., 2022; Jezequel et al., 2022), see also Van Erven et al. (2020) and the references within. However, none of these methods consider the contextual version of the problem and are considerably different from the oracle-efficient approach. On the other hand, work studying portfolio selection with contexts (Cover and Ordentlich, 1996; Cross and Barron, 2003; Gyorfi et al., 2006; Bhatt et al., 2023) does not take oracle-efficiency into account.\n\n**Concurrent Work:**Wu et al. (2023) also study the problem of sequential probability assignment (and general mixable losses) and for VC classes achieve the optimal regret of \\(O(d\\log(T/\\sigma))\\). In addition to the smooth adversaries, they also studied general models capturing the setting where the base measures are not known. They work primarily in the information theoretical setting and do not present any results regarding efficient algorithms. A major contribution of our work is the study of oracle-efficient online learning for the log-loss, which has not been considered by Wu et al. (2023) Indeed, to the best of knowledge, oracle efficiency has not been considered at all for losses such as the log loss even for worst-case settings. Considering oracle-efficiency for log-loss is an important and natural line of research as the ERM oracle corresponds to the maximum likelihood estimator (MLE) that is commonly used in practice. Our paper presents sublinear regret oracle efficient algorithms for the log loss (which does not have an analog in Wu et al. (2023)).\n\nDeferred Proof from Section 3\n\nIn order to obtain an upper bound on \\(\\mathcal{R}_{T}(\\mathcal{F},\\sigma)\\) in terms of \\(\\underline{\\mathcal{R}}_{T}^{kT}(\\mathcal{F})\\) for some \\(k\\), we will consider (2) and proceed inductively. The main idea is to note that since \\(\\mathcal{D}_{i}\\) is \\(\\sigma\\)-smoothed, conditioned on the history thus far, we can invoke the coupling lemma given in Theorem 2.1.\n\nFor the sake of illustration, first consider the simple case of \\(T=1\\). Let \\(X_{1},Z_{1}\\ldots Z_{k}\\) denote the coupling alluded to in Theorem 2.1. Recall that \\(X_{1}\\sim\\mathcal{D}_{1}\\) and \\(Z_{1:k}\\sim\\mu^{k}\\). Defining the event \\(E_{1}:=\\{X_{1}\\in Z_{1:k}\\}\\), we have\n\n\\[\\mathcal{R}_{1}(\\mathcal{F},\\mathcal{D}) =\\mathbb{E}_{X_{1}\\sim\\mathcal{D}_{1}}\\inf_{a_{1}}\\sup_{y_{1}} \\mathcal{R}_{1}(\\mathcal{F},X_{1},y_{1},a_{1})\\] \\[=\\mathbb{E}_{X_{1},Z_{1:k}}\\left[\\inf_{a_{1}}\\sup_{y_{1}} \\mathcal{R}_{1}(\\mathcal{F},X_{1},y_{1},a_{1})\\right]\\] \\[=\\mathbb{E}_{X_{1},Z_{1:k}}\\left[\\mathds{1}\\{E_{1}\\}\\inf_{a_{1}} \\sup_{y_{1}}\\mathcal{R}_{1}(\\mathcal{F},X_{1},y_{1},a_{1})\\right]\\] \\[\\qquad\\qquad+\\mathbb{E}_{X_{1},Z_{1:k}}\\left[\\mathds{1}\\{E_{1}^{C }\\}\\inf_{a_{1}}\\sup_{y_{1}}\\mathcal{R}_{1}(\\mathcal{F},X_{1},y_{1},a_{1})\\right]\\] \\[\\leq\\mathbb{E}_{X_{1},Z_{1:k}}\\left[\\mathds{1}\\{E_{1}\\}\\inf_{a_{1 }}\\sup_{y_{1}}\\mathcal{R}_{1}(\\mathcal{F},X_{1},y_{1},a_{1})\\right]+\\mathbb{P} (E_{1}^{c}) \\tag{3}\\] \\[\\leq\\mathbb{E}_{Z_{1:k}}\\left[\\max_{X_{1}\\in Z_{1:k}}\\inf_{a_{1}} \\sup_{y_{1}}\\mathcal{R}_{1}(\\mathcal{F},X_{1},y_{1},a_{1})\\right]+(1-\\sigma)^{k}\\] (4) \\[=\\underline{\\mathcal{R}}_{T}^{kT}(\\mathcal{F})+(1-\\sigma)^{k}, \\tag{5}\\]\n\nwhere (3) uses that \\(\\inf_{a_{1}}\\sup_{y_{1}}\\mathcal{R}_{1}(\\mathcal{F},X_{1},y_{1},a_{1})\\leq 1\\)4, (4) follows by the coupling lemma and (5) follows from the definition of transductive learning regret. The next step is to generalize this to arbitrary \\(T\\). The key aspect that makes this possible is that for all \\(t\\leq T\\), we have \\(D_{t}\\in\\Delta_{\\sigma}\\left(\\mu\\right)\\), even conditioned on the past, allowing us to apply the coupling lemma. Furthermore, we need that \\(\\mathcal{R}_{T}\\leq T\\) for arbitrary sequences which is indeed guaranteed for reasonable losses such as the log-loss as noted above.\n\nFootnote 4: This holds for the log-loss by using the trivial strategy of using a uniform probability assignment at each step.\n\nWe now move to general case. We will prove this inductively. Assume that we have used the coupling lemma till time \\(t-1\\) and replaced the samples from the smooth distributions with samples from the uniform. That is assume the induction hypothesis, for time \\(t\\) as\n\n\\[\\mathcal{R}_{T}\\leq\\mathop{\\mathbb{E}}_{\\{Z_{1:k}\\}\\sim\\mu}\\max_ {X_{1}\\in Z_{1}^{k}}\\inf_{a_{1}}\\sup_{y_{1}}\\ldots\\sup_{\\mathcal{D}_{t}}\\mathop {\\mathbb{E}}_{X_{t}\\sim\\mathcal{D}_{t}}\\inf_{a_{t}}\\sup_{y_{t}}\\ldots\\] \\[\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad \\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad \\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad \\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad \\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad \\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad \\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad \\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad \\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad \\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad \\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad \\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad \\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad \\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad \\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad \\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad \\\\[\\leq\\mathbb{E}_{Z_{1}\\sim\\mu^{k}}\\max_{X_{1}\\in Z_{1}}\\inf_{a_{1}}\\sup_{ y_{1}}\\ldots\\sup_{\\mathcal{D}_{t}}\\mathbb{E}_{X_{t},Z_{t}\\sim\\Pi_{t}}\\Bigg{[} \\mathds{1}[E_{t}]\\left(\\inf_{a_{t}}\\sup_{y_{t}}\\ldots\\sup_{\\mathcal{D}_{T}} \\mathbb{E}_{X_{T}\\sim\\mathcal{D}_{T}}\\inf_{a_{T}}\\sup_{y_{T}}\\mathcal{R}( \\mathcal{F},X_{1:T},y_{1:T},a_{1:T})\\right)\\Bigg{]}\\] \\[\\quad+T\\left(1-\\sigma\\right)^{k}\\] \\[\\leq\\mathbb{E}_{Z_{1}\\sim\\mu^{k}}\\max_{X_{1}\\in Z_{1}}\\inf_{a_{1} }\\sup_{y_{1}}\\ldots\\sup_{\\mathcal{D}_{t}}\\mathbb{E}_{Z_{t}\\sim\\Pi_{t}}\\max_{X_ {t}\\in Z_{t}}\\left(\\inf_{a_{t}}\\sup_{y_{t}}\\ldots\\sup_{\\mathcal{D}_{T}}\\mathbb{ E}_{X_{T}\\sim\\mathcal{D}_{T}}\\inf_{a_{T}}\\sup_{y_{T}}\\mathcal{R}(\\mathcal{F},X_{1:T },y_{1:T},a_{1:T})\\right)\\] \\[\\quad\\quad+T\\left(1-\\sigma\\right)^{k}\\] \\[=\\mathbb{E}_{Z_{1}\\sim\\mu^{k}}\\max_{X_{1}\\in Z_{1}}\\inf_{a_{1}} \\sup_{y_{1}}\\ldots\\mathbb{E}_{Z_{t}\\sim\\mu^{k}}\\max_{X_{t}\\in Z_{t}}\\left(\\inf_ {a_{t}}\\sup_{y_{t}}\\ldots\\sup_{\\mathcal{D}_{T}}\\mathbb{E}_{X_{T}\\sim\\mathcal{D} _{T}}\\inf_{a_{T}}\\sup_{y_{T}}\\mathcal{R}(\\mathcal{F},X_{1:T},y_{1:T},a_{1:T}) \\right)\\] \\[\\quad\\quad+T\\left(1-\\sigma\\right)^{k}\\]\n\nCombining with the induction hypothesis, gives us the induction hypothesis for the next \\(t\\) as required. The desired result follows by upper bounding the average with the supremum over all subsets of size \\(kT\\).\n\n## Appendix C Proof of Theorem 3.2\n\nFirst, recall the notion of the global sequential covering for a class Wu et al. (2022).\n\n**Definition C.1** (Global Sequential Covering Wu et al. (2022)).: For any class \\(\\mathcal{F}\\), we say that \\(\\mathcal{F}^{\\prime}_{\\alpha}\\subset\\mathcal{X}^{*}\\rightarrow[0,1]\\) is a global sequential \\(\\alpha\\)-covering of \\(\\mathcal{F}\\) at scale \\(T\\) if for any sequence \\(x_{1:T}\\) and \\(h\\in\\mathcal{F}\\), there is a \\(h^{\\prime}\\in\\mathcal{F}^{\\prime}\\) such that for all \\(i\\),\n\n\\[\\big{|}h(x_{i})-h^{\\prime}(x_{1:i})\\big{|}\\leq\\alpha.\\]\n\n**Theorem C.1** (Wu et al. (2022)).: _If \\(\\mathcal{F}^{\\prime}_{\\alpha}\\) is a global sequential \\(\\alpha\\)-covering of \\(\\mathcal{F}\\) at scale \\(T\\), then_\n\n\\[\\mathcal{R}_{T}\\left(\\mathcal{F}\\right)\\leq\\inf_{\\alpha>0}\\left\\{2\\alpha T+ \\log\\!\\big{|}\\mathcal{F}^{\\prime}_{\\alpha}\\big{|}\\right\\}.\\]\n\nTo finish the proof note that a \\(\\epsilon\\)-cover in the sense of Definition 3.1 gives a global sequential cover in the sense of Definition C.1.\n\n## Appendix D VC Classes\n\nIn this section, we construct a probability assignment for the case when \\(\\mathcal{F}\\subset\\{\\mathcal{X}\\rightarrow[0,1]\\}\\) is a _VC class_. To motivate this probability assignment, consider the no-context case, which is a classic problem in information theory, where the (asymptotically) optimal probability assignment is known to be the Krichevsky and Trofimov (1981) (KT) probability assignment which is a Bayesian mixture of the form\n\n\\[q_{\\mathrm{KT}}(y_{1:T})=\\int_{0}^{1}p_{\\theta}(y_{1:T})w(\\theta)d\\theta\\]\n\nfor a particular prior \\(w(\\theta)\\). This can be written sequentially as \\(q_{\\mathrm{KT}}(1|y_{1:t-1})=\\frac{\\sum_{i=1}^{t-1}y_{i}+1/2}{t-1+1}\\) leading to it sometimes being called the add-1/2 probability assignment; by choosing \\(w(\\theta)\\) to be \\(\\mathrm{Beta}(\\beta,\\beta)\\) prior one can achieve a corresponding add-\\(\\beta\\) probability assignment. We extend the mixture idea to the contextual case. In particular, for functions \\(f_{1},\\ldots,f_{m}\\in\\mathcal{F}\\), one can choose a mixture probability assignment as 5\n\nFootnote 5: Note that once a mixture \\(q(y_{1:t}\\|x_{1:t})\\) has been defined for arbitrary \\(x_{1:t},y_{1:t}\\), the probability assignment at time \\(t\\) (or equivalently, the predicted probability with which the upcoming bit is \\(1\\)) can be defined as \\(q(1|x_{1:t},y_{1:t-1})=\\frac{q(y_{1:t-1}\\|x_{1:t})}{q(y_{1:t-1}\\|x_{1:t-1}\\|x_{1: t-1})}\\); in particular, this prediction depends only on the observed history \\(x_{1:t},y_{1:t-1}\\) and not the future \\(y_{t}\\).\n\n\\[\\prod_{i=1}^{t}q(y_{i}|x_{1:i},y_{1:i-1})=:q(y_{1:t}\\|x_{1:t})=\\frac{1}{m}\\sum_{ j=1}^{m}\\prod_{i=1}^{t}\\left(\\frac{p_{f_{j}}(y_{i}|x_{i})+\\beta}{1+2\\beta} \\right).\\]This is the approach employed presently with a carefully chosen \\(f_{1},\\ldots,f_{m}\\). We remark that for VC classes this mixture approach may be extended to any _mixable_(Cesa-Bianchi and Lugosi, 2006, Chapter 3) loss.\n\nFirst consider VC classes more carefully: i.e. each \\(f\\in\\mathcal{F}^{\\mathrm{VC}}\\) is characterized by three things: a set \\(A\\subseteq\\mathcal{X}\\), where \\(A\\in\\mathcal{A}\\subset 2^{\\mathcal{X}}\\) with the VC dimension of the collection \\(\\mathcal{A}\\) being \\(d<\\infty\\); as well as two numbers \\(\\theta_{0},\\theta_{1}\\in[0,1]\\). Then, we have\n\n\\[f_{A,\\theta_{0},\\theta_{1}}(x)=\\theta_{0}\\mathds{1}\\{x\\in A\\}+\\theta_{1} \\mathds{1}\\{x\\in A^{C}\\}.\\]\n\nThe following equivalent representation of this hypothesis class is more convenient to use. We consider each \\(f\\) to be characterized by a tuple \\(f=(g,\\theta_{0},\\theta_{1})\\) where\n\n1. \\(\\theta_{0},\\theta_{1}\\in[0,1]\\)\n2. \\(g\\in\\mathcal{G}\\subset\\{\\mathcal{X}\\rightarrow\\{0,1\\}\\}\\).\n\nIn other words, \\(g\\) belongs to a class \\(\\mathcal{G}\\) of binary functions--this is simply the class of functions \\(\\{x\\mapsto\\mathds{1}\\{x\\notin A\\}|A\\in\\mathcal{A}\\}\\) in the original notation; so that clearly \\(\\mathrm{VCdim}(\\mathcal{G})=d\\). Then, we have \\(p_{f}(\\cdot|x)=p_{g,\\theta_{0},\\theta_{1}}(\\cdot|x)=\\mathrm{Bernoulli}(\\theta _{0})\\) if \\(g(x)=0\\); and \\(p_{g,\\theta_{0},\\theta_{1}}(\\cdot|x)=\\mathrm{Bernoulli}(\\theta_{1})\\) otherwise.\n\nRecalling the definition of regret against a particular \\(f=(g,\\theta_{0},\\theta_{1})\\) for a sequential probability assignment strategy \\(\\mathscr{Q}=\\{q(\\cdot|x_{1:t},y_{1:t-1})\\}_{t=1}^{T}\\)\n\n\\[\\mathcal{R}_{T}(f,x_{1:T},y_{1:T},\\mathscr{Q}) =\\sum_{t=1}^{T}\\log\\frac{1}{q(y_{t}|x_{1:t},y_{1:t-1})}-\\sum_{t=1 }^{T}\\log\\frac{1}{p_{f}(y_{t}|x_{t})} \\tag{6}\\] \\[=\\log\\frac{p_{f}(y_{1:T}|x_{1:T})}{q(y_{1:T}\\|x_{1:T})}\\]\n\nwhere \\(q(y_{1:T}\\|x_{1:T}):=\\prod_{t=1}^{T}q(y_{t}|x_{1:t},y_{1:t-1})\\).\n\nIn the smoothed analysis case, we have \\(X_{t}\\sim\\mathcal{D}_{t}\\) where \\(\\mathcal{D}_{t}\\) for all \\(t\\) is \\(\\sigma\\)-smoothed. Recall that in this case, we are concerned with the regret\n\n\\[\\mathcal{R}_{T}(\\mathcal{F},\\sigma,\\mathscr{Q}) =\\max_{\\mathscr{D}:\\sigma\\text{-smoothed}}\\mathbb{E}_{X_{1:T}} \\left[\\max_{y_{1:T}}\\sup_{f\\in\\mathcal{F}}\\frac{p_{f}(y_{1:T}|X_{1:T})}{q(y_{ 1:T}\\|X_{1:T})}\\right]\\] \\[=\\max_{\\mathscr{D}:\\sigma\\text{-smoothed}}\\mathbb{E}_{X_{1:T}} \\left[\\max_{y_{1:T}}\\sup_{g\\in\\mathcal{G}}\\max_{\\theta_{0},\\theta_{1}}\\frac{p_ {g,\\theta_{0},\\theta_{1}}(y_{1:T}|X_{1:T})}{q(y_{1:T}\\|X_{1:T})}\\right].\\]\n\n### Proposed probability assignment\n\nLet \\(\\mu\\) be the dominating measure for the \\(\\sigma\\)-smoothed distribution of \\(X_{1:T}\\). Let \\(g_{1},\\ldots,g_{m_{\\epsilon}}\\in\\mathcal{G}\\) be an \\(\\epsilon\\)-cover of the function class \\(\\mathcal{G}\\) under the metric \\(\\delta_{\\mu}(g_{1},g_{2})=\\Pr_{X\\sim\\mu}(g_{1}(X)\\neq g_{2}(X))\\). The following lemma bounds \\(m_{\\epsilon}\\).\n\n**Lemma D.1** (Covering number of VC classes under the metric \\(\\delta\\), Vershynin (2018)).: \\[m_{\\epsilon}\\leq\\left(\\frac{1}{\\epsilon}\\right)^{cd}\\]\n\n_for an absolute constant \\(c\\)._\n\nFollowing the idea of using a mixture probability assignment, we take a uniform mixture over \\(g_{1},\\ldots,g_{m_{\\epsilon}}\\) and \\(\\theta_{0},\\theta_{1}\\) so that\n\n\\[q(y_{1:t}\\|x_{1:t})=\\frac{1}{m_{\\epsilon}}\\sum_{i=1}^{m_{\\epsilon}}\\int_{0}^{ 1}\\int_{0}^{1}p_{g_{i},\\theta_{0},\\theta_{1}}(y_{1:t}|x_{1:t})d\\theta_{0}d \\theta_{1}\\]\n\nand consequently the sequential probability assignment (or equivalently, the probability assigned to 1) is\n\n\\[q(1|x_{1:t},y_{1:t-1})=\\frac{q(y_{1:t-1}\\mathds{1}\\|x_{1:t})}{q(y_{1:t-1}\\|x_{ 1:t-1})}.\\]\n\nOne can observe that \\(q(0|x_{1:t},y_{1:t-1}),q(1|x_{1:t},y_{1:t-1})>0\\) and \\(q(0|x_{1:t},y_{1:t-1})+q(1|x_{1:t},y_{1:t-1})=1\\) so that \\(q\\) is a legitimate probability assignment. Let the strategy induced by this uniform mixture be called \\(\\mathscr{Q}^{\\mathrm{VC}}\\).\n\n### Analysis of \\(\\mathcal{Q}^{\\mathrm{VC}}\\) for smoothed adversaries\n\nWe note from (6) that for the \\(\\mathcal{Q}^{\\mathrm{VC}}\\) as defined in the last section, we have\n\n\\[\\mathcal{R}_{T}((g^{*},\\theta_{0}^{*},\\theta_{1}^{*}),x_{1:T},y_{1: T},\\mathcal{Q}^{\\mathrm{VC}}) =\\log m_{\\epsilon}+\\log\\frac{p_{g^{*},\\theta_{0}^{*},\\theta_{1}^{ *}}(y_{1:T}|x_{1:T})}{\\sum_{i=1}^{m_{*}}\\int_{0}^{1}\\int_{0}^{1}p_{g_{i},\\theta _{0},\\theta_{1}}(y_{1:T}|x_{1:T})d\\theta_{0}d\\theta_{1}}\\] \\[\\leq\\log m_{\\epsilon}+\\log\\frac{p_{g^{*},\\theta_{0}^{*},\\theta_{1} ^{*}}(y_{1:T}|x_{1:T})}{\\int_{0}^{1}\\int_{0}^{1}p_{g_{i},\\cdot\\theta_{0},\\theta _{1}}(y_{1:T}|x_{1:T})d\\theta_{0}d\\theta_{1}} \\tag{7}\\]\n\nwhere \\(g_{i^{*}}\\in\\{g_{1},\\ldots,g_{m_{*}}\\}\\) is the function \\(i^{*}\\in[m]\\) that minimizes the Hamming distance between the binary strings \\((g_{i^{*}}(x_{1}),\\ldots,g_{i^{*}}(x_{T}))\\) and \\((g^{*}(x_{1}),\\ldots,g^{*}(x_{T}))\\).\n\nWe now take a closer look at the second term of (7). Firstly, note that for any \\((g,\\theta_{0},\\theta_{1})\\) we have \\(p_{g,\\theta_{0},\\theta_{1}}(y_{1:T}|x_{1:T})=\\)\n\n\\[\\prod_{t=1}^{T}p_{g,\\theta_{0},\\theta_{1}}(y_{t}|x_{t}) =\\prod_{t=1}^{T}\\theta_{g(x_{t})}^{y_{t}}(1-\\theta_{g(x_{t})})^{1 -y_{t}}=\\prod_{t:g(x_{t})=0}\\theta_{0}^{y_{t}}(1-\\theta_{0})^{1-y_{t}}\\prod_{t: g(x_{t})=1}\\theta_{1}^{y_{t}}(1-\\theta_{1})^{1-y_{t}}\\] \\[=\\theta_{0}^{k_{0}(g;x_{1:T},y_{1:T})}(1-\\theta_{0})^{n_{0}(g;x_{ 1:T})-k_{0}(g;x_{1:T},y_{1:T})}\\] \\[\\theta_{1}^{k_{1}(g;x_{1:T},y_{1:T})}(1-\\theta_{1})^{n_{1}(g;x_{1 :T})-k_{1}(g;x_{1:T},y_{1:T})},\\]\n\nwhere for \\(j\\in\\{0,1\\}\\)\n\n\\[k_{j}(g;x_{1:T},y_{1:T}) =|\\{t:y_{t}=1,g(x_{t})=j\\}|\\] \\[n_{0}(g;x_{1:T}) =|\\{t:g(x_{t})=j\\}|.\\]\n\nNext, we note that for any \\(g\\in\\mathcal{G}\\)\n\n\\[\\int_{0}^{1}\\int_{0}^{1}p_{g,\\theta_{0},\\theta_{1}}(y_{1:T}|x_{1 :T})d\\theta_{0}d\\theta_{1}\\] \\[=\\left(\\int_{0}^{1}\\theta_{0}^{k_{0}(g;x_{1:T},y_{1:T})}(1- \\theta_{0})^{n_{0}(g;x_{1:T})-k_{0}(g;x_{1:T},y_{1:T})}d\\theta_{0}\\right).\\] \\[\\qquad\\qquad\\left(\\int_{0}^{1}\\theta_{1}^{k_{1}(g;x_{1:T},y_{1:T} )}(1-\\theta_{1})^{n_{1}(g;x_{1:T})-k_{0}(g;x_{1:T},y_{1:T})}d\\theta_{1}\\right)\\] \\[=\\frac{1}{\\binom{n_{0}(g;x_{1:T})}{k_{0}(g;x_{1:T},y_{1:T})}(n_{0 }(g;x_{1:T})+1)}\\frac{1}{\\binom{n_{1}(g;x_{1:T})}{k_{1}(g;x_{1:T},y_{1:T})}(n_ {1}(g;x_{1:T})+1)} \\tag{8}\\] \\[\\geq\\frac{1}{n^{2}\\binom{n_{0}(g;x_{1:T})}{k_{0}(g;x_{1:T},y_{1:T} )}\\binom{n_{1}(g;x_{1:T})}{k_{1}(g;x_{1:T},y_{1:T})}}\\]\n\nwhere (8) follows from properties of the Laplace probability assignment (or that of the Beta/Gamma functions), captured by Lemma D.2.\n\n**Lemma D.2**.: _For \\(k\\leq n\\in\\mathbb{N}\\),_\n\n\\[\\int_{0}^{1}t^{k}(1-t)^{n-k}dt=\\frac{\\Gamma(k+1)\\Gamma(n-k+1)}{ \\Gamma(n+2)}=\\frac{1}{(n+1)\\binom{n}{k}}\\]\n\n_where \\(\\Gamma(\\cdot)\\) represents the Gamma function._\n\nPutting this back into (7) (and rearranging), we have\n\n\\[\\mathcal{R}_{T}((g^{*},\\theta_{0}^{*},\\theta_{1}^{*}),x_{1:T},y_{1: T},\\mathcal{Q}^{\\mathrm{VC}})-\\log m_{\\epsilon}-2\\log n\\] \\[\\leq\\sum_{j\\in\\{0,1\\}}\\log\\left(\\binom{n_{j}(g_{i^{*}};x_{1:T},y_ {1:T})}{k_{j}(g_{i^{*}};x_{1:T},y_{1:T})}(\\theta_{j}^{*})^{k_{j}(g^{*};x_{1:T},y _{1:T})}(1-\\theta_{j}^{*})^{n_{j}(g^{*};x_{1:T})-k_{j}(g^{*};x_{1:T},y_{1:T})}\\right)\\] \\[=\\sum_{j\\in\\{0,1\\}}\\log\\left(\\binom{n_{j}(g_{i^{*}};x_{1:T})}{k_ {j}(g_{i^{*}};x_{1:T},y_{1:T})}\\binom{n_{j}(g^{*};x_{1:T})}{k_{j}(g^{*};x_{1:T}, y_{1:T})}\\right)^{-1}.\\]\\[\\binom{n_{j}(g^{*};x_{1:T})}{k_{j}(g^{*};x_{1:T},y_{1:T})}\\binom{g^{*} }{k_{j}(g^{*};x_{1:T},y_{1:T})}(1-\\theta^{*}_{j})^{n_{j}(g^{*};x_{1:T})-k_{j}(g ^{*};x_{1:T},y_{1:T})}\\] \\[\\leq\\sum_{j\\in\\{0,1\\}}\\log\\left(\\binom{n_{j}(g_{i^{*}};x_{1:T})}{k _{j}(g_{i^{*}};x_{1:T},y_{1:T})}\\binom{n_{j}(g^{*};x_{1:T})}{k_{j}(g^{*};x_{1: T},y_{1:T})}\\right)^{-1} \\tag{9}\\]\n\nwhere (9) follows since for any natural numbers \\(k\\leq n\\) and \\(\\theta\\in[0,1]\\) we have \\(\\binom{n}{k}\\theta^{k}(1-\\theta)^{n-k}\\leq 1\\). Now, note that\n\n\\[\\log\\frac{\\binom{n}{k}}{\\binom{n^{\\prime}}{k^{\\prime}}} =\\log\\frac{n!}{n^{\\prime}!}+\\log\\frac{k^{\\prime}!}{k!}+\\log\\frac {(n^{\\prime}-k^{\\prime})!}{(n-k)!}\\] \\[\\leq\\log\\frac{(n^{\\prime}+|n-n^{\\prime}|)!}{n^{\\prime}!}+\\log \\frac{(k+|k-k^{\\prime}|)!}{k!}+\\log\\frac{((n-k)+|n-n^{\\prime}|+|k-k^{\\prime}|)! }{(n-k)!}\\]\n\nIf \\(|k-k^{\\prime}|,|n-n^{\\prime}|\\leq\\delta\\), and \\(\\max\\{n,n^{\\prime}\\}\\leq N\\) then by for example [1, Proposition 6] we have that\n\n\\[\\log\\frac{\\binom{n}{k}}{\\binom{n^{\\prime}}{k^{\\prime}}} \\leq 2\\delta\\log(n^{\\prime}+2\\delta)+2\\delta\\log(k+2\\delta)+4 \\delta\\log((n-k)+4\\delta)\\] \\[\\leq 16\\delta\\log N. \\tag{10}\\]\n\nWe now wish to use this bound in (9). For this, we will recall the definitions of \\(n_{0}(g;x_{1:T})\\) and \\(k_{0}(g;x_{1:T},y_{1:T})\\) for a particular function \\(g\\) and observe that for two functions \\(g,g^{\\prime}\\) we have that both \\(|n_{0}(g;x_{1:T})-n_{0}(g^{\\prime};x_{1:T})|,|k_{0}(g;x_{1:T},y_{1:T})-k_{0}(g ^{\\prime};x_{1:T},y_{1:T})|\\leq d_{H}(g(x_{1:T}),g^{\\prime}(x_{1:T}))\\) where \\(d_{H}(\\cdot,\\cdot)\\) denotes the Hamming distance and \\(g(x_{1:T}):=(g(x_{1}),\\ldots,g(x_{T}))\\in\\{0,1\\}^{T}\\). Thus, by using (10) in (9) with \\(\\delta=d_{H}(g^{*}(x_{1:T}),g_{i^{*}}(x_{1:T})),N=T\\), we get\n\n\\[\\mathcal{R}_{T}((g^{*},\\theta^{*}_{0},\\theta^{*}_{1}),x_{1:T},y_{1:T},\\mathcal{ Q}^{\\text{VC}})\\leq\\log m_{\\epsilon}+2\\log T+32d_{H}(g^{*}(x_{1:T}),g_{i^{*}}(x_ {1:T}))\\log T. \\tag{11}\\]\n\nNote that (11) has effectively removed any dependence on \\(y,\\theta^{*}_{0},\\theta^{*}_{1}\\). We then have for some absolute constant \\(C\\), (recalling the definition of \\(i^{*}\\) and \\(\\mathcal{F}\\) from earlier)\n\n\\[\\mathcal{R}_{T}(\\mathcal{F},\\sigma,\\mathcal{Q}^{\\text{VC}})\\leq C\\log m_{ \\epsilon}+C\\log T\\max_{\\mathcal{D}:\\sigma\\text{-smoothed}}\\mathbb{E}\\left[ \\sup_{g^{*}\\in\\mathcal{G}}\\min_{i\\in[m_{\\epsilon}]}d_{H}(g^{*}(X_{1:T}),g_{i}(X _{1:T}))\\right]. \\tag{12}\\]\n\nFinally, we can control the last term in (12) by the following result, which follows from the coupling lemma and variance sensitive upper bounds on suprema over VC classes.\n\n**Lemma D.3** (Lemma 3.3 of [1]).: \\[\\mathbb{E}\\left[\\sup_{g^{*}\\in\\mathcal{G}}\\min_{i\\in[m]}d_{H}(g^{*}(X_{1:T}), g_{i}(X_{1:T}))\\right]\\leq\\sqrt{\\frac{\\epsilon}{\\sigma}T\\log Td\\log\\left( \\frac{1}{\\epsilon}\\right)}+T\\log T\\frac{\\epsilon}{\\sigma}\\]\n\nPlugging the above into (12) and taking \\(\\epsilon=\\frac{\\sigma}{T^{2}}\\) gives us\n\n\\[\\mathcal{R}_{T}(\\mathcal{F},\\sigma,\\mathcal{Q}^{\\text{VC}})\\leq O\\left(d\\log \\left(\\frac{T}{\\sigma}\\right)\\right).\\]\n\n## Appendix E Proof of Lemma 4.2\n\nProof.: Note that this proof holds for general loss functions. Let \\(\\mathcal{R}_{T}\\) denote the regret.\n\n\\[\\mathcal{R}_{T} \\leq\\mathbb{E}\\left[\\sum_{i=1}^{T}\\ell(h_{t},s_{t})-\\inf_{h\\in \\mathcal{F}}\\sum_{i=1}^{T}\\ell(h,s_{t})\\right]\\] \\[=\\mathbb{E}\\left[\\sum_{i=1}^{T}\\ell(h_{t},s_{t})-\\sum_{t=1}^{T} \\ell(h_{t+1},s_{t})+\\sum_{t=1}^{T}\\ell(h_{t+1},s_{t})-\\inf_{h\\in\\mathcal{F}} \\sum_{i=1}^{T}\\ell(h,s_{t})\\right]\\]\\[=\\mathbb{E}\\left[\\sum_{i=1}^{T}\\ell(h_{t},s_{t})-\\sum_{t=1}^{T}\\ell(h_{t+1},s_{t}) \\right]+\\mathbb{E}\\left[\\sum_{t=1}^{T}\\ell(h_{t+1},s_{t})-\\inf_{h\\in\\mathcal{F}} \\sum_{i=1}^{T}\\ell(h,s_{t})\\right]\\]\n\nLet us focus on the second term.\n\n\\[\\mathbb{E}\\left[\\sum_{t=1}^{T}\\ell(h_{t+1},s_{t})-\\inf_{h\\in \\mathcal{F}}\\sum_{i=1}^{T}\\ell(h,s_{t})\\right]\\] \\[\\leq\\mathbb{E}\\left[\\sum_{t=1}^{T}\\ell(h_{t+1},s_{t})-\\inf_{h\\in \\mathcal{F}_{\\alpha}}\\sum_{i=1}^{T}\\ell(h,s_{t})+\\inf_{h\\in\\mathcal{F}_{\\alpha} }\\sum_{i=1}^{T}\\ell(h,s_{t})-\\inf_{h\\in\\mathcal{F}}\\sum_{i=1}^{T}\\ell(h,s_{t})\\right]\\] \\[\\leq 2\\alpha T+\\mathbb{E}\\left[\\sum_{t=1}^{T}\\ell(h_{t+1},s_{t})- \\inf_{h\\in\\mathcal{F}_{\\alpha}}\\sum_{i=1}^{T}\\ell(h,s_{t})\\right] \\tag{13}\\] \\[\\leq 2\\alpha T+\\mathbb{E}\\left[\\sum_{t=1}^{N}\\ell(h_{t},\\tilde{s} _{t})-\\ell\\left(h^{*},\\tilde{s}_{t}\\right)\\right]\\] (14) \\[\\leq 2\\alpha T+\\mathbb{E}\\left[\\sup_{h\\in\\mathcal{F}_{\\alpha}} \\sum_{t=1}^{N}\\ell(h,\\tilde{s}_{t})-\\ell\\left(h^{*},\\tilde{s}_{t}\\right)\\right]\\]\n\nwhere \\(h^{*}=\\inf_{h\\in\\mathcal{F}_{\\alpha}}\\sum_{i=1}^{T}\\ell(h,s_{t})\\). (13) follows by comparing the optimal of the truncated class with the whole class, see [Cesa-Bianchi and Lugosi, 2006, Lemma 9.5]. (14) follows from the Be-the-leader lemma Cesa-Bianchi and Lugosi (2006). \n\n## Appendix F Proof of Lemma 4.3\n\nDenote by \\(R^{(t)}=(N^{(t)},\\{\\widetilde{s}_{i}\\}_{i\\in N^{(t)}})\\) the fresh randomness generated at the beginning of time \\(t\\), which is independent of \\(\\{s_{\\tau}\\}_{\\tau<t}\\) generated by the adversary. Let \\(\\mathcal{Q}_{t}\\) be the distribution of the learner's action \\(h_{t}\\in\\mathcal{H}\\) in Algorithm 1, Formally,\n\n\\[r^{t}(x)=\\sum_{i=1}^{N^{(t+1)}}\\widetilde{y}_{i}^{\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\, \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\, \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\, \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\, \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\, \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\, \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\, \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\, \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\, \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\, \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\, \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\, \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\, \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\, \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\, \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\, \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\, \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\, \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\, \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\, \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\, \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\, \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\, \\,\\Here we use the notation \\(\\widetilde{\\mathcal{Q}}_{t+1}=\\mathbb{E}_{s_{t}\\sim\\mathcal{D}_{t}}[\\mathcal{Q}_{t+1}]\\) for the mixture distribution. In order to bound this, we look a variational interpretation of the \\(\\chi^{2}\\) distance between two distributions \\(P\\) and \\(Q\\).\n\n**Lemma F.1** (Hammersley-Chapman-Robbins bound).: _For any pair of measures \\(P\\) and \\(Q\\) and any measurable function \\(h:\\mathcal{X}\\rightarrow\\mathbb{R}\\), we have_\n\n\\[\\big{|}\\mathbb{E}_{X\\sim P}[h(X)]-\\mathbb{E}_{X\\sim Q}[h(X)]\\big{|} \\leq\\sqrt{\\chi^{2}\\left(P,Q\\right)\\cdot\\mathrm{Var}_{X\\sim Q} \\left(h\\left(X\\right)\\right)}\\] \\[\\leq\\sqrt{\\frac{1}{2}\\chi^{2}\\left(P,Q\\right)\\cdot\\mathbb{E}_{X,X ^{\\prime}\\sim Q}\\left(h\\left(X\\right)-h\\left(X^{\\prime}\\right)\\right)^{2}}.\\]\n\nApplying this to (17), we get\n\n\\[\\underset{h_{t}\\sim\\mathcal{Q}_{t}}{\\mathbb{E}}\\left[\\underset{s_ {t}^{\\prime}\\sim\\mathcal{D}_{t}}{\\mathbb{E}}[\\ell(h_{t},s_{t}^{\\prime})]\\right] -\\underset{h_{t+1}\\sim\\widetilde{\\mathcal{Q}}_{t+1}}{\\mathbb{E}}\\left[ \\underset{s_{t}^{\\prime}\\sim\\mathcal{D}_{t}}{\\mathbb{E}}[\\ell(h_{t+1},s_{t}^ {\\prime})]\\right]\\] \\[\\leq\\sqrt{\\frac{1}{2}\\chi^{2}(\\underset{s_{t}\\sim\\mathcal{D}_{t}}{ \\mathbb{E}}[\\mathcal{Q}_{t+1}],\\mathcal{Q}_{t})\\cdot\\mathbb{E}_{h_{t},h_{t}^{ \\prime}\\sim\\mathcal{Q}_{t}}\\left(\\mathbb{E}_{s_{t}\\sim\\mathcal{D}_{t}}\\left[ \\ell(h_{t},s_{t})-\\ell(h_{t}^{\\prime},s_{t})\\right]\\right)^{2}}.\\]\n\nas required. As noted before, for the particular use in our analysis a simpler version of the lemma similar to Haghtalab et al. (2022) suffices but we include the general version since we believe such a version is useful in providing improved regret bounds for the problem.\n\n## Appendix G Upper Bounding \\(\\chi^{2}\\) Distance: Proof of Lemma 4.4\n\nIn this section, we will focus on bounding the \\(\\chi^{2}\\) distance between the distribution of actions at time steps. The reasoning in this section closely follows Haghtalab et al. (2022). We reproduce it here for completeness.\n\nWe assume that \\(\\mathcal{X}\\) is discrete. Define\n\n\\[n_{0}(x)=\\sum_{i=1}^{N}\\mathbf{1}(\\widetilde{x}_{i}=x,\\widetilde{y}_{i}=0)\\ \\ \\text{and}\\ \\ n_{1}(x)=\\sum_{i=1}^{N}\\mathbf{1}(\\widetilde{x}_{i}=x,\\widetilde{y}_{i}=1).\\]\n\nAs each \\(\\widetilde{x}_{i}\\) is uniformly distributed on \\(\\mathcal{X}\\) and \\(\\widetilde{y}_{i}\\sim\\mathcal{U}(\\{0,1\\})\\), by the subsampling property of the Poisson distribution, the \\(2|\\mathcal{X}|\\) random variables \\(\\{n_{0}(x),n_{1}(x)\\}_{x\\in\\mathcal{X}}\\) are i.i.d. distributed as \\(\\text{Poi}(n/2|\\mathcal{X}|)\\).\n\nSince the historic data is only a translation, it suffices to consider the distributions at time \\(t=0\\) and \\(t=1\\). Let \\(n_{0}^{1}(x)=n_{0}(x)+\\mathbf{1}(x_{1}=x,y_{1}=0)\\) with \\(n_{1}^{1}\\) defined similarly. Let \\(P\\) and \\(Q\\) be the probability distributions of \\(\\{n_{0}(x),n_{1}(x)\\}_{x\\in\\mathcal{X}}\\) and \\(\\{n_{0}^{1}(x),n_{1}^{1}(x)\\}_{x\\in\\mathcal{X}}\\), respectively. Note that the output of the oracle depends only on this vector and thus by the data processing inequality it suffices to bound \\(\\chi^{2}(P,Q)\\).\n\nNote that the distribution \\(P\\) is a product Poisson distribution:\n\n\\[P(\\{n_{0}(x),n_{1}(x)\\})=\\prod_{x\\in\\mathcal{X}}\\prod_{y\\in\\{0,1\\}}\\mathbb{P}( \\text{Poi}(n/2|\\mathcal{X}|)=n_{y}(x)).\\]\n\nAs for the distribution \\(Q\\), it could be obtained from \\(P\\) in the following way: the smooth adversary draws \\(x^{*}\\sim\\mathcal{D}\\), independent of \\(\\{n_{0}(x),n_{1}(x)\\}_{x\\in\\mathcal{X}}\\sim P\\), for some \\(\\sigma\\)-smooth distribution \\(\\mathcal{D}\\in\\Delta_{\\sigma}(\\mathcal{X})\\). He then chooses a label \\(y^{*}=y(x^{*})\\in\\{0,1\\}\\) as a function of \\(x^{*}\\), and sets\n\n\\[n_{y(x^{*})}^{1}(x^{*})=n_{y(x^{*})}(x^{*})+1,\\qquad\\text{and}\\qquad n_{y}^{1} (x)=n_{y}(x),\\quad\\forall(x,y)\\neq(x^{*},y(x^{*})).\\]\n\nConsequently, given a \\(\\sigma\\)-smooth distribution \\(\\mathcal{D}\\) and a labeling function \\(y:\\mathcal{X}\\rightarrow\\{0,1\\}\\) used by the adversary, the distribution \\(Q\\) is a mixture distribution \\(Q=\\mathbb{E}_{x^{*}\\sim\\mathcal{D}^{\\mathcal{X}}}[Q_{x^{*}}]\\), with\n\n\\[Q_{x^{*}}(\\{n_{0}^{1}(x),n_{1}^{1}(x)\\})=\\mathbb{P}(\\text{Poi}(n/2|\\mathcal{X }|)=n_{y(x^{*})}(x^{*})-1)\\times\\prod_{(x,y)\\neq(x^{*},y(x^{*}))}\\mathbb{P}( \\text{Poi}(n/2|\\mathcal{X}|)=n_{y}(x)).\\]\n\nWe will use the Ingster method to control the \\(\\chi^{2}\\) between the mixture distribution \\(Q\\) and the base distribution \\(P\\).\n\n**Lemma G.1** (Ingster's \\(\\chi^{2}\\) method).: _For a mixture distribution \\(\\mathbb{E}_{\\theta\\sim\\pi}[Q_{\\theta}]\\) and a generic distribution \\(P\\), the following identity holds:_\n\n\\[\\chi^{2}\\left(\\operatorname*{\\mathbb{E}}_{\\theta\\sim\\pi}[Q_{\\theta}],P\\right)= \\operatorname*{\\mathbb{E}}_{\\theta,\\theta^{\\prime}\\sim\\pi}\\left[\\operatorname*{ \\mathbb{E}}_{z\\sim P}\\left(\\frac{Q_{\\theta}(x)Q_{\\theta^{\\prime}}(x)}{P(x)^{2} }\\right)\\right]-1,\\]\n\n_where \\(\\theta^{\\prime}\\) is an independent copy of \\(\\theta\\)._\n\nLet \\(x_{1}^{\\star},x_{2}^{\\star}\\) be an arbitrary pair of instance. Using the closed-form expressions of distributions \\(P\\) and \\(Q_{x^{\\star}}\\), it holds that\n\n\\[\\frac{Q_{x_{1}^{\\star}}Q_{x_{2}^{\\star}}}{P^{2}}=\\frac{2|\\mathcal{X}|n_{y(x_{1 }^{\\star})}(x_{1}^{\\star})}{n}\\cdot\\frac{2|\\mathcal{X}|n_{y(x_{2}^{\\star})}(x_ {2}^{\\star})}{n}.\\]\n\nUsing the fact that \\(\\{n_{0}(x),n_{1}(x)\\}_{x\\in\\mathcal{X}}\\) are i.i.d. distributed as \\(\\text{Poi}(n/2|\\mathcal{X}|)\\) under \\(P\\), we have\n\n\\[\\operatorname*{\\mathbb{E}}_{\\{n_{0}(x),n_{1}(x)\\}\\sim P}\\left(\\frac{Q_{x_{1}^{ \\star}}(\\{n_{0}^{1}(x),n_{1}^{1}(x)\\})}{P(\\{n_{0}(x),n_{1}(x)\\})^{2}}\\right)=1 +\\frac{2|\\mathcal{X}|}{n}\\cdot\\mathbf{1}(x_{1}^{\\star}=x_{2}^{\\star}).\\]\n\nWe will use the fact that the probability of collision between two independent draws \\(x_{1}^{\\star},x_{2}^{\\star}\\sim\\mathcal{D}\\) is small. That is using the Lemma G.1, we have\n\n\\[\\sqrt{\\frac{\\chi^{2}(Q,P)}{2}}=\\sqrt{\\frac{\\chi^{2}(\\mathbb{E}_{x ^{\\star}\\sim\\mathcal{D}}[Q_{x^{\\star}}],P)}{2}}=\\sqrt{\\frac{|\\mathcal{X}|}{n} \\cdot\\operatorname*{\\mathbb{E}}_{x_{1}^{\\star},x_{2}^{\\star}\\sim\\mathcal{D}}[ \\mathbf{1}(x_{1}^{\\star}=x_{2}^{\\star})]}\\] \\[=\\sqrt{\\frac{|\\mathcal{X}|}{n}\\sum_{x\\in\\mathcal{X}}\\mathcal{D}(x )^{2}}{\\leq}\\sqrt{\\frac{|\\mathcal{X}|}{n}\\sum_{x\\in\\mathcal{X}}\\mathcal{D}(x) \\cdot\\frac{1}{\\sigma|\\mathcal{X}|}}=\\frac{1}{\\sqrt{\\sigma n}},\\]\n\nwhere the last inequality follows from the definition of a \\(\\sigma\\)-smooth distribution.\n\n## Appendix H Upper Bounding Generalization Error: Proof of Lemma 4.5\n\nThe proof of the theorem is similar to the [14, Section 4.2.2]. In our setting, we need to deal with general losses. We shall need the following property of smooth distributions which is a slightly strengthened version of the coupling lemma in Theorem 2.1 shown in [14].\n\n**Lemma H.1**.: _Let \\(X_{1},\\cdots,X_{m}\\sim Q\\) and \\(P\\) be another distribution with a bounded likelihood ratio: \\(dP/dQ\\leq 1/\\sigma\\). Then using external randomness \\(R\\), there exists an index \\(I=I(X_{1},\\cdots,X_{m},R)\\in[m]\\) and a success event \\(E=E(X_{1},\\cdots,X_{m},R)\\) such that \\(\\Pr[E^{c}]\\leq(1-\\sigma)^{m}\\), and_\n\n\\[(X_{I}\\mid E,X_{\\setminus I})\\sim P.\\]\n\nFix any realization of the Poissonized sample size \\(N\\sim\\text{Poi}(n)\\). Choose \\(m\\) in Lemma H.1. Since for any \\(\\sigma\\)-smooth \\(\\mathcal{D}_{t}\\), it holds that\n\n\\[\\frac{\\mathcal{D}_{t}(s)}{\\mathcal{U}(\\mathcal{X}\\times\\{0,1\\})(s)}=\\frac{ \\mathcal{D}_{t}(x)}{\\mathcal{U}(\\mathcal{X})(x)}\\cdot\\frac{\\mathcal{D}_{t}(y \\mid x)}{\\mathcal{U}(\\{0,1\\})(y)}\\leq\\frac{2}{\\sigma},\\]\n\nthe premise of Lemma H.1 holds with parameter \\(\\sigma/2\\) for \\(P=\\mathcal{D}_{t},Q=\\mathcal{U}(\\mathcal{X}\\times\\{0,1\\})\\). Consequently, dividing the self-generated samples \\(\\widetilde{s}_{1},\\cdots,\\widetilde{s}_{N}\\) into \\(N/m\\) groups each of size \\(m\\), and running the procedure in Lemma H.1, we arrive at \\(N/m\\) independent events \\(E_{1},\\cdots,E_{N/m}\\), each with probability at least \\(1-(1-\\sigma/2)^{m}\\geq 1-T^{-2}\\). Moreover, conditioned on each \\(E_{j}\\), we can pick an element \\(u_{j}\\in\\{\\widetilde{s}_{(j-1)m+1},\\cdots,\\widetilde{s}_{jm}\\}\\) such that\n\n\\[(u_{j}\\mid E_{j},\\{\\widetilde{s}_{(j-1)m+1},\\cdots,\\widetilde{s}_{jm}\\} \\backslash\\{u_{j}\\})\\sim\\mathcal{D}_{t}.\\]\n\nFor notational simplicity we denote the set of unpicked samples \\(\\{\\widetilde{s}_{(j-1)m+1},\\cdots,\\widetilde{s}_{jm}\\}\\backslash\\{u_{j}\\}\\) by \\(v_{j}\\). As a result, thanks to the mutual independence of different groups and \\(s_{t}\\sim\\mathcal{D}_{t}\\) conditioned on \\(s_{1:t-1}\\) (note that we draw fresh randomness at every round), for \\(E\\triangleq\\cap_{j\\in[N/m]}E_{j}\\) we have\n\n\\[(u_{1},\\cdots,u_{N/m},s_{t})\\mid(E,s_{1:t-1},v_{1},\\cdots,v_{N/m})\\stackrel{{ \\text{iid}}}{{\\sim}}\\mathcal{D}_{t}.\\]Let us denote \\(h_{t+1}=O_{t+1}(\\widetilde{s}_{1},\\cdots,\\widetilde{s}_{N},s_{1:t-1},s_{t})\\) the output of the algorithm at time \\(t\\) when \\(\\widetilde{s}_{1},\\cdots,\\widetilde{s}_{N}\\) denotes the hallucinated data points and \\(s_{1:t-1},s_{t}\\) denotes the observed data points. We will use the fact that \\(O_{t+1}\\) is a permutation invariant function. Consequently, for each \\(j\\in[N/m]\\) we have\n\n\\[\\mathop{\\mathbb{E}}_{s_{t}\\sim\\mathcal{D}_{t},R^{(t+1)}}[\\ell(h_{ t+1},s_{t})\\mid E] \\tag{19}\\] \\[=\\mathop{\\mathbb{E}}_{s_{t}\\sim\\mathcal{D}_{t},\\widetilde{s}_{1}, \\cdots,\\widetilde{s}_{N}}\\left[\\ell(O_{t+1}(\\widetilde{s}_{1},\\cdots, \\widetilde{s}_{N},s_{1:t-1},s_{t}),s_{t})\\mid E\\right]\\] \\[=\\mathop{\\mathbb{E}}_{v,s_{1:t-1}\\mid E}\\left(\\mathop{\\mathbb{E}} _{s_{t},u_{1},\\cdots,u_{N/m}}\\left[\\ell(O_{t+1}(s_{1:t-1},v,u_{1},\\cdots,u_{N/ m},s_{t}),s_{t})\\mid E,s_{1:t-1},v\\right]\\right)\\] \\[=\\mathop{\\mathbb{E}}_{v,s_{1:t-1}\\mid E}\\left(\\mathop{\\mathbb{E}} _{s_{t},u_{1},\\cdots,u_{N/m}}\\left[\\ell(O_{t+1}(s_{1:t-1},v,u_{1},\\cdots,u_{N/ m},s_{t}),u_{j})\\mid E,s_{1:t-1},v\\right]\\right)\\] \\[=\\mathop{\\mathbb{E}}_{s_{t}\\sim\\mathcal{D}_{t},R^{(t+1)}}[\\ell(h_ {t+1},u_{j})\\mid E],\\]\n\nwhere (18) follows from the conditional iid (and thus exchangeable) property of \\((u_{1},\\cdots,u_{N/m},s_{t})\\) after the conditioning, and (19) is due to the invariance of the \\(O_{t+1}\\) after any permutation of the inputs. On the other hand, if \\(s^{\\prime}_{t},u^{\\prime}_{1},\\cdots,u^{\\prime}_{N/m}\\) are independent copies of \\(s_{t}\\sim\\mathcal{D}_{t}\\), by independence it is clear that\n\n\\[\\mathop{\\mathbb{E}}_{s_{t},s^{\\prime}_{t}\\sim\\mathcal{D}_{t},R^{(t+1)}}[\\ell(h _{t+1},s^{\\prime}_{t})\\mid E]=\\mathop{\\mathbb{E}}_{s_{t},s^{\\prime}_{t}\\sim \\mathcal{D}_{t},R^{(t+1)}}[\\ell(h_{t+1},u^{\\prime}_{j})\\mid E],\\quad\\forall j \\in[N/m].\\]\n\nConsequently, using the shorthand \\(u_{0}=s_{t},u^{\\prime}_{0}=s^{\\prime}_{t}\\), we have\n\n\\[\\mathop{\\mathbb{E}}_{s_{t},s^{\\prime}_{t}\\sim\\mathcal{D}_{t},R^{ (t+1)}}[\\ell(h_{t+1},s^{\\prime}_{t})-\\ell(h_{t+1},s_{t})\\mid E]\\] \\[=\\frac{1}{N/m+1}\\mathop{\\mathbb{E}}_{s_{t},s^{\\prime}_{t}\\sim \\mathcal{D}_{t},R^{(t+1)}}\\left[\\sum_{j=0}^{N/m}(\\ell(h_{t+1},u^{\\prime}_{j})- \\ell(h_{t+1},u_{j}))\\,\\bigg{|}\\,E\\right]\\] \\[\\leq\\frac{1}{N/m+1}\\mathop{\\mathbb{E}}_{u_{0},\\cdots,u_{N/m},u^{ \\prime}_{0},\\cdots,u^{\\prime}_{N/m}\\sim\\mathcal{D}_{t}}\\left[\\sup_{h\\in \\mathcal{F}_{\\alpha}}\\sum_{j=0}^{N/m}(\\ell(h,u^{\\prime}_{j})-\\ell(h,u_{j}))\\right]\\] \\[\\leq\\frac{2\\alpha}{N/m+1}\\mathop{\\mathbb{E}}_{u_{0},\\cdots,u_{N/ m}\\sim\\mathcal{D}_{t}}\\mathop{\\mathbb{E}}_{\\epsilon_{1}\\ldots\\epsilon_{N/m}} \\left[\\sup_{h\\in\\mathcal{F}_{\\alpha}}\\sum_{j=0}^{N/m}\\epsilon_{j}h(u_{j})\\right]\\] \\[\\leq\\frac{1}{\\alpha}\\mathrm{Rad}\\left(\\mathcal{F}_{\\alpha},N/m \\right).\\]\n\nThe last inequality uses the fact that the algorithm always outputs a function in \\(\\mathcal{F}_{\\alpha}\\). Further, we have used the Ledoux-Talagrand contraction inequality.\n\n**Theorem H.2** (Ledoux-Talagrand Contraction).: _Let \\(g:\\mathbb{R}\\rightarrow\\mathbb{R}\\) be a \\(L\\)-Lipschitz function. For a function class \\(\\mathcal{F}\\), denote by \\(g\\circ\\mathcal{F}\\) the compositions of function in \\(\\mathcal{F}\\) with \\(g\\). Then, for all \\(n\\),_\n\n\\[\\mathrm{Rad}\\left(g\\circ\\mathcal{F},n\\right)\\leq L\\cdot\\mathrm{Rad}\\left( \\mathcal{F},n\\right).\\]\n\nLast inequality follows from the fact that the derivative of the log loss is bounded by \\(1/\\alpha\\) when truncated at level \\(\\alpha\\). Note that the union bound gives\n\n\\[\\Pr[E^{c}]\\leq\\sum_{j=1}^{N/m}\\Pr[E^{c}_{j}]\\leq\\frac{N\\left(1-\\sigma\\right)^{ m}}{m}.\\]\n\nThus,the law of total expectation gives\n\n\\[\\mathop{\\mathbb{E}}_{s_{t},s^{\\prime}_{t}\\sim\\mathcal{D}_{t},R^{(t+1)}}[\\ell(h _{t+1},s^{\\prime}_{t})-\\ell(h_{t+1},s_{t})]\\]\\[\\leq\\operatorname*{\\mathbb{E}}_{s_{t},s^{\\prime}_{t}\\sim\\mathcal{D}_{t},R^{(t+1)}} [\\ell(h_{t+1},s^{\\prime}_{t})-\\ell(h_{t+1},s_{t})\\mid E]+\\Pr[E^{c}]\\log(1/\\alpha)\\]\n\n\\[\\leq\\frac{1}{\\alpha}\\mathrm{Rad}\\left(\\mathcal{F}_{\\alpha},N/m\\right)+\\frac{N \\left(1-\\sigma\\right)^{m}\\log\\left(1/\\alpha\\right)}{m}.\\]\n\nThe last equation follows from the fact that the output of the algorithm has loss always bounded by \\(\\log\\left(1/\\alpha\\right)\\).\n\nWe get the desired result by taking the expectation of \\(N\\sim\\text{Poi}(n)\\), and using \\(\\Pr[N>n/2]\\geq 1-e^{-n/8}\\) in the above inequality completes the proof.\n\n## Appendix I Bound on the Perturbation Term\n\n**Lemma I.1** (Perturbation).: \\[\\operatorname*{\\mathbb{E}}\\left[\\sum_{i=1}^{N}L(\\hat{h},\\tilde{s}_{t})-L(h^{ *},\\tilde{s}_{t})\\right]\\leq n\\log\\alpha\\]\n\nProof.: Note from the truncation step in Algorithm 1, we have that \\(L(\\hat{h},\\tilde{s}_{t})\\leq\\log\\left(\\alpha\\right)\\). We get the desired bound by taking expectations.", "response": "## Review\n\n### Summary\n\nThis paper investigates the sequential probability assignment problem under a smoothed adversarial setting. The authors demonstrate how to achieve sublinear regret by reducing the problem to a transductive learning framework and deriving upper bounds based on the scale-sensitive VC dimension. They propose an oracle-efficient algorithm based on the Follow-the-Perturbed-Leader approach, showing that it can attain a regret of order T^{4/5} for VC classes, thereby providing significant insights into the interplay between adversarial contexts and probability assignments. The paper's results contribute to the understanding of regret minimization in online learning with adversarial setups, making it a notable addition to the literature.\n\n### Strengths\n\n- The paper presents a strong contribution to the literature on sequential probability assignment, particularly by applying a smoothed analysis framework.\n- It provides essentially tight bounds on regret for VC-classes and introduces an oracle-efficient algorithm achieving sublinear regret.\n- The results are well-presented, clear, and motivate further research within the context of smoothed analysis.\n- The paper creatively combines earlier ideas and provides a novel approach to existing problems.\n\n### Weaknesses\n\n- The proposed oracle-efficient algorithm may not be practically efficient due to the high complexity of the ERM oracle under adversarial label selection.\n- Many technical contributions are not clearly articulated in the main text, leaving gaps in understanding how the results build upon prior work.\n- The novelty of the techniques used in the paper is questionable, as they closely resemble previous works, particularly those by Haghtalab et al.\n- There are multiple presentation issues, including inconsistent notation, unclear proofs, and potential typos that detract from the overall quality.\n\n### Questions\n\n- Can the authors elaborate on any simple function classes where the ERM oracle can be efficiently implemented under the given assumptions?\n- What are the significant technical hurdles when applying prior techniques to this setting, and how do they differ from existing work?\n- Could the authors clarify the differences and similarities with the results from Wu et al. (2023)?\n- How does Algorithm 1 perform if the oracle is not exactly optimal and relies on approximate solutions?\n- Is there a connection between the scale-sensitive VC dimension and the fat-shattering dimension?\n\n### Soundness\n\n**Score:** 3\n\n**Description:** 3 = good; the results are generally sound, but they heavily rely on existing work and lack clarity on key technical contributions.\n\n### Presentation\n\n**Score:** 3\n\n**Description:** 3 = good; while the paper is generally clear, there are notable issues with notation and presentation that could be improved.\n\n### Contribution\n\n**Score:** 3\n\n**Description:** 3 = good; the paper presents solid contributions but lacks sufficient novelty to distinguish it significantly from related prior work.\n\n### Rating\n\n**Score:** 7\n\n**Description:** 7 = accept, but needs minor improvements; the paper is technically solid and has a high impact potential in its area, but it requires clearer articulation of its contributions.\n\n### Paper Decision\n\n**Decision:** Accept\n\n**Reasons:** The paper provides valuable insights into the sequential probability assignment problem under a smoothed adversarial setting, with technically solid results and a new algorithmic approach. However, some issues related to clarity, presentation, and the novelty of the techniques used need to be addressed. Overall, the contributions are significant enough to warrant acceptance.\n"}
{"query": "You are a highly experienced, conscientious, and fair academic reviewer, please help me review this paper. The review should be organized into nine sections: \n1. Summary: A summary of the paper in 100-150 words.\n2. Strengths/Weaknesses/Questions: The Strengths/Weaknesses/Questions of paper, which should be listed in bullet points, with each point supported by specific examples from the article where possible.\n3. Soundness/Contribution/Presentation: Rate the paper's Soundness/Contribution/Presentation, and match this score to the corresponding description from the list below and provide the result. The possible scores and their descriptions are: \n    1 poor\n    2 fair\n    3 good\n    4 excellent\n4. Rating: Give this paper an appropriate rating, match this rating to the corresponding description from the list below and provide the result. The possible Ratings and their descriptions are: \n    1 strong reject\n    2 reject, significant issues present\n    3 reject, not good enough\n    4 possibly reject, but has redeeming facets\n    5 marginally below the acceptance threshold\n    6 marginally above the acceptance threshold\n    7 accept, but needs minor improvements \n    8 accept, good paper\n    9 strong accept, excellent work\n    10 strong accept, should be highlighted at the conference   \n5. Paper Decision: It must include the Decision itself(Accept or Reject) and the reasons for this decision, based on the criteria of originality, methodological soundness, significance of results, and clarity and logic of presentation.\n\nHere is the template for a review format, you must follow this format to output your review result:\n**Summary:**\nSummary content\n\n**Strengths:**\n- Strength 1\n- Strength 2\n- ...\n\n**Weaknesses:**\n- Weakness 1\n- Weakness 2\n- ...\n\n**Questions:**\n- Question 1\n- Question 2\n- ...\n\n**Soundness:**\nSoundness result\n\n**Presentation:**\nPresentation result\n\n**Contribution:**\nContribution result\n\n**Rating:**\nRating result\n\n**Paper Decision:**\n- Decision: Accept/Reject\n- Reasons: reasons content\n\n\nPlease ensure your feedback is objective and constructive. The paper is as follows:\n\nVariational Inference with Gaussian Score Matching\n\n**Chirag Modi**\n\nCenter for Computational Astrophysics,\n\nCenter for Computational Mathematics,\n\nFlatiron Institute, New York\n\ncmodi@flatironinstitute.org\n\n**Charles C. Margossian**\n\nCenter for Computational Mathematics,\n\nFlatiron Institute, New York\n\ncmargossian@flatironinstitute.org\n\n**Yuling Yao**\n\nCenter for Computational Mathematics,\n\nFlatiron Institute, New York\n\nyyao@flatironinstitute.org\n\n**Robert M. Gower**\n\nCenter for Computational Mathematics,\n\nFlatiron Institute, New York\n\nrgower@flatironinstitute.org\n\n**David M. Blei**\n\nDepartment of Computer Science, Statistics,\n\nColumbia University, New York\n\ndavid.blei@columbia.edu\n\n**Lawrence K. Saul**\n\nCenter for Computational Mathematics,\n\nFlatiron Institute, New York\n\nlsaul@flatironinstitute.org\n\n###### Abstract\n\nVariational inference (VI) is a method to approximate the computationally intractable posterior distributions that arise in Bayesian statistics. Typically, VI fits a simple parametric distribution to be close to the target posterior, optimizing an appropriate objective such as the evidence lower bound (ELBO). In this work, we present a new approach to VI. Our method is based on the principle of score matching--namely, that if two distributions are equal then their score functions (i.e., gradients of the log density) are equal at every point on their support. With this principle, we develop score-matching VI, an iterative algorithm that seeks to match the scores between the variational approximation and the exact posterior. At each iteration, score-matching VI solves an inner optimization, one that minimally adjusts the current variational estimate to match the scores at a newly sampled value of the latent variables. We show that when the variational family is a Gaussian, this inner optimization enjoys a closed-form solution, which we call Gaussian score matching VI (GSM-VI). GSM-VI is a \"black box\" variational algorithm in that it only requires a differentiable joint distribution, and as such it can be applied to a wide class of models. We compare GSM-VI to black box variational inference (BBVI), which has similar requirements but instead optimizes the ELBO. We first study how GSM-VI behaves as a function of the problem dimensionality, the condition number of the target covariance matrix (when the target is Gaussian), and the degree of mismatch between the approximating and exact posterior distribution. We then study GSM-VI on a collection of real-world Bayesian inference problems from the posteriorDB database of datasets and models. We find that GSM-VI is faster than BBVI  and equally or more accurate. Specifically, over a wide range of target posteriors, GSM-VI requires 10-100x fewer gradient evaluations than BBVI to obtain a comparable quality of approximation.1Introduction\n\nThis paper is about variational inference for approximate Bayesian computation. Consider a statistical model \\(p(\\mathbf{\\theta},\\mathbf{x})\\) of parameters \\(\\mathbf{\\theta}\\in\\mathbb{R}^{d}\\) and observations \\(\\mathbf{x}\\). Bayesian inference aims to infer the posterior distribution \\(p(\\mathbf{\\theta}\\,|\\,\\mathbf{x})\\), which is often intractable to compute. Variational inference is an optimization-based approach to approximate the posterior [5; 18].\n\nThe idea behind VI is to approximate the posterior with a member of a _variational family_ of distributions \\(q_{\\mathbf{w}}(\\mathbf{\\theta})\\), parameterized by _variational parameters_\\(\\mathbf{w}\\)[5; 18]. Specifically, VI methods establish a measure of closeness between \\(q_{\\mathbf{w}}(\\mathbf{\\theta})\\) and the posterior, and then minimize it with an optimization algorithm. Researchers have explored many aspects of VI, including different objectives [8; 9; 20; 26; 27; 29; 34] and optimization strategies [1; 15; 28].\n\nIn its modern form, VI typically minimizes \\(\\text{KL}\\left(q_{\\mathbf{w}}(\\theta)||p(\\theta\\,|\\,x)\\right)\\) with stochastic optimization, and further satisfies the so-called \"black-box\" criteria [1; 28; 33]. Black-box VI (BBVI) only requires the practitioner to specify the log joint \\(\\log p(\\mathbf{\\theta},\\mathbf{x})\\) and (often) its gradient \\(\\nabla_{\\mathbf{\\theta}}\\log p(\\theta,\\mathbf{x})\\), which for many models can be obtained by automatic differentiation [14; 25]. BBVI has been widely implemented, and it is available in many probabilistic programming systems [4; 21; 31].\n\nIn this paper, we propose a new approach to VI. We begin with the principle of _score matching_[16], that when two densities are equal then their gradients are equal as well, and we use this principle to derive a new way to fit a variational distribution to be close to the exact posterior. The result is _score-matching VI_. Rather than explicitly minimize a divergence, score-matching VI iteratively projects the variational distribution onto the exact score-matching constraint. This strategy enables a new black-box VI algorithm.\n\nScore-matching VI relies on the same ingredients as reparameterization BBVI [21]--a differentiable variational family and a differentiable log joint--and so it can be as easily incorporated into probabilistic programming systems as well. Further, when the variational family is a Gaussian, score-matching VI is particularly efficient: each iteration is computable in closed form. We call the resulting algorithm Gaussian score matching VI (GSM-VI).\n\nUnlike BBVI, GSM-VI does not rely on stochastic gradient descent (SGD) for its core optimization. Though SGD has the appeal of simplicity, it requires careful tuning of learning rates. GSM-VI was inspired by a different tradition of constraint-based algorithms for online learning [3; 7; 12; 13; 23]. These algorithms have been extensively developed and analyzed for problems in classification, and under the right conditions, they have been observed to outperform SGD. This paper shows how to extend this constraint-based framework--and the powerful machinery behind it--from the problem of classification to the workhorse of Gaussian VI. The key insight is that score-matching (unlike ELBO maximization) lends itself naturally to a constraint-based formulation.\n\nWe empirically compared GSM-VI to reparameterization BBVI on several classes of models, and with both synthetic and real-world data. In general, we found that GSM requires 10-100x fewer gradient evaluations to converge to an equally good approximation. When the exact posterior is Gaussian, we found that GSM-VI scales significantly better with respect to dimensionality and is insensitive to the condition number of the target covariance. When the exact posterior is non-Gaussian, we found that GSM-VI converges more quickly without sacrificing the quality of the final approximation.\n\nThis paper makes the following contributions:\n\n* We introduce _score-matching variational inference_, a new black-box approach to fitting \\(q_{\\mathbf{w}}(\\mathbf{\\theta})\\) to be close to \\(p(\\mathbf{\\theta}\\,|\\,\\mathbf{x})\\). Score-matching VI requires no tunable optimization hyperparameters, to which BBVI can be sensitive.\n* When the variational family is Gaussian, we develop _Gaussian score-matching variational inference_ (GSM-VI). It establishes efficient closed-form iterates for score-matching VI.\n* We empirically compare GSM-VI to reparameterization BBVI. Across many models and datasets, we found that GSM-VI enjoys faster convergence to an equally good approximation.\n\nWe develop score-matching VI in Section 2 and study its performance in Section 3.\n\n**Related work.** This paper introduces a new method for black-box variational inference that relies only on having access to the gradients of the variational distribution and the log joint. GSM-VI has similar goals to automatic-differentiation variational inference (ADVI) [21] and Pathfinder [35], which also fit multivariate Gaussian variational families, but do so by maximizing the ELBO using stochastic optimization. As in GSM-VI, the algorithm of ref. [32] also seeks to match the scores of the variational and the target posterior, but it does so by minimizing the L2 loss between them.\n\nA novel aspect of GSM-VI is how it fits the variational parameters. Rather than minimize a loss function, it aims to solve a set of nonlinear equations. Similar ideas have been pursued in the context of fitting a model to data using empirical risk minimization (ERM). For example, passive agressive (PA) methods [7] and the stochastic polyak stepsize (SPS) are also derived via projections onto sampled nonlinear equations [3; 13; 23]. A probabilistic extension of PA methods is known as confidence-weighted (CW) learning [12]. In CW learning, the learner maintains a multivariate Gaussian distribution over the weight vector of a linear classifier. Similarly, the second step of GSM-VI also minimizes a KL divergence between multivariate Gaussians. Unlike CW learning, however, the projection in GSM-VI enforces a score-matching constraint as opposed to a margin-based constraint for linear classification.\n\n## 2 Score Matching Variational Inference\n\nSuppose for the moment that the variational family \\(q_{\\mathbf{w}}(\\mathbf{\\theta})\\) is rich enough to perfectly capture the posterior \\(p(\\mathbf{\\theta}\\,|\\,\\mathbf{x})\\). In other words, there exists a \\(\\mathbf{w}^{*}\\) such that\n\n\\[\\log\\,q_{\\mathbf{w}^{*}}(\\mathbf{\\theta})=\\log\\,p(\\mathbf{\\theta}|\\mathbf{x}),\\qquad\\forall\\bm {\\theta}\\,\\in\\Theta. \\tag{1}\\]\n\nIf we could solve Eq. 1 for \\(\\mathbf{w}^{*}\\), the resulting variational distribution would be a perfect fit. The challenge is that the posterior on the right side is intractable to compute.\n\nTo help, we appeal to the idea of score matching [16]. We define the score of a distribution to be the gradient of its log with respect to the variable2, e.g., \\(\\nabla_{\\mathbf{\\theta}}\\log q_{\\mathbf{w}}(\\mathbf{\\theta})\\). The principle of score matching is that if two distributions are equal at every point in their support then their score functions are also equal.\n\nFootnote 2: We make this clear because, in some literature, the score is defined as the gradient with respect to the parameter.\n\nTo use score matching for VI, we first write the log posterior as the log joint minus the normalizing constant, i.e., the marginal distribution of \\(\\mathbf{x}\\),\n\n\\[\\log\\,p(\\mathbf{\\theta}\\,|\\,\\mathbf{x})=\\log\\,p(\\mathbf{\\theta},\\mathbf{x})-\\log\\,p(\\mathbf{x}). \\tag{2}\\]\n\nWith this expression, the principle of score matching leads to the following Lemma.\n\n**Lemma 2.1**.: Let \\(\\Theta\\subset\\mathbb{R}^{d}\\) be a set that contains the support of \\(p(\\mathbf{\\theta},\\mathbf{x})\\) and \\(q_{\\mathbf{w}^{*}}(\\mathbf{\\theta})\\) for some \\(\\mathbf{w}^{*}\\). That is \\(p(\\mathbf{\\theta},\\mathbf{x})=q_{\\mathbf{w}^{*}}(\\mathbf{\\theta})=0\\) for every \\(\\theta\\not\\in\\Theta\\). Furthermore, suppose that the support of \\(p(\\mathbf{\\theta},\\mathbf{x})\\) and \\(q_{\\mathbf{w}^{*}}(\\mathbf{\\theta})\\), and \\(\\Theta\\) is path-connected. The parameter \\(\\mathbf{w}^{*}\\) satisfies\n\n\\[\\nabla_{\\mathbf{\\theta}}\\log\\,q_{\\mathbf{w}^{*}}(\\mathbf{\\theta})=\\nabla_{\\mathbf{\\theta}}\\log \\,p(\\mathbf{\\theta},\\mathbf{x}),\\qquad\\forall\\mathbf{\\theta}\\in\\Theta, \\tag{3}\\]\n\nif and only if \\(\\mathbf{w}^{*}\\) also satisfies Eq. 1.\n\nWhat is notable about Eq. 3 is that the right side is the gradient of the log joint. Unlike the posterior, the gradient of the log joint is tractable to compute for a large class of probabilistic models. (The proof of the lemma is in the appendix.)\n\nThis lemma motivates a new algorithm, _score-matching VI._ The idea is to iteratively refine the variational parameters \\(\\mathbf{w}\\) in order to satisfy (as much as possible) the system of equations in Eq. 3. At each iteration \\(t\\), the algorithm first samples a new \\(\\mathbf{\\theta}_{t}\\) from the current variational approximation and then minimally adjusts \\(\\mathbf{w}\\) to satisfy Eq. 3 for that value of \\(\\mathbf{\\theta}_{t}\\), see (4). We want this adjustment to be minimal because, eventually \\(q_{\\mathbf{w}_{t}}\\) will learn a good fit--one that increasingly matches the scores over the support of the target posterior--and we want to preserve this good fit as much as possible.\n\n**Score matching variational inference**\n\nAt iteration \\(t\\):\n\nSample \\(\\mathbf{\\theta}_{t}\\sim q_{\\mathbf{w}_{t}}(\\mathbf{\\theta})\\).\n\nUpdate the variational parameters:\n\n\\[\\begin{split}\\mathbf{w}_{t+1}=\\arg\\min_{\\mathbf{w}}\\text{KL}\\left(q_{\\mathbf{w} _{t}}(\\mathbf{\\theta})\\,||\\,q_{\\mathbf{w}}(\\mathbf{\\theta})\\right)\\\\ \\text{such that}&\\nabla_{\\mathbf{\\theta}}\\,\\log q_{\\mathbf{w} }(\\mathbf{\\theta}_{t})=\\nabla_{\\mathbf{\\theta}}\\,\\log p(\\mathbf{\\theta}_{t},\\mathbf{x}).\\end{split} \\tag{4}\\]\n\nThis algorithm for score matching VI was inspired by earlier online algorithms for learning a classifier from a stream of labeled examples. One particularly elegant algorithm in this setting is known as passive-aggressive (PA) learning [7]; this algorithm incrementally updates its model by the minimal amount to classify each example correctly by a large margin. This approach was subsequently extended to a probabilistic setting, known as confidence-weighted (CW) learning [12] in which one minimally updates a _distribution_ over classifiers. Our algorithm is similar in that it minimally updates an approximating distribution for VI, but it is different in that it enforces constraints for score matching instead of large margin classification.\n\nAt a high level, what makes this approach to VI likely to succeed or fail? Certainly it is necessary that there are more variational parameters than elements of the latent variable \\(\\mathbf{\\theta}\\); when this is not the case, it may be impossible to satisfy a _single_ score matching constraint in Eq. 4. That said, it is standard (as in, e.g., a factorized or mean-field variational family) to have at least as many variational parameters as there are elements of the latent variable. It is also apparent that the algorithm may never converge if the target posterior is not contained in the variational family, or it may converge to a degenerate solution if the variational approximation collapses to a point mass, thus terminating the updates altogether. While we cannot dismiss these possibilities out of hand, we did not observe erratic or pathological outcomes in any of the empirical studies of Section 3.\n\nFor more intuition, Figure 0(a) illustrates the effect of the update in Eq. 4 when both the target and approximating distribution are 1d Gaussian. The target posterior \\(p(\\mathbf{\\theta}\\,|\\,\\mathbf{x})\\) is shaded blue. The plot shows the initial variational distribution \\(q_{\\mathbf{w}_{0}}\\) (light grey curve) and its update to \\(q_{\\mathbf{w}_{1}}\\) (medium grey) so that the gradient of the updated distribution matches the gradient of the target at the sampled \\(\\mathbf{\\theta}_{0}\\) (dotted red tangent line). It also shows the update from \\(\\mathbf{w}_{1}\\) to \\(\\mathbf{w}_{2}\\), now matching the gradient at \\(\\mathbf{\\theta}_{1}\\). With these two updates, \\(q_{\\mathbf{w}_{2}}\\) (dark grey) is very close to the target \\(p(\\mathbf{\\theta}\\,|\\,\\mathbf{x})\\). With this picture in mind, we now develop the details of this algorithm for a more widely applicable setting.\n\n**Gaussian Score-Matching VI.** Suppose the variational distribution belongs to a multivariate Gaussian family \\(q_{\\mathbf{w}}(\\mathbf{\\theta}):=\\mathcal{N}(\\mathbf{\\mu},\\mathbf{\\Sigma})\\); this is a common choice especially in systems for automated approximate inference [1, 21]. One of our main contributions is to show in this case that Eq. 4 has a closed-form solution. The solution \\(\\mathbf{w}_{t+1}=(\\mathbf{\\mu}_{t+1},\\mathbf{\\Sigma}_{t+1})\\) has the following form:\n\n\\[\\mathbf{\\mu}_{t+1} =\\ \\mathbf{\\mu}_{t}+\\mathbf{A}_{t}\\,\\left(\\nabla_{\\mathbf{\\theta}}\\,\\log p(\\mathbf{ \\theta}_{t},\\mathbf{x})-\\nabla_{\\mathbf{\\theta}}\\,\\log q_{\\mathbf{w}_{t}}(\\mathbf{\\theta}_{t})\\right) \\tag{5}\\] \\[\\mathbf{\\Sigma}_{t+1} =\\ \\mathbf{\\Sigma}_{t}+(\\mathbf{\\mu}_{t}-\\mathbf{\\theta}_{t})(\\mathbf{\\mu}_{t}- \\mathbf{\\theta}_{t})^{\\top}-(\\mathbf{\\mu}_{t+1}-\\mathbf{\\theta}_{t})(\\mathbf{\\mu}_{t+1}-\\mathbf{ \\theta}_{t})^{\\top} \\tag{6}\\]\n\nwhere \\(\\mathbf{A}_{t}\\in\\mathbb{R}^{d\\times d}\\) is a matrix defined in the theorem below. Note that these updates, which exactly solve the optimization in Eq. 4, only require the score of the log joint \\(\\nabla_{\\mathbf{\\theta}}\\log p(\\mathbf{\\theta},\\mathbf{x})\\) and the score of the variational distribution \\(\\nabla_{\\mathbf{\\theta}}\\log q_{\\mathbf{w}}(\\mathbf{\\theta})\\).\n\nBefore deriving Eqs. 5 and 6 in full, we highlight the intuitive form of this closed-form solution. Consider the approximation at the \\(t\\)th iteration \\(q_{\\mathbf{w}_{t}}\\) and the current sample \\(\\mathbf{\\theta}_{t}\\). First suppose the scores already match at this sample, that is \\(\\nabla_{\\mathbf{\\theta}}\\,\\log p(\\mathbf{\\theta}_{t},\\mathbf{x})=\\nabla_{\\mathbf{\\theta}}\\, \\log q_{\\mathbf{w}_{t}}(\\mathbf{\\theta}_{t})\\). Then the mean does not change \\(\\mathbf{\\mu}_{t+1}=\\mathbf{\\mu}_{t}\\) and, similarly, the two rank-one terms in the covariance update in Eq. 6 cancel out so \\(\\mathbf{\\Sigma}_{t+1}=\\mathbf{\\Sigma}_{t}\\). This shows that when \\(q_{\\mathbf{w}_{t}}(\\mathbf{\\theta})=p(\\mathbf{\\theta},\\mathbf{x})\\) for all \\(\\mathbf{\\theta}\\), the method stops. On the other hand, if the scores do not match, then the mean is updated proportionally to the difference between the scores, and the covariance is updated by a rank-two correction. Figure 0(b) illustrates the vector field of updates for a one dimensional target \\(p(\\mathbf{\\theta},\\mathbf{x})=\\mathcal{N}(0,1)\\). The vector field points to the solution (green star), which is also the only fixed point of the updates.\n\nWe now formalize this result and give the complete expression for the matrix \\(\\mathbf{A}_{t}\\) that appears in the update for the mean of the variational approximation in Eq. 5.\n\n**Theorem 2.2**.: **(GSM-VI updates)** Let \\(p(\\mathbf{\\theta},\\mathbf{x})\\) be given for some \\(\\mathbf{\\theta}\\in\\mathbb{R}^{d}\\), and let \\(q_{\\mathbf{w}^{\\epsilon}}(\\mathbf{\\theta})\\) and \\(q_{\\mathbf{w}}(\\mathbf{\\theta})\\) be multivariate normal distributions with means \\(\\mathbf{\\mu}_{t}\\) and \\(\\mathbf{\\mu}\\) and covariance matrices \\(\\mathbf{\\Sigma}_{t}\\) and \\(\\mathbf{\\Sigma}\\), respectively. As shorthand, let \\(\\mathbf{g}_{t}:=\\nabla_{\\mathbf{\\theta}}\\log p(\\mathbf{\\theta}_{t},\\mathbf{x})\\) and let\n\n\\[\\mathbf{\\mu}_{t+1},\\mathbf{\\Sigma}_{t+1}\\ =\\ \\operatorname*{argmin}_{\\mathbf{\\mu},\\mathbf{ \\Sigma}\\succeq 0}\\left[\\operatorname{KL}(q_{t},q)\\right]\\quad\\text{such that}\\quad\\nabla_{\\mathbf{\\theta}}\\log q(\\mathbf{\\theta}_{t})=\\nabla_{\\mathbf{\\theta}}\\log p(\\mathbf{\\theta}_{t},\\mathbf{x}). \\tag{7}\\]\n\nThe solution to eq. (7) is given by Eqs. 5 and 6 where\n\n\\[\\mathbf{A}_{t} := \\frac{1}{1+\\rho}\\left[\\mathbf{I}-\\frac{(\\mathbf{\\mu}_{t}-\\mathbf{\\theta}_ {t})\\mathbf{g}_{t}^{\\top}}{1+\\rho+(\\mathbf{\\mu}_{t}-\\mathbf{\\theta}_{t})^{\\top}\\mathbf{g}_{t} }\\right]\\mathbf{\\Sigma}_{t}\\,, \\tag{8}\\]\n\nand \\(\\rho\\) is the positive root of the quadratic equation\n\n\\[\\rho(1+\\rho)\\ =\\ \\mathbf{g}_{t}^{\\top}\\mathbf{\\Sigma}_{t}\\mathbf{g}_{t}+\\left[(\\mathbf{\\mu}_{t}-\\mathbf{ \\theta}_{t})^{\\top}\\mathbf{g}_{t}\\right]^{2}. \\tag{9}\\]\n\nWith the definition of \\(\\mathbf{A}_{t}\\) in Eq. 8 we can see that the computational complexity of updating \\(\\mathbf{\\mu}\\) and \\(\\mathbf{\\Sigma}\\) via Eqs. 5 and 6 is \\(\\mathcal{O}(d^{2})\\), where \\(\\mathbf{\\theta}\\in\\mathbb{R}^{d}\\) and we assume the cost of computing the gradients is \\(\\mathcal{O}(d)\\). Note this is the best possible iteration complexity we can hope for, since we store and maintain the full covariance matrix of \\(d^{2}\\) elements. (The proof is in the appendix.)\n\nAlgorithm 1 presents the full GSM-VI algorithm. Here we also use mini-batching, where we average over \\(B\\in\\mathbb{N}\\) independently sampled updates of Eqs. 5 and 6 before updating the mean and covariance.\n\n## 3 Empirical Studies\n\nWe evaluate how well GSM-VI can approximate a variety of target posterior distributions. Recall that GSM-VI uses a multivariate Gaussian distribution as its variational family. We investigate two separate cases--one when the target posterior is in this family, and the other when it is not.\n\n**Algorithmic details and comparisons.** We compare GSM-VI with a reparameterization variant of BBVI as the baseline, similar to [21]. BBVI uses the same multivariate Gaussian variational family, which we fit by maximizing the ELBO. (Maximizing the ELBO is equivalent to minimizing KL).\n\n```\nInput :Initial mean estimate \\(\\mathbf{\\mu}_{0}\\), initial covariance estimate \\(\\mathbf{\\Sigma}_{0}\\), target distribution \\(p(\\mathbf{\\theta}|\\mathbf{x})\\), number of iterations \\(N\\in\\mathbb{N}\\), batch size \\(B\\in\\mathbb{N}\\). Output :Multivariate normal variational distribution \\(q_{\\mathbf{w}}(\\mathbf{\\theta}):=\\mathcal{N}(\\mathbf{\\mu},\\mathbf{\\Sigma})\\) for\\(i=0,\\ldots,N-1\\)\\(\\triangleright\\) iteration loop do for\\(j=0,\\ldots,B-1\\)\\(\\triangleright\\) batch loop do  Sample \\(\\mathbf{\\theta}^{(j)}\\sim\\mathcal{N}(\\mathbf{\\mu}_{i},\\mathbf{\\Sigma}_{i})\\) \\(\\mathbf{g}\\leftarrow\\nabla_{\\mathbf{\\theta}}\\log p(\\mathbf{\\theta}^{(j)}|\\mathbf{x})\\) \\(\\mathbf{\\varepsilon}\\leftarrow\\mathbf{\\Sigma}_{i}\\mathbf{g}-\\mathbf{\\mu}_{i}+\\mathbf{\\theta}\\)  Solve \\(\\rho(1+\\rho)\\ =\\ \\mathbf{g}^{\\top}\\mathbf{\\Sigma}_{i}\\mathbf{g}+\\left[(\\mathbf{\\mu}_{i}-\\mathbf{ \\theta})^{\\top}\\mathbf{g}\\right]^{2}\\) for \\(\\rho>0\\) \\(\\mathbf{\\delta}\\mathbf{\\mu}^{(j)}\\leftarrow\\frac{1}{1+\\rho}\\left[\\mathbf{I}-\\frac{( \\mathbf{\\mu}_{i}-\\mathbf{\\theta})\\mathbf{g}^{\\top}}{1+\\rho+(\\mathbf{\\mu}_{i}-\\mathbf{\\theta})^{ \\top}\\mathbf{g}}\\right]\\mathbf{\\varepsilon}\\) \\(\\mathbf{\\mu}^{(j)}_{i}\\leftarrow\\mathbf{\\mu}_{i}+\\mathbf{\\delta}\\mathbf{\\mu}^{(j)}\\) \\(\\mathbf{\\delta}\\mathbf{\\Sigma}^{(j)}\\leftarrow(\\mathbf{\\mu}_{i}-\\mathbf{\\theta})(\\mathbf{\\mu}_{i}- \\mathbf{\\theta})^{\\top}-(\\mathbf{\\mu}^{(j)}_{i}-\\mathbf{\\theta})(\\mathbf{\\mu}^{(j)}_{i}-\\mathbf{ \\theta})^{\\top}\\)  end for  Update \\(\\mathbf{\\mu}_{i+1}\\leftarrow\\mathbf{\\mu}_{i}+\\sum_{j}\\mathbf{\\delta}\\mathbf{\\mu}^{(j)}/B\\)  Update \\(\\mathbf{\\Sigma}_{i+1}-\\mathbf{\\Sigma}_{i}+\\sum_{j}\\mathbf{\\delta}\\mathbf{\\Sigma}^{(j)}/B\\)  end for \\(q_{\\mathbf{w}}(\\mathbf{\\theta})\\leftarrow\\mathcal{N}(\\mathbf{\\mu}_{N},\\mathbf{\\Sigma}_{N})\\)\n```\n\n**Algorithm 1**Gaussian Score Matching VI\n\nWe use the ADAM optimizer [19] with default settings but vary the learning rate between \\(10^{-1}\\) and \\(10^{-3}\\). We report results only for the best performing setting. The full BBVI algorithm is shown in Algorithm 2.\n\nThe only free parameter in GSM-VI is the batch size \\(B\\). We find that for Gaussian targets, GSM performs equally well for all \\(B\\geq 1\\). There are marginal gains for larger batches for high dimensional targets, but \\(B=2\\) is a good conservative default. For non-Gaussian targets, we find that all batch sizes converge to the same solution, but we also note that larger batch sizes help to dampen the oscillations in KL divergence around the optimal solution. Notwithstanding these effects, it is not generally necessary to fine-tune this hyperparameter. In all studies of this section we report results for \\(B=2\\) and show that it is a good default baseline. We use the same batch size for BBVI.\n\nBoth GSM-VI and BBVI require an initial distribution for the variational approximation. Unless specified otherwise, we initialize the variational approximation as a Gaussian distribution with zero mean and identity covariance matrix.\n\n**Evaluation metric.** GSM-VI does not explicitly minimize any loss function. Hence to compare its performance against BBVI, we estimate empirical divergences between the variational and the target distribution and show their evolution with the number of gradient evaluations. We measure performance in terms of gradient evaluations as the actual running time can be sensitive to implementation details; moreover, we note that in real-world problems, the computation is often bottlenecked by evaluations of the target log density and its gradient. For completeness, we also measure running times of GSM-VI and BBVI for a full rank Gaussian target with 2048 dimensions using sample size of 1 in Jax after JIT-compilation. The GSM update takes 230 ms/update while BBVI takes 227 ms/update.\n\nIn the experiments with synthetic models in Sections 3.1, and 3.2 we have access to the true distribution; hence for these experiments, we estimate the forward KL divergence (FKL) empirically by \\(\\sum_{\\mathbf{\\theta}_{i}\\sim p(\\mathbf{\\theta})}\\log p(\\mathbf{\\theta}_{i})-\\log q(\\mathbf{ \\theta}_{i})\\). To reduce stochasticity, we always use the same pre-generated set of 1000 samples from the target distribution. In the experiments with real-world models in Section 3.3, we do not have access to the samples from the target distribution; for these experiments we monitor the negative ELBO. In all experiments, we show the results for 10 independent runs.\n\n### GSM-VI for Gaussian approximation\n\nWe begin by studying GSM-VI where the target distribution is also a multivariate Gaussian.\n\n**Scaling with dimensions.** How does GSM-VI scale with respect to the dimensionality, \\(D\\), of the latent variable? Figure 2 shows the convergence of FKL for GSM-VI and BBVI for different dimensionalities. Empirically, for GSM-VI, we find that the number of iterations required for convergence increases almost linearly with dimensionality. The scaling for BBVI is worse: even for small problems (\\(D<64\\)) it requires 100 times more iterations while also converging to a sub-optimal solution (as measured by the FKL metric). The figure also shows the evolution of the _reverse_ KL divergence, and it is noteworthy that GSM-VI also outperforms BBVI in this metric.\n\nFor completeness, we also experimented on these problems3 with methods for variational inference based on natural gradient descent (NGD-VI) [2, 22]. Natural gradient descent is a stochastic gradient method that uses the Fisher matrix as a preconditioner. We find NGD-VI is slower by factor of\n\nFigure 2: Scaling with dimensionality: evolution of forward (a) and reverse KL (b) with the number of gradient evaluations of the target distribution, which here is a Gaussian distribution with a dense \\(D\\times D\\) covariance matrix. Each panel shows the results for a different dimensionality \\(D\\) (indicated in the title). Translucent lines show the scatter of 10 different runs, and the solid line shows the average. GSM-VI and BBVI use batch sizes of \\(B=2\\), while the batch sizes for natural gradient descent are shown in the legend.\n\n5-10x than GSM-VI. While NGD-VI outperforms BBVI, its performance is sensitive to the choice of batch size, with small batch sizes prone to divergence. Furthermore, we note that that per-iteration-complexity of NGD-VI is cubic in the dimensionality (\\(D^{3}\\)) as each iteration requires inverting a Hessian approximation; this is in contrast to the quadratic (\\(D^{2}\\)) per-iteration-complexities of GSM-VI and BBVI. For these reasons, we do not investigate NGD-VI further in this paper.\n\n**Impact of condition number.** What is the impact of the shape of the target distribution? We again consider a Gaussian target distribution, but vary the condition number of its covariance matrix by fixing its smallest eigenvalue to \\(0.1\\) and scaling the largest eigenvalue to \\(0.1\\times c\\). Figure 3 shows the results for a 10 dimensional Gaussian where we vary the condition number \\(c\\) from 1 to 1000. Convergence of GSM-VI seems to be largely insensitive to the condition number of the covariance matrix. BBVI on the other hand struggles with poorly conditioned problems, and it does not converge for \\(c>100\\) even with 100 times more iterations than GSM.\n\n### GSM-VI for non-Gaussian target distributions\n\nGSM-VI was designed to solve the exact score-matching equations in Eq. 3, but these equations only have a solution when the family of variational distributions contains the target distribution (see Lemma 2.1). Here we investigate the sensitivity of GSM-VI to this assumption by fitting non-Gaussian target distributions with varying degrees on non-Gaussianity. Specifically, we suppose that\n\nFigure 4: Impact of non-Gaussianity: evolution of FKL with the number of gradient evaluations for Sinh-arcsinh distributions with 10-dimensional dense Gaussian as the base distribution. Gaussian distribution has \\(s=0,\\,t=1\\). In the left panel, we vary skewness \\(s\\) while fixing \\(t=1\\), and in the right panel we vary the tail-weight \\(t\\) with skewness fixed to \\(s=0\\). Solid lines are the results for GSM, dashed for BBVI.\n\nFigure 3: Impact of condition number: evolution of FKL with the number of gradient evaluations of the target distribution. The target is a 10-dimensional Gaussian albeit with a dense covariance matrix of different condition numbers \\(c\\) specified in the title of different panels. Translucent lines show the scatter of 10 different runs and the solid line shows the average.\n\nthe target has a multivariate Sinh-arcsinh normal distribution [17]\n\n\\[\\mathbf{z}\\sim\\mathcal{N}(\\mathbf{\\mu},\\mathbf{\\Sigma});\\quad\\mathbf{x}=\\sinh\\left(\\frac{1}{t} \\left[\\sinh^{-1}(\\mathbf{z})+s\\right]\\right) \\tag{10}\\]\n\nwhere the scalar parameters \\(s\\) and \\(t\\) control, respectively, the skewness and the heaviness of the tails. Note that Eq. 10 reduces as a special case to a Gaussian distribution for the choices \\(s=0\\) and \\(t=1\\) of these parameters.\n\nFigure 4 shows the result for fitting the variational Gaussian to a 10-dimensional Sinh-arcsinh normal distribution for different values of \\(s\\) and \\(t\\). As the target departs further from Gaussianity, the quality of variational fit worsens for both GSM-VI (solid lines) and BBVI (dashed lines), but they converge to a fit of similar quality in terms of average FKL. GSM converges to this solution at least 10 times faster than BBVI. For highly non-Gaussian targets (\\(s\\geq 1\\) or \\(|t-1|\\geq 0.8\\)), we have found that GSM-VI does not converge to a fixed point, and it can experience oscillations that are larger in amplitude than BBVI; see for instance \\(s=1.8\\) and \\(t=0.1\\) on the left and right of Figure 4, respectively.\n\n### GSM-VI on real-world data\n\nWe evaluate GSM-VI for approximate on real-world data with 8 models from the posterioridb database [24]. The database provides the Stan code, data and reference posterior samples, and we use bridgestan to access the gradients of these models [6, 30]. We study the following statistical models, each with a different complexity and dimensionality (D): diamonds (generalized linear models, D=26), hudson-lynx-bare (differential equation dynamics, D=8), bball-drive (hidden Markov models, D=8) and arK (time-series, D=7), eight-schools-centered and non-centered (hierarchical meta-analysis, D=10), gp-pois-regr (Gaussian processes, D=13), low-dim-gauss-mix (Gaussian mixture, D=5).\n\nFor each model (except hudson-lynx-bare), we initialize the variational parameter \\(\\mathbf{\\mu}_{0}\\) at the mode of the distribution, and we set \\(\\mathbf{\\Sigma}_{0}=0.1\\,\\mathbf{I}_{d}\\) where \\(\\mathbf{I}_{d}\\) is the identity matrix of dimension \\(d\\)\n\nFigure 5: Models from posterioridb: Convergence of the ELBO for four models with multivariate normal posteriors. We show results for 10 runs.\n\nFigure 6: Models from posterioridb: Convergence of ELBO for four models with non-Gaussian posteriors. We show results for 10 runs.\n\nFor Hudson-lynx-hare, we initialize the variational distribution as standard normal. We also experimented with other initializations. We find that they do not qualitatively change the conclusions, but can have larger variance between different runs.\n\nWe show the evolution of the ELBO for 10 runs of these models. Four of the models have posteriors that can be fit with multivariate normal distribution: diamonds, hudson-lynx-hare, bball-drive, and arK. Figure 5 shows the result for these models. The other models have non-Gaussian posteriors: eight-schools-centered, eight-schools-non-centered, gp-pois-regr,, and low-dim-gauss-mix. Figure 6 shows the results.\n\nOverall, GSM-VI outperforms BBVI by a factor of 10-100x. When the target posterior is Gaussian, GSM-VI leads to more stable solutions. When the target is non-Gaussian, it converges to the same quality of variational approximation as BBVI. Finally, we note that while GSM-VI yields noisy estimates of the ELBO, it does yield stable estimates of the 1-D marginals and their moments.\n\n## 4 Conclusion and Future Work\n\nIn this paper we have proposed Gaussian score matching VI (GSM-VI), a new approach for VI when the variational family is multivariate Gaussian. GSM-VI does not explicitly minimize a divergence between the variational and target distribution; instead, it repeatedly enforces score-matching constraints with closed-form updates for the mean and covariance matrix of the variational distribution. This is in contrast to stochastic gradient based methods, such as in BBVI, that rely on first-order Taylor approximations of their objective function.\n\nGSM-VI is implemented in an open-source Python code at [https://github.com/modichirag/GSM-VI](https://github.com/modichirag/GSM-VI).\n\nUnlike approaches that are rooted in stochastic gradient descent, GSM-VI does not require the tuning of step-size hyper-parameters. Rather, it is able to adaptively make large jumps in the initial iterations (see Fig. 1a) and make smaller adjustments as the approximation converges to the target. GSM-VI has only one free parameter, the batch size, and we found a batch-size of 2 to perform competitively across all experiments. Another choice is how to initialize the variational distribution. For the experiments in this paper, we initialized the covariance matrix as the identity matrix, but additional gains could potentially be made with more informed choices derived from a Laplace approximation or L-BFGS Hessian approximation [35].\n\nWe evaluated the performance of GSM-VI on synthetic targets and real-world models from posterioriD. In general, we found that it requires 10-100x fewer gradient evaluations than BBVI for the target distribution to converge. When the target distribution is itself multivariate Gaussian, we observed that GSM-VI scales nearly _linearly_ with dimensionality, which is significantly better than BBVI, and that GSM-VI is fairly insensitive to the condition number of the target covariance matrix. We also found that GSM-VI converges more quickly than BBVI--and also to a solution with a larger ELBO, which is surprising given that BBVI explicitly maximizes the ELBO.\n\nAn important avenue for future work is to provide a proof that GSM-VI converges. We note that good convergence results have been obtained for analogous methods that project onto interpolation equations for empirical risk minimization. For instance the Stochastic Polyak Step achieves the min-max optimal rates of convergence for SGD [23]. Note that convergence of VI is a generally challenging problem, with no known rates of convergence even for BBVI [10; 11]. This and others directions are left to future work.\n\n## References\n\n* Agrawal et al. [2020] A. Agrawal, D. R. Sheldon, and J. Domke. Advances in black-box vi: normalizing flows, importance weighting, and optimization. _Neural Information Processing Systems_, 2020.\n* Arenz et al. [2023] O. Arenz, P. Dahlinger, Z. Ye, M. Volpp, and G. Neumann. A unified perspective on natural gradient variational inference with gaussian mixture models. _Transactions on Machine Learning Research_, 2023. ISSN 2835-8856. URL [https://openreview.net/forum?id=tLbjsX4tjs](https://openreview.net/forum?id=tLbjsX4tjs).\n\n* Berrada et al. [2020] L. Berrada, A. Zisserman, and M. P. Kumar. Training neural networks for and by interpolation. In _Machine Learning_, volume 119 of _Proceedings of Machine Learning Research_, 2020.\n* Bingham et al. [2018] E. Bingham, J. Chen, M. Jankowiak, F. Obermeyer, N. Pradhan, T. Karaletsos, R. Singh, P. Szerlip, P. Horsfall, and N. Goodman. Pyro: Deep universal probabilistic programming. _arXiv:1810.09538_, 2018.\n* Blei et al. [2017] D. M. Blei, A. Kucukelbir, and J. D. McAuliffe. Variational inference: A review for statisticians. _Journal of the American statistical Association_, 112(518):859-877, 2017.\n* Carpenter et al. [2017] B. Carpenter, A. Gelman, M. D. Hoffman, D. Lee, B. Goodrich, M. Betancourt, M. Brubaker, J. Guo, P. Li, and A. Riddell. Stan: A probabilistic programming language. _Journal of Statistical Software_, 76(1):1-32, 2017.\n* Crammer et al. [2006] K. Crammer, O. Dekel, J. Keshet, S. Shalev-Shwartz, and Y. Singer. Online passive-aggressive algorithms. _Journal of Machine Learning Research_, 7:551-585, 2006.\n* Dhaka et al. [2021] A. K. Dhaka, A. Catalina, M. Welandawe, M. Riis Andersen, J. Huggins, and A. Vehtari. Challenges and Opportunities in High-dimensional Variational Inference. art. arXiv:2103.01085, March 2021.\n* Dieng et al. [2017] A. Dieng, D. Tran, R. Ranganath, J. Paisley, and D. Blei. Variational inference via \\(\\chi\\) upper bound minimization. In _Neural Information Processing Systems_, 2017.\n* Domke [2019] J. Domke. Provable gradient variance guarantees for black-box variational inference. In _Neural Information Processing Systems_, volume 32, 2019.\n* Domke [2020] J. Domke. Provable smoothness guarantees for black-box variational inference. In _International Conference on Machine Learning_, volume 119, pages 2587-2596. PMLR, 2020.\n* Dredze et al. [2008] M. Dredze, K. Crammer, and F. Pereira. Confidence-weighted linear classification. In _International Conference on Machine Learning,_, volume 307, pages 264-271, 2008.\n* Gower et al. [2021] R. M. Gower, A. Defazio, and M. Rabbat. Stochastic polyak stepsize with a moving target. _arXiv:2106.11851_, 2021.\n* Gunes Baydin et al. [2015] A. Gunes Baydin, B. A. Pearlmutter, A. Andreyevich Radul, and J. M. Siskind. Automatic differentiation in machine learning: a survey. _arXiv e-prints_, art. arXiv:1502.05767, February 2015. doi: 10.48550/arXiv.1502.05767.\n* Hoffman et al. [2013] M. Hoffman, D. Blei, C. Wang, and J. Paisley. Stochastic variational inference. _Journal of Machine Learning Research_, 14:1303-1347, 2013.\n* Hyvarinen [2005] A. Hyvarinen. Estimation of non-normalized statistical models by score matching. _Journal of Machine Learning Research_, 6:695-709, 2005.\n* Jones and Pewsey [2019] C. Jones and A. Pewsey. The Sinh-Arcsinh Normal Distribution. _Significance_, 16(2):6-7, 04 2019. ISSN 1740-9705.\n* Jordan et al. [1999] M. I. Jordan, Z. Ghahramani, T. S. Jaakkola, and L. K. Saul. An introduction to variational methods for graphical models. _Machine learning_, 37:183-233, 1999.\n* Kingma and Ba [2015] D. P. Kingma and J. Ba. Adam: A method for stochastic optimization. In _ICLR_, 2015.\n* Knoblauch et al. [2022] J. Knoblauch, J. Jewson, and T. Damoulas. An optimization-centric view on Bayes' rule: Reviewing and generalizing variational inference. _Journal of Machine Learning Research_, 23:1-109, 2022.\n* Kucukelbir et al. [2016] A. Kucukelbir, D. Tran, R. Ranganath, A. Gelman, and D. M. Blei. Automatic Differentiation Variational Inference. art. arXiv:1603.00788, March 2016.\n* Lin et al. [2019] W. Lin, M. Emtiyaz Khan, and M. Schmidt. Fast and Simple Natural-Gradient Variational Inference with Mixture of Exponential-family Approximations. _arXiv e-prints_, art. arXiv:1906.02914, June 2019. doi: 10.48550/arXiv.1906.02914.\n* Loizou et al. [2021] N. Loizou, S. Vaswani, I. Laradji, and S. Lacoste-Julien. Stochastic polyak step-size for sgd: An adaptive learning rate for fast convergence. _International Conference on Artificial Intelligence and Statistics_, 2021.\n* Magnusson et al. [2022] M. Magnusson, P. Burkner, and A. Vehtari. posterioridb: a set of posteriors for Bayesian inference and probabilistic programming, November 2022. URL [https://github.com/stan-dev/posteriordb](https://github.com/stan-dev/posteriordb).\n\n* [25] Charles C. Margossian. A review of automatic differentiation and its efficient implementation. _Wiley interdisciplinary reviews: data mining and knowledge discovery_, 9, 3 2019. doi: 10.1002/WIDM.1305.\n* [26] T. Minka. Expectation propagation for approximate Bayesian inference. In _Uncertainty in Artificial Intelligence (UAI)_, pages 362-369, 2001.\n* [27] C. Naesseth, F. Lindsten, and D. Blei. Markovian score climbing: Variational inference with \\(\\text{KL}(\\text{p}\\,\\|\\,\\text{q})\\). In _Neural Information Processing Systems_, 2020.\n* [28] R. Ranganath, S. Gerrish, and D. M. Blei. Black Box Variational Inference. art. arXiv:1401.0118, December 2013.\n* [29] R. Ranganath, J. Altosaar, D. Tran, and D. Blei. Operator variational inference. In _Neural Information Processing Systems_. 2016.\n* [30] E. Roualdes, B. Ward, S. Axen, and B. Carpenter. BridgeStan: Efficient in-memory access to Stan programs through Python, Julia, and R, March 2023. URL [https://github.com/roualdes/bridgestan](https://github.com/roualdes/bridgestan).\n* [31] J. Salvatier, Wiecki T., and C. Fonnesbeck. Probabilistic programming in Python using PyMC3. _PeerJ Computer Science_, 2016.\n* [32] U. Seljak and B. Yu. Posterior inference unchained with EL_2O. art. arXiv:1901.04454, January 2019.\n* [33] M. W., M. R. Andersen, A. Vehtari, and J. H. Huggins. Robust, automated, and accurate black-box variational inference. _arXiv:2203.15945_, 2022.\n* [34] Y. Yang, R. Martin, and H. Bondell. Variational approximations using Fisher divergence. _arXiv e-prints_, art. arXiv:1905.05284, May 2019.\n* [35] L. Zhang, B. Carpenter, A. Gelman, and A. Vehtari. Pathfinder: Parallel quasi-newton variational inference. _Journal of Machine Learning Research_, 23(306):1-49, 2022.\n\n## Appendix A Proof of Lemma 2.1\n\n**Lemma 2.1**.: Let \\(\\Theta\\subset\\mathbb{R}^{d}\\) be a set that contains the support of \\(p(\\mathbf{\\theta},\\mathbf{x})\\) and \\(q_{\\mathbf{w}^{*}}(\\mathbf{\\theta})\\) for some \\(\\mathbf{w}^{*}\\). That is \\(p(\\mathbf{\\theta},\\mathbf{x})=q_{\\mathbf{w}^{*}}(\\mathbf{\\theta})=0\\) for every \\(\\theta\\not\\in\\Theta\\). Furthermore, suppose that the support of \\(p(\\mathbf{\\theta},\\mathbf{x})\\) and \\(q_{\\mathbf{w}^{*}}(\\mathbf{\\theta})\\), and \\(\\Theta\\) is path-connected. The parameter \\(\\mathbf{w}^{*}\\) satisfies\n\n\\[\\nabla_{\\mathbf{\\theta}}\\log\\,q_{\\mathbf{w}^{*}}(\\mathbf{\\theta})=\\nabla_{\\mathbf{\\theta}}\\log \\,p(\\mathbf{\\theta},\\mathbf{x}),\\qquad\\forall\\mathbf{\\theta}\\in\\Theta, \\tag{3}\\]\n\nif and only if \\(\\mathbf{w}^{*}\\) also satisfies Eq. 1.\n\nProof.: (1) \\(\\implies\\)(3): Differentiating both sides of (1) in \\(\\mathbf{\\theta}\\) gives\n\n\\[\\nabla_{\\theta}\\log q_{\\mathbf{w}^{*}}(\\mathbf{\\theta}) =\\nabla_{\\mathbf{\\theta}}\\log p(\\mathbf{\\theta}|\\mathbf{x})=\\nabla_{\\theta} \\left(\\log p(\\mathbf{\\theta},\\mathbf{x})-\\log p(\\mathbf{x})\\right)\\] \\[=\\nabla_{\\theta}\\log p(\\mathbf{\\theta},\\mathbf{x}),\\qquad\\forall\\mathbf{ \\theta}\\in\\Theta.\\]\n\n(3) \\(\\implies\\)(1): Let \\(\\mathbf{\\theta}_{0}\\) be any arbitrary point in \\(\\Theta\\). Because \\(\\Theta\\) is path-connected, every \\(\\mathbf{\\theta}\\in\\Theta\\) is connected to \\(\\mathbf{\\theta}_{0}\\) via a differentiable path \\(\\mathbf{r}(t)\\) where \\(\\mathbf{r}(0)=\\mathbf{\\theta}_{0}\\) and \\(\\mathbf{r}(1)=\\mathbf{\\theta}\\). Integrating both sides of (3) along this path \\(\\mathbf{r}(t)\\), using again that \\(\\nabla_{\\theta}\\log p(\\mathbf{\\theta}|\\mathbf{x})=\\nabla_{\\theta}\\log p(\\mathbf{\\theta}, \\mathbf{x})\\), and using the Fundamental Theorem of Calculus gives\n\n\\[\\log q_{\\mathbf{w}^{*}}(\\mathbf{\\theta})-\\log q_{\\mathbf{w}^{*}}(\\mathbf{\\theta}_ {0}) =\\int_{0}^{1}\\left\\langle\\nabla_{\\theta}\\log q_{\\mathbf{w}^{*}}( \\mathbf{r}(t)),\\mathbf{r}^{\\prime}(t)\\right\\rangle dt\\] \\[=\\int_{0}^{1}\\left\\langle\\nabla_{\\theta}\\log p((\\mathbf{r}(t))| \\mathbf{x}),\\mathbf{r}^{\\prime}(t)\\right\\rangle dt\\] \\[=\\log p(\\mathbf{\\theta}|\\mathbf{x})-\\log p(\\mathbf{\\theta}_{0}|\\mathbf{x}),\\quad \\forall\\mathbf{\\theta}\\in\\Theta.\\]\n\nRearranging and defining \\(C:=\\log q_{\\mathbf{w}^{*}}(\\mathbf{\\theta}_{0})-\\log p(\\mathbf{\\theta}_{0}|\\mathbf{x})\\) gives\n\n\\[\\log q_{\\mathbf{w}^{*}}(\\mathbf{\\theta})=\\log p(\\mathbf{\\theta}|\\mathbf{x})+C,\\qquad\\forall \\mathbf{\\theta}\\in\\Theta,\\]By exponentiating both sides and integrating in \\(\\mathbf{\\theta}\\) over \\(\\Theta\\) we have that\n\n\\[1=\\int_{\\mathbf{\\theta}\\in\\Theta}q_{\\mathbf{w}^{*}}(\\mathbf{\\theta})d\\mathbf{\\theta}=e^{C}\\int_{ \\mathbf{\\theta}\\in\\Theta}p(\\mathbf{\\theta}|\\mathbf{x})d\\mathbf{\\theta}=e^{C},\\]\n\nwhere we used that \\(\\Theta\\) contains the support of both \\(p(\\mathbf{\\theta}|\\mathbf{x})\\) and \\(q_{\\mathbf{w}^{*}}(\\mathbf{\\theta})\\). Consequently \\(C=0\\), which gives our result. \n\n## Appendix B Proof of Theorem 2.2\n\nHere we give the proof for Theorem 2.2. We also re-introduce the theorem with a simplified notation, where we use \\((\\mathbf{\\mu}_{0},\\mathbf{\\Sigma}_{0})\\) to denote the mean and covariance at the previous time step of the method, thus dropping the iteration counter \\(t\\).\n\n**Theorem B.1**.: **(GSM updates)** Let \\(p(\\mathbf{\\theta},\\mathbf{x})\\) be given for some \\(\\mathbf{\\theta}\\in\\mathbb{R}^{d}\\), and let \\(q_{0}(\\mathbf{\\theta})\\) and \\(q(\\mathbf{\\theta})\\) be the multivariate normal distributions, respectively, with means \\(\\mathbf{\\mu}_{0}\\) and \\(\\mathbf{\\mu}\\) and covariance matrices \\(\\mathbf{\\Sigma}_{0}\\) and \\(\\mathbf{\\Sigma}\\). We seek the distribution\n\n\\[\\operatorname*{arg\\,min}_{\\mathbf{\\mu},\\mathbf{\\Sigma}=\\mathbf{\\Sigma}^{\\top}}\\left[ \\operatorname{KL}(q_{0},q)\\right]\\quad\\text{such that}\\quad\\nabla_{\\mathbf{\\theta }}\\log q(\\mathbf{\\theta})=\\nabla_{\\mathbf{\\theta}}\\log p(\\mathbf{\\theta},\\mathbf{x}). \\tag{11}\\]\n\nAs shorthand, let \\(\\mathbf{g}\\;:=\\;\\nabla_{\\mathbf{\\theta}}\\log p(\\mathbf{\\theta},\\mathbf{x}),\\) and let \\(\\rho\\) be the positive root of the quadratic equation\n\n\\[\\rho(1\\!+\\!\\rho)\\;=\\;\\mathbf{g}^{\\top}\\mathbf{\\Sigma}_{0}\\mathbf{g}+\\left[(\\mathbf{\\mu}_{0}\\!- \\!\\mathbf{\\theta})^{\\top}\\!\\mathbf{g}\\right]^{2}. \\tag{12}\\]\n\nThen the solution is given by the following closed-form updates:\n\n\\[\\mathbf{\\mu} = \\mathbf{\\mu}_{0}\\;+\\frac{1}{1+\\rho}\\left[\\mathbf{I}-\\frac{(\\mathbf{\\mu}_{ 0}-\\mathbf{\\theta})\\mathbf{g}^{\\top}}{1+\\rho+(\\mathbf{\\mu}_{0}-\\mathbf{\\theta})^{\\top}\\mathbf{g}} \\right]\\mathbf{\\Sigma}_{0}\\left(\\mathbf{g}-\\nabla_{\\mathbf{\\theta}}\\log q_{0}(\\mathbf{\\theta}) \\right), \\tag{13}\\] \\[\\mathbf{\\Sigma} = \\mathbf{\\Sigma}_{0}+(\\mathbf{\\mu}_{0}-\\mathbf{\\theta})(\\mathbf{\\mu}_{0}-\\mathbf{ \\theta})^{\\top}-(\\mathbf{\\mu}-\\mathbf{\\theta})(\\mathbf{\\mu}-\\mathbf{\\theta})^{\\top}. \\tag{14}\\]\n\nFurthermore, if \\(\\mathbf{\\Sigma}_{0}\\) is symmetric positive definite then so is \\(\\mathbf{\\Sigma}\\).\n\nProof.: The constraint in this optimization is given by\n\n\\[\\mathbf{g} = \\nabla_{\\mathbf{\\theta}}\\log q(\\mathbf{\\theta}) \\tag{15}\\] \\[= \\nabla_{\\mathbf{\\theta}}\\left[-\\tfrac{1}{2}(\\mathbf{\\theta}-\\mathbf{\\mu})\\mathbf{ \\Sigma}^{-1}(\\mathbf{\\theta}-\\mathbf{\\mu})-\\tfrac{1}{2}\\log\\left((2\\pi)^{d}|\\mathbf{ \\Sigma}|\\right)\\right]\\] (16) \\[= -\\mathbf{\\Sigma}^{-1}(\\mathbf{\\theta}-\\mathbf{\\mu}). \\tag{17}\\]\n\nThe KL divergence is given by\n\n\\[\\operatorname{KL}(q_{0},q)\\;=\\;\\frac{1}{2}\\left\\{\\mathbf{tr}[\\mathbf{\\Sigma}^{-1} \\mathbf{\\Sigma}_{0}]+\\log\\frac{|\\mathbf{\\Sigma}|}{|\\mathbf{\\Sigma}_{0}|}+(\\mathbf{\\mu}-\\mathbf{\\mu }_{0})^{\\top}\\mathbf{\\Sigma}^{-1}(\\mathbf{\\mu}-\\mathbf{\\mu}_{0})-d\\right\\}. \\tag{18}\\]\n\nDropping irrelevant terms from the optimization, we obtain the Lagrangian\n\n\\[\\mathcal{L}(\\mathbf{\\mu},\\mathbf{\\Sigma},\\mathbf{\\lambda})\\;=\\;\\frac{1}{2}\\left\\{\\mathbf{ tr}[\\mathbf{\\Sigma}^{-1}\\mathbf{\\Sigma}_{0}]-\\log|\\mathbf{\\Sigma}^{-1}|+(\\mathbf{\\mu}-\\mathbf{\\mu}_{0 })^{\\top}\\mathbf{\\Sigma}^{-1}(\\mathbf{\\mu}-\\mathbf{\\mu}_{0})\\right\\}+\\mathbf{\\lambda}^{\\top} \\big{(}\\mathbf{g}-\\mathbf{\\Sigma}^{-1}(\\mathbf{\\mu}-\\mathbf{\\theta})\\big{)}. \\tag{19}\\]\n\nIt is easier to optimize the matrix \\(\\mathbf{\\Sigma}^{-1}\\) instead of \\(\\mathbf{\\Sigma}\\). We can enforce the symmetry of \\(\\mathbf{\\Sigma}^{-1}\\) by writing\n\n\\[\\mathbf{\\Sigma}^{-1}\\;=\\;\\tfrac{1}{2}\\left(\\mathbf{\\Phi}+\\mathbf{\\Phi}^{\\top}\\right) \\tag{20}\\]\n\nand performing an unconstrained optimization over \\(\\mathbf{\\Phi}\\). With respect to the latter, the gradients of the Lagrangian are given by\n\n\\[\\frac{\\partial\\mathcal{L}}{\\partial\\Phi_{ij}}\\;=\\;\\sum_{kl}\\left(\\frac{ \\partial\\mathcal{L}}{\\partial\\Sigma_{kl}^{-1}}\\right)\\left(\\frac{\\partial \\Sigma_{kl}^{-1}}{\\partial\\Phi_{ij}}\\right)\\;=\\;\\sum_{kl}\\left(\\frac{\\partial \\mathcal{L}}{\\partial\\Sigma_{kl}^{-1}}\\right)\\left(\\frac{1}{2}\\delta_{ki} \\delta_{lj}+\\frac{1}{2}\\delta_{kj}\\delta_{li}\\right)\\;=\\;\\frac{1}{2}\\left( \\frac{\\partial\\mathcal{L}}{\\partial\\Sigma_{ij}^{-1}}+\\frac{\\partial\\mathcal{L }}{\\partial\\Sigma_{ji}^{-1}}\\right). \\tag{21}\\]Next we examine where the gradients of the Lagrangian vanish:\n\n\\[\\mathbf{0}=\\frac{\\partial\\mathcal{L}}{\\partial\\mathbf{\\mu}}\\;\\;\\Longrightarrow\\;\\; \\mathbf{0}\\;=\\;\\mathbf{\\Sigma}^{-1}(\\mathbf{\\mu}-\\mathbf{\\mu}_{0})\\,-\\,\\mathbf{\\Sigma}^{-1} \\mathbf{\\lambda}\\;\\;\\;\\Longrightarrow\\;\\;\\;\\boxed{\\mathbf{\\lambda}=\\mathbf{\\mu}-\\mathbf{\\mu}_{0 }} \\tag{22}\\]\n\n\\[\\mathbf{0}=\\frac{\\partial\\mathcal{L}}{\\partial\\mathbf{\\lambda}}\\;\\;\\Longrightarrow\\; \\;\\mathbf{0}\\;=\\;\\mathbf{g}-\\mathbf{\\Sigma}^{-1}(\\mathbf{\\mu}-\\mathbf{\\theta})\\;\\;\\;\\Longrightarrow \\;\\;\\;\\boxed{\\mathbf{\\mu}-\\mathbf{\\theta}=\\mathbf{\\Sigma}\\mathbf{g}} \\tag{23}\\]\n\n\\[\\mathbf{0}=\\frac{\\partial\\mathcal{L}}{\\partial\\mathbf{\\Phi}}\\;\\;\\Longrightarrow\\; \\;\\mathbf{0}\\;=\\;\\mathbf{\\Sigma}_{0}-\\mathbf{\\Sigma}+(\\mathbf{\\mu}-\\mathbf{\\mu}_{0})(\\mathbf{\\mu} -\\mathbf{\\mu}_{0})^{\\top}-\\left[\\mathbf{\\lambda}(\\mathbf{\\mu}-\\mathbf{\\theta})^{\\top}+(\\mathbf{\\mu }-\\mathbf{\\theta})\\mathbf{\\lambda}^{\\top}\\right], \\tag{24}\\]\n\n\\[\\Longrightarrow\\;\\;\\boxed{\\mathbf{\\Sigma}\\;=\\;\\mathbf{\\Sigma}_{0}\\,+\\,(\\mathbf{\\mu}-\\mathbf{ \\mu}_{0})(\\mathbf{\\mu}-\\mathbf{\\mu}_{0})^{\\top}\\,-\\,\\mathbf{\\lambda}(\\mathbf{\\mu}-\\mathbf{\\theta} )^{\\top}\\,-\\,(\\mathbf{\\mu}-\\mathbf{\\theta})\\mathbf{\\lambda}^{\\top}} \\tag{25}\\]\n\nWe claim that these equations (though nonlinear) can be solved in closed form. The first step is to eliminate \\(\\mathbf{\\lambda}\\) from eq. (25) using eq. (22). In this way we find\n\n\\[\\mathbf{\\Sigma} = \\mathbf{\\Sigma}_{0}+(\\mathbf{\\mu}-\\mathbf{\\mu}_{0})(\\mathbf{\\mu}-\\mathbf{\\mu}_{0})^{ \\top}-(\\mathbf{\\mu}-\\mathbf{\\mu}_{0})(\\mathbf{\\mu}-\\mathbf{\\theta})^{\\top}-(\\mathbf{\\mu}-\\mathbf{\\theta })(\\mathbf{\\mu}-\\mathbf{\\mu}_{0})^{\\top} \\tag{26}\\] \\[= \\mathbf{\\Sigma}_{0}-\\mathbf{\\mu}\\mathbf{\\mu}^{\\top}+\\mathbf{\\mu}\\mathbf{\\theta}^{\\top }+\\mathbf{\\theta}\\mathbf{\\mu}^{\\top}+\\mathbf{\\mu}_{0}\\mathbf{\\mu}_{0}^{\\top}-\\mathbf{\\mu}_{0}\\mathbf{ \\theta}^{\\top}-\\mathbf{\\theta}\\mathbf{\\mu}_{0}^{\\top}\\] (27) \\[= \\mathbf{\\Sigma}_{0}+(\\mathbf{\\mu}_{0}-\\mathbf{\\theta})(\\mathbf{\\mu}_{0}-\\mathbf{ \\theta})^{\\top}-(\\mathbf{\\mu}-\\mathbf{\\theta})(\\mathbf{\\mu}-\\mathbf{\\theta})^{\\top}. \\tag{28}\\]\n\nIt is worth highlighting the form of this equation:\n\n\\[\\mathbf{\\Sigma}\\;=\\;\\mathbf{\\Sigma}_{0}+(\\mathbf{\\mu}_{0}-\\mathbf{\\theta})(\\mathbf{\\mu}_{0}-\\mathbf{ \\theta})^{\\top}-(\\mathbf{\\mu}-\\mathbf{\\theta})(\\mathbf{\\mu}-\\mathbf{\\theta})^{\\top}\\]\n\nThis is a simple rank-two update for \\(\\mathbf{\\Sigma}\\). Note that \\(\\mathbf{\\Sigma}=\\mathbf{\\Sigma}_{0}\\) if \\(\\mathbf{\\mu}=\\mathbf{\\mu}_{0}\\); also, the solution for \\(\\mathbf{\\Sigma}\\) is determined by the solution for \\(\\mathbf{\\mu}\\).\n\nUltimately we must solve for \\(\\mathbf{\\mu}\\), but first it is useful to solve for the intermediate quantity \\(\\mathbf{g}^{\\top}\\mathbf{\\Sigma}\\mathbf{g}>0\\). From eq. (28), we obtain\n\n\\[\\mathbf{g}^{\\top}\\mathbf{\\Sigma}\\mathbf{g}\\;=\\;\\mathbf{g}^{\\top}\\mathbf{\\Sigma}_{0}\\mathbf{g}+\\left[( \\mathbf{\\mu}_{0}-\\mathbf{\\theta})^{\\top}\\mathbf{g}\\right]^{2}-\\left[(\\mathbf{\\mu}-\\mathbf{\\theta })^{\\top}\\mathbf{g}\\right]^{2}, \\tag{29}\\]\n\nand from eq. (23), we obtain\n\n\\[\\mathbf{g}^{\\top}\\mathbf{\\Sigma}\\mathbf{g}\\;=\\;\\mathbf{g}^{\\top}\\mathbf{\\Sigma}_{0}\\mathbf{g}+\\left[( \\mathbf{\\mu}_{0}-\\mathbf{\\theta})^{\\top}\\mathbf{g}\\right]^{2}-\\left(\\mathbf{g}^{\\top}\\mathbf{ \\Sigma}\\mathbf{g}\\right)^{2}. \\tag{30}\\]\n\nAs shorthand, let \\(\\rho=\\mathbf{g}^{\\top}\\mathbf{\\Sigma}\\mathbf{g}\\). Then from eq. (30) we see that \\(\\rho\\) satisfies the quadratic equation\n\n\\[\\rho(1\\!+\\!\\rho)\\;=\\;\\mathbf{g}^{\\top}\\mathbf{\\Sigma}_{0}\\mathbf{g}+\\left[(\\mathbf{\\mu}_{0}- \\mathbf{\\theta})^{\\top}\\mathbf{g}\\right]^{2}.\\]\n\nNote that there are no unknowns on the right side of this equation. The correct solution is given by the positive root since \\(\\rho=\\mathbf{g}^{\\top}\\mathbf{\\Sigma}\\mathbf{g}>0\\). Also note that \\(\\rho=(\\mathbf{\\mu}-\\mathbf{\\theta})^{\\top}\\mathbf{g}\\) from eq. (23).\n\nIt is useful to define one final intermediate quantity before solving for \\(\\mathbf{\\mu}\\). Let\n\n\\[\\mathbf{\\varepsilon}_{0}\\;=\\;\\mathbf{\\Sigma}_{0}\\mathbf{g}-\\mathbf{\\mu}_{0}+\\mathbf{\\theta}.\\]\n\nNote that \\(\\mathbf{\\varepsilon}_{0}\\) simply measures the degree to which the parameters of \\(q_{0}(\\mathbf{\\theta})\\) violate the desired constraint \\(\\nabla_{\\mathbf{w}}\\log q(\\mathbf{\\theta})=\\nabla_{\\mathbf{w}}\\log p(\\mathbf{\\theta},\\mathbf{y})\\). Put another way, if \\(\\mathbf{\\varepsilon}_{0}=\\mathbf{0}\\), then we have the trivial solution \\(\\mathbf{\\mu}=\\mathbf{\\mu}_{0}\\) and \\(\\mathbf{\\Sigma}=\\mathbf{\\Sigma}_{0}\\).\n\nNow we have everything to express the solution for \\(\\mathbf{\\mu}\\) in a highly intuitive form; in particular, it will be immediately evident that \\(\\mathbf{\\mu}\\rightarrow\\mathbf{\\mu}_{0}\\) as \\(\\mathbf{\\varepsilon}_{0}\\rightarrow\\mathbf{0}\\). Starting from eqs. (23) and (28), we find\n\n\\[\\mathbf{\\mu}-\\mathbf{\\mu}_{0} = \\mathbf{\\theta}-\\mathbf{\\mu}_{0}+\\mathbf{\\Sigma}\\mathbf{g}, \\tag{31}\\] \\[= \\mathbf{\\theta}-\\mathbf{\\mu}_{0}+\\left[\\mathbf{\\Sigma}_{0}+(\\mathbf{\\mu}_{0}-\\bm {\\theta})(\\mathbf{\\mu}_{0}-\\mathbf{\\theta})^{\\top}-(\\mathbf{\\mu}-\\mathbf{\\theta})(\\mathbf{\\mu}-\\mathbf{ \\theta})^{\\top}\\right]\\mathbf{g},\\] (32) \\[= \\mathbf{\\varepsilon}_{0}\\,+\\,(\\mathbf{\\mu}_{0}-\\mathbf{\\theta})(\\mathbf{\\mu}_{0}- \\mathbf{\\theta})^{\\top}\\mathbf{g}-(\\mathbf{\\mu}-\\mathbf{\\theta})(\\mathbf{\\mu}-\\mathbf{\\theta})^{\\top} \\mathbf{g},\\] (33) \\[= \\mathbf{\\varepsilon}_{0}\\,+\\,(\\mathbf{\\mu}_{0}-\\mathbf{\\theta})(\\mathbf{\\mu}_{0}- \\mathbf{\\theta})^{\\top}\\mathbf{g}\\,-\\,(\\mathbf{\\mu}-\\mathbf{\\mu}_{0}+\\mathbf{\\mu}_{0}-\\mathbf{\\theta})( \\mathbf{\\mu}-\\mathbf{\\theta})^{\\top}\\mathbf{g},\\] (34) \\[= \\mathbf{\\varepsilon}_{0}\\,-\\,\\rho(\\mathbf{\\mu}-\\mathbf{\\mu}_{0})\\,+\\,(\\mathbf{\\mu}_ {0}-\\mathbf{\\theta})[(\\mathbf{\\mu}_{0}-\\mathbf{\\theta})-(\\mathbf{\\mu}-\\mathbf{\\theta})^{\\top}\\mathbf{g},\\] (35) \\[= \\mathbf{\\varepsilon}_{0}\\,-\\,\\rho(\\mathbf{\\mu}-\\mathbf{\\mu}_{0})\\,+\\,(\\mathbf{\\mu} _{0}-\\mathbf{\\theta})(\\mathbf{\\mu}_{0}-\\mathbf{\\mu})^{\\top}\\mathbf{g},\\] (36) \\[= \\mathbf{\\varepsilon}_{0}\\,-\\,(\\rho\\mathbf{\\Gamma}+(\\mathbf{\\mu}_{0}-\\mathbf{\\theta} )\\mathbf{g}^{\\top})(\\mathbf{\\mu}_{0}-\\mathbf{\\mu}). \\tag{37}\\]Note what has happened here: eq. (32) is a system of _nonlinear_ equations for \\(\\mu\\), but in eq. (37), all the nonlinearity has been expressed in terms of \\(\\rho\\). Since \\(\\rho\\) can be determined via eq. (30), we arrive effectively at a system of _linear_ equations for \\(\\mu\\). Collecting terms, we obtain\n\n\\[\\left[(1+\\rho)\\mathbf{I}+(\\mathbf{\\mu}_{0}-\\mathbf{\\theta}) \\mathbf{g}^{\\top}\\right](\\mathbf{\\mu}-\\mathbf{\\mu} _{0})\\;=\\;\\mathbf{\\varepsilon}_{0}. \\tag{38}\\]\n\nWe thus arrive at the closed-form update\n\n\\[\\mathbf{\\mu}\\;=\\;\\mathbf{\\mu}_{0}\\,+\\,\\left[(1+\\rho)\\mathbf{ I}+(\\mathbf{\\mu}_{0}-\\mathbf{\\theta})\\mathbf{g}^{\\top} \\right]^{-1}\\mathbf{\\varepsilon}_{0} \\tag{39}\\]\n\nIt is evident from this update that \\(\\mathbf{\\mu}\\rightarrow\\mathbf{\\mu}_{0}\\) as \\(\\mathbf{\\varepsilon}_{0}\\rightarrow\\mathbf{0}\\). The matrix inverse in this update can also be computed efficiently from the Woodbury matrix identity.\n\nIn sum, the joint update for \\(\\mu\\) and \\(\\Sigma\\) can be efficiently computed as follows:\n\n1. Set \\(\\mathbf{g}=\\nabla_{\\mathbf{w}}\\log p(\\mathbf{\\theta },\\mathbf{y})\\) and \\(\\mathbf{\\varepsilon}_{0}\\;=\\;\\mathbf{\\Sigma}_{0}\\mathbf{g}-\\mathbf{\\mu}_{0}+\\mathbf{\\theta}\\).\n2. Solve \\(\\rho(1+\\rho)\\;=\\;\\mathbf{g}^{\\top}\\mathbf{\\Sigma}_{0}\\mathbf{g}+\\left[(\\mathbf{\\mu}_{0}-\\mathbf{\\theta})^{\\top }\\mathbf{g}\\right]^{2}\\) for \\(\\rho>0\\).\n3. Compute \\(\\mathbf{\\mu}\\;=\\;\\mathbf{\\mu}_{0}\\,+\\,\\left[(1+\\rho)\\mathbf{ I}+(\\mathbf{\\mu}_{0}-\\mathbf{\\theta})\\mathbf{g}^{\\top} \\right]^{-1}\\mathbf{\\varepsilon}_{0}\\).\n4. Compute \\(\\mathbf{\\Sigma}\\;=\\;\\mathbf{\\Sigma}_{0}+(\\mathbf{ \\mu}_{0}-\\mathbf{\\theta})(\\mathbf{\\mu}_{0}-\\mathbf{ \\theta})^{\\top}-(\\mathbf{\\mu}-\\mathbf{\\theta})(\\mathbf{\\mu}-\\mathbf{\\theta})^{\\top}\\).\n\nNote that it straightforward to solve for the value of \\(\\rho\\) in step 2 of this procedure. In particular, from the quadratic formula, we obtain the positive root\n\n\\[\\boxed{\\rho\\;=\\;\\frac{\\sqrt{1+4(\\mathbf{g}^{\\top}\\mathbf{ \\Sigma}_{0}\\mathbf{g}+\\left[(\\mathbf{\\mu}_{0}-\\mathbf{\\theta})\\mathbf{\\overline{g}}\\right]^{2})-1}}{2}.}\\]\n\nWe can solve the linear equation for \\(\\mu\\) in step 3 of this procedure most efficiently by using the Sherman-Morrison formula\n\n\\[\\left[a\\mathbf{I}+\\mathbf{u}\\mathbf{g}^{\\top}\\right]^{-1}= \\frac{1}{a}\\left(\\mathbf{I}-\\frac{\\mathbf{u}\\mathbf{g}^{\\top }}{a+\\mathbf{u}^{\\top}\\mathbf{g}}\\right),\\quad\\mbox{for every }\\mathbf{u},\\mathbf{g},a.\\]\n\nApplying this formula in step 3, we find that\n\n\\[\\mathbf{\\mu}\\;=\\;\\mathbf{\\mu}_{0}\\,+\\frac{1}{1+\\rho}\\left[ \\mathbf{I}-\\frac{(\\mathbf{\\mu}_{0}-\\mathbf{\\theta})\\mathbf{g}^{\\top}}{1+\\rho+(\\mathbf{\\mu}_{0}-\\mathbf{ \\theta})^{\\top}\\mathbf{g}}\\right]\\mathbf{\\varepsilon}_{0}. \\tag{40}\\]\n\nNext we relate this update to the difference in the scores of the target distribution and the current estimate of the variational approximation. Since \\(\\nabla_{\\mathbf{\\theta}}\\log q_{0}(\\mathbf{\\theta})=-\\mathbf{\\Sigma}_{0}^{-1}(\\mathbf{\\theta}-\\mathbf{\\mu}_{0})\\), we have that\n\n\\[\\mathbf{\\varepsilon}_{0}=\\mathbf{\\Sigma}_{0}\\left(\\mathbf{g}-\\mathbf{\\Sigma}_{0}^{-1}(\\mathbf{\\mu}_{0}+ \\mathbf{\\theta})\\right)=\\mathbf{\\Sigma}_{0}\\left(\\mathbf{g}-\\nabla_{\\mathbf{\\theta}}\\log q_{0}(\\mathbf{ \\theta})\\right).\\]\n\nIt follows, by substituting the above expression into Eq. 40, that\n\n\\[\\boxed{\\mathbf{\\mu}\\;=\\;\\mathbf{\\mu}_{0}\\,+\\frac{1}{1+\\rho} \\left[\\mathbf{I}-\\frac{(\\mathbf{\\mu}_{0}-\\mathbf{\\theta}) \\mathbf{g}^{\\top}}{1+\\rho+(\\mathbf{\\mu}_{0}-\\mathbf{ \\theta})^{\\top}\\mathbf{g}}\\right]\\mathbf{\\Sigma}_{0}\\left( \\mathbf{\\nabla}_{\\mathbf{\\theta}}\\log p(\\mathbf{x}, \\mathbf{\\theta})-\\nabla_{\\mathbf{\\theta}}\\log q_{0}(\\mathbf{\\theta})\\right),}\\]\n\nwhich gives the update in the statement of the theorem.\n\nFinally we prove that if \\(\\mathbf{\\Sigma}_{0}\\) is positive definite, then so is \\(\\Sigma\\). To do so, we show explicitly that all of the eigenvalues of \\(\\Sigma\\) are positive. We begin by rewriting our results for \\(\\Sigma\\) in eq. (28) and \\(\\rho\\) in eq. (30) in a more convenient form. As shorthand, let\n\n\\[\\mathbf{M}_{0}\\;=\\;\\mathbf{\\Sigma}_{0}+(\\mathbf{\\mu}_{0}- \\mathbf{\\theta})(\\mathbf{\\mu}_{0}-\\mathbf{\\theta})^{ \\top}, \\tag{41}\\]\n\nso that \\(\\mathbf{M}_{0}\\) captures the first two terms on the right side of eq. (28). Note that \\(\\mathbf{M}_{0}\\) is positive-definite, a fact that we will exploit repeatedly in what follows. In addition, recall that \\(\\mathbf{\\mu}-\\mathbf{\\theta}=\\mathbf{\\Sigma}\\mathbf{g}\\) from eq. (23). Thus with this notation we can rewrite eqs. (28) and (30) as\n\n\\[\\mathbf{\\Sigma} = \\mathbf{M}_{0}-(\\mathbf{\\Sigma}\\mathbf{g})( \\mathbf{\\Sigma}\\mathbf{g})^{\\top}, \\tag{42}\\] \\[\\rho(1+\\rho) = \\mathbf{g}^{\\top}\\mathbf{M}_{0}\\mathbf{g}. \\tag{43}\\]Now let \\(\\mathbf{e}\\) be any normalized eigenvector of \\(\\mathbf{\\Sigma}\\); we want to show that its corresponding eigenvalue \\(\\lambda_{e}\\) is positive. From eq. (42), it follows that\n\n\\[\\lambda_{e} = \\mathbf{e}^{\\top}\\mathbf{\\Sigma}\\mathbf{e} \\tag{44}\\] \\[= \\mathbf{e}^{\\top}\\left[\\mathbf{M}_{0}-(\\mathbf{\\Sigma}\\mathbf{g})(\\mathbf{\\Sigma} \\mathbf{g})^{\\top}\\right]\\mathbf{e}\\] (45) \\[= \\mathbf{e}^{\\top}\\mathbf{M}_{0}\\mathbf{e}-\\lambda_{e}^{2}(\\mathbf{e}^{\\top} \\mathbf{g})^{2}. \\tag{46}\\]\n\nNote that if \\(\\mathbf{e}^{\\top}\\mathbf{g}=0\\), then it follows trivially that \\(\\lambda_{e}=\\mathbf{e}^{\\top}\\mathbf{M}_{0}\\mathbf{e}>0\\). So we only need to consider the non-trivial case \\(\\mathbf{e}^{\\top}\\mathbf{g}\\neq 0\\). To proceed, we note that\n\n\\[(\\mathbf{e}^{\\top}\\mathbf{M}_{0}\\mathbf{g})^{2}\\ =\\ (\\mathbf{e}^{\\top}\\mathbf{M}_{0}^{\\frac{1} {2}}\\mathbf{M}_{0}^{\\frac{1}{2}}\\mathbf{g})^{2}\\ \\leq\\ (\\mathbf{e}^{\\top}\\mathbf{M}_{0} \\mathbf{e})(\\mathbf{g}^{\\top}\\mathbf{M}_{0}\\mathbf{g}), \\tag{47}\\]\n\nwhere we have used the Cauchy-Schwartz inequality to bound \\((\\mathbf{e}^{\\top}\\mathbf{M}_{o}\\mathbf{g})\\) in terms of \\((\\mathbf{e}^{\\top}\\mathbf{M}_{0}\\mathbf{e})\\), the latter of which appears in eq. (46). Substituting this inequality into eq. (46), we find that\n\n\\[\\lambda_{e}\\ \\geq\\ \\frac{(\\mathbf{e}^{\\top}\\mathbf{M}_{0}\\mathbf{g})^{2}}{\\mathbf{g}^{ \\top}\\mathbf{M}_{0}\\mathbf{g}}\\ -\\ \\lambda_{e}^{2}(\\mathbf{e}^{\\top}\\mathbf{g})^{2}. \\tag{48}\\]\n\nTo prove that \\(\\lambda_{e}>0\\) we need one more intermediate result. Focusing on the rightmost term in this equality, we note that\n\n\\[\\lambda_{e}(\\mathbf{e}^{\\top}\\mathbf{g})\\ =\\ \\mathbf{e}^{\\top}\\Sigma\\mathbf{g}\\ =\\ \\mathbf{e}^{\\top} \\left[\\mathbf{M}_{0}-(\\mathbf{\\Sigma}\\mathbf{g})(\\mathbf{\\Sigma}\\mathbf{g})^{\\top}\\right]\\mathbf{g} \\ =\\ \\mathbf{e}^{\\top}\\mathbf{M}_{0}\\mathbf{g}-\\lambda_{e}(\\mathbf{e}^{\\top}\\mathbf{g})(\\mathbf{g}^{ \\top}\\mathbf{\\Sigma}\\mathbf{g}), \\tag{49}\\]\n\nand rearranging the terms in this equation, we find\n\n\\[\\mathbf{e}^{\\top}\\mathbf{M}_{0}\\mathbf{g}\\ =\\ \\lambda_{e}(\\mathbf{e}^{\\top}\\mathbf{g})(1+\\mathbf{g}^{ \\top}\\mathbf{\\Sigma}\\mathbf{g}). \\tag{50}\\]\n\nThis intermediate result is useful because it relates the two terms on the right side of eq. (48). In particular, using eq. (50) to eliminate the term \\(\\mathbf{e}^{\\top}\\mathbf{M}_{0}\\mathbf{g}\\) in eq. (48), we find:\n\n\\[\\lambda_{e} \\geq \\frac{\\left[\\lambda_{e}(\\mathbf{e}^{\\top}\\mathbf{g})(1+\\mathbf{g}^{\\top}\\mathbf{ \\Sigma}\\mathbf{g})\\right]^{2}}{\\mathbf{g}^{\\top}\\mathbf{M}_{0}\\mathbf{g}}\\ -\\ \\lambda_{e}^{2}(\\mathbf{e}^{ \\top}\\mathbf{g})^{2}\\] \\[= \\lambda_{e}^{2}(\\mathbf{e}^{\\top}\\mathbf{g})^{2}\\left[\\frac{(1+\\mathbf{g}^{ \\top}\\mathbf{\\Sigma}\\mathbf{g})^{2}}{\\mathbf{g}^{\\top}\\mathbf{M}_{0}\\mathbf{g}}\\ -\\ 1\\right]\\] \\[= \\lambda_{e}^{2}(\\mathbf{e}^{\\top}\\mathbf{g})^{2}\\left[\\frac{(1+\\rho)^{2} }{\\rho(1+\\rho)}\\ -\\ 1\\right]\\] \\[= \\frac{\\lambda_{e}^{2}(\\mathbf{e}^{\\top}\\mathbf{g})^{2}}{\\rho},\\] \\[> 0,\\]\n\nwhere the final inequality follows because the individual terms \\(\\lambda_{e}^{2}\\), \\((\\mathbf{e}^{\\top}\\mathbf{g})^{2}\\), and \\(\\rho\\) are all strictly positive; note that \\(\\lambda_{e}\\) cannot be equal to zero because this contradicts the equality in eq. (46). This completes the proof. Perhaps it is useful that this derivation also gives upper bounds on \\(\\lambda_{e}\\), namely\n\n\\[\\frac{1}{\\lambda_{e}}\\ \\geq\\ \\frac{(\\mathbf{e}^{\\top}\\mathbf{g})^{2}}{\\rho}\\quad \\Longrightarrow\\quad\\lambda_{e}\\ \\leq\\ \\frac{\\rho}{(\\mathbf{e}^{\\top}\\mathbf{g})^{2}}\\ =\\ \\frac{\\mathbf{g}^{\\top}\\mathbf{\\Sigma}\\mathbf{g}}{(\\mathbf{e}^{\\top}\\mathbf{g})^{2}}. \\tag{51}\\]", "response": "## Review\n\n### Summary\n\nThe paper presents a novel method for black-box variational inference (VI) based on score matching, specifically targeting Gaussian variational families. The approach derives a framework where the iterative updates of the variational distribution are guided by the alignment of gradients between the target and the variational distribution. This results in a method termed Gaussian Score Matching VI (GSM-VI), which is shown to outperform traditional BBVI in several experimental setups, particularly in terms of convergence speed and efficiency. However, the paper lacks a thorough theoretical analysis and detailed exploration of the methodology's performance beyond Gaussian distributions, with potential limitations regarding convergence in high-dimensional or non-Gaussian settings.\n\n### Strengths\n\n- The score matching approach is novel and effectively addresses variational inference.\n- The paper is well-structured and clearly communicates complex ideas.\n- GSM-VI shows significant performance improvements over standard BBVI in terms of convergence rates.\n- The experimental results are robust and demonstrate the method's practical applicability.\n- The availability of code promotes reproducibility.\n\n### Weaknesses\n\n- The paper is primarily empirical and lacks theoretical guarantees for the proposed method.\n- There is insufficient explanation for the strong empirical results, particularly regarding scaling with dimension and batch size.\n- The evaluation is limited to Gaussian distributions without exploring other potential variational families.\n- Important experimental details, including the dimensionality of benchmarks and hyperparameter settings, are not well-documented.\n- The paper does not adequately discuss the oscillatory behavior observed in experiments or its implications.\n\n### Questions\n\n- Why does GSM-VI demonstrate such effective scaling properties with dimensionality?\n- What are the measured running times for the experiments, considering the O(d2) time complexity of the update step?\n- Could the O(d2) update step negatively affect performance in very high-dimensional cases?\n- What were the batch sizes used for BBVI in the benchmarks?\n- How were the examples from the posteriordb selected and what is their dimensionality?\n- Could the authors elaborate on the scenarios where the optimization could result in local minima?\n- Can the paper clarify the implications of only reporting the Forward KL when comparing with other VI methods?\n\n### Soundness\n\n**Score:** 3\n\n**Description:** Good - The methodology is promising and demonstrates empirical effectiveness, but lacks theoretical backing and comprehensive analysis of limitations, impacting the overall soundness.\n\n### Presentation\n\n**Score:** 3\n\n**Description:** Good - The paper is generally well-written and clear, but could benefit from additional explanations and organization of results to enhance understanding.\n\n### Contribution\n\n**Score:** 3\n\n**Description:** Good - The approach presents a novel insight into variational inference, but the limited exploration of non-Gaussian settings somewhat diminishes the overall contribution.\n\n### Rating\n\n**Score:** 7\n\n**Description:** Accept: The paper demonstrates solid technical merit with a significant impact potential, but requires revisions to address theoretical gaps and clarify experimental details.\n\n### Paper Decision\n\n**Decision:** Accept (poster)\n\n**Reasons:** The paper presents an innovative framework for variational inference that shows promising empirical results. While it has some limitations, particularly regarding the theoretical analysis and exploration of non-Gaussian families, the overall contribution is significant. The clarity of presentation and the reproducibility of results further support the decision to accept the paper with minor revisions.\n"}
{"query": "You are a highly experienced, conscientious, and fair academic reviewer, please help me review this paper. The review should be organized into nine sections: \n1. Summary: A summary of the paper in 100-150 words.\n2. Strengths/Weaknesses/Questions: The Strengths/Weaknesses/Questions of paper, which should be listed in bullet points, with each point supported by specific examples from the article where possible.\n3. Soundness/Contribution/Presentation: Rate the paper's Soundness/Contribution/Presentation, and match this score to the corresponding description from the list below and provide the result. The possible scores and their descriptions are: \n    1 poor\n    2 fair\n    3 good\n    4 excellent\n4. Rating: Give this paper an appropriate rating, match this rating to the corresponding description from the list below and provide the result. The possible Ratings and their descriptions are: \n    1 strong reject\n    2 reject, significant issues present\n    3 reject, not good enough\n    4 possibly reject, but has redeeming facets\n    5 marginally below the acceptance threshold\n    6 marginally above the acceptance threshold\n    7 accept, but needs minor improvements \n    8 accept, good paper\n    9 strong accept, excellent work\n    10 strong accept, should be highlighted at the conference   \n5. Paper Decision: It must include the Decision itself(Accept or Reject) and the reasons for this decision, based on the criteria of originality, methodological soundness, significance of results, and clarity and logic of presentation.\n\nHere is the template for a review format, you must follow this format to output your review result:\n**Summary:**\nSummary content\n\n**Strengths:**\n- Strength 1\n- Strength 2\n- ...\n\n**Weaknesses:**\n- Weakness 1\n- Weakness 2\n- ...\n\n**Questions:**\n- Question 1\n- Question 2\n- ...\n\n**Soundness:**\nSoundness result\n\n**Presentation:**\nPresentation result\n\n**Contribution:**\nContribution result\n\n**Rating:**\nRating result\n\n**Paper Decision:**\n- Decision: Accept/Reject\n- Reasons: reasons content\n\n\nPlease ensure your feedback is objective and constructive. The paper is as follows:\n\n# Approximately Equivariant Graph Networks\n\n Ningyuan (Teresa) Huang\n\nJohns Hopkins University\n\nnhuang19@jhu.edu\n\n&Ron Levie\n\nTechnion - Israel Institute of Technology\n\nlevieron@technion.ac.il\n\n&Soledad Villar\n\nJohns Hopkins University\n\nsvillar3@jhu.edu\n\n###### Abstract\n\nGraph neural networks (GNNs) are commonly described as being permutation equivariant with respect to node relabeling in the graph. This symmetry of GNNs is often compared to the translation equivariance of Euclidean convolution neural networks (CNNs). However, these two symmetries are fundamentally different: The translation equivariance of CNNs corresponds to symmetries of the fixed domain acting on the image signals (sometimes known as _active symmetries_), whereas in GNNs any permutation acts on both the graph signals and the graph domain (sometimes described as _passive symmetries_). In this work, we focus on the active symmetries of GNNs, by considering a learning setting where signals are supported on a fixed graph. In this case, the natural symmetries of GNNs are the automorphisms of the graph. Since real-world graphs tend to be asymmetric, we relax the notion of symmetries by formalizing approximate symmetries via graph coarsening. We present a bias-variance formula that quantifies the tradeoff between the loss in expressivity and the gain in the regularity of the learned estimator, depending on the chosen symmetry group. To illustrate our approach, we conduct extensive experiments on image inpainting, traffic flow prediction, and human pose estimation with different choices of symmetries. We show theoretically and empirically that the best generalization performance can be achieved by choosing a suitably larger group than the graph automorphism, but smaller than the permutation group.\n\n## 1 Introduction\n\nGraph Neural Networks (GNNs) are popular tools to learn functions on graphs. They are commonly designed to be permutation equivariant since the node ordering can be arbitrary (in the matrix representation of a graph). Permutation equivariance serves as a strong geometric prior and allows GNNs to generalize well [1, 2, 3]. Yet in many applications, the node ordering across different graphs is matched or fixed a priori, such as a time series of social networks where the nodes identify the same users, or a set of skeleton graphs where the nodes represent the same joints. In such settings, the natural symmetries arise from graph automorphisms, which effectively only act on the graph signals; This is inherently different from the standard equivariance in GNNs that concerns all possible permutations acting on both the signals and the graph domain. A permutation of both the graph and the graph signal can be seen as a _change of coordinates_ since it does not change the object it represents, just the way to express it. This parallels the _passive symmetries_ in physics, where physical observables are independent of the coordinate system one uses to express them [4, 5]. In contrast, a permutation of the graph signal on a fixed graph potentially transforms the object itself, not only its representation. This parallels the _active symmetries_ in physics, where the coordinate system (in this case, the graph or domain) is fixed but a group transformation on the signal results in a predictabletransformation of the outcome (e.g., permuting left and right joints in the human skeleton changes a left-handed person to right-handed). The active symmetries on a fixed graph are similar to the translation equivariance symmetries in Euclidean convolutional neural networks (CNNs), where the domain is a fixed-size grid and the signals are images.\n\nIn this work, we focus on active symmetries in GNNs. Specifically, we consider a fixed graph domain \\(G\\) with \\(N\\) nodes, an adjacency matrix \\(A\\in\\mathbb{R}^{N\\times N}\\), and input graph signals \\(X\\in\\mathbb{R}^{N\\times d}\\). We are interested in learning equivariant functions \\(f\\) that satisfy (approximate) active symmetries\n\n\\[f(\\Pi X)\\approx\\Pi f(X)\\text{ for permutations }\\Pi\\in\\mathcal{G}\\subseteq\\mathcal{S} _{N}, \\tag{1}\\]\n\nwhere \\(\\mathcal{G}\\) is a subgroup of the permutation group \\(\\mathcal{S}_{N}\\) that depends on \\(G\\). For example, \\(\\mathcal{G}\\) can be the graph automorphism group \\(\\mathcal{A}_{G}=\\{\\Pi:\\Pi A=A\\Pi\\}\\). Each choice of \\(\\mathcal{G}\\) induces a hypothesis class \\(\\mathcal{H}_{\\mathcal{G}}\\) for \\(f\\): the smaller the group \\(\\mathcal{G}\\), the larger the class \\(\\mathcal{H}_{\\mathcal{G}}\\). We aim to select \\(\\mathcal{G}\\) so that the learned function \\(f\\in\\mathcal{H}_{\\mathcal{G}}\\) generalizes well, also known as the _model selection_ problem [6, Chp.4]. In contrast, standard graph learning methods (GNNs, spectral methods) are defined to satisfy passive symmetries, by treating \\(A\\) as input and requiring \\(f(\\Pi A\\Pi^{\\top},\\Pi X)=\\Pi f(A,X)\\) for all permutations \\(\\Pi\\in\\mathcal{S}_{N}\\) and all \\(N\\). But for the fixed graph setting, we argue that active symmetries are more relevant. Thus we use \\(A\\) to define the hypothesis class rather than treating it an input. By switching from passive symmetries to active symmetries, we will show how to design GNNs for signals supported on a fixed graph with different levels of expressivity and generalization properties.\n\nWhile enforcing symmetries has been shown to improve generalization when the symmetry group is known a priori [7, 8, 9, 10, 11, 12, 13], the problem of _symmetry model selection_ is not completely solved, particularly when the data lacks exact symmetries (see for instance [14, 15, 16] and references therein). Motivated by this, we study the symmetry model selection problem for learning on a fixed graph domain. This setting is particularly interesting since (1) the graph automorphism group serves as the natural _oracle_ symmetry; (2) real-world graphs tend to be asymmetric, but admit cluster structure or local symmetries. Therefore, we define _approximate symmetries of graphs_ using the cut distance between graphs from graphon analysis. An approximate symmetry of a graph \\(G\\) is a symmetry of any other graph \\(G^{\\prime}\\) that approximates \\(G\\) in the cut distance. In practice, we take \\(G^{\\prime}\\) as coarse-grainings (or clusterings) of \\(G\\), as these are typically guaranteed to be close in cut distance to \\(G\\). We show how to induce approximate symmetries for \\(G\\) via the automorphisms of \\(G^{\\prime}\\). Our main contributions include:\n\n1. We formalize the notion of active symmetries and approximate symmetries of GNNs for signals supported on a fixed graph domain, which allows us to study the symmetry group model selection problem. (See Sections 2, 4)\n2. We theoretically characterize the statistical risk depending on the hypothesis class induced from the symmetry group, and show a bias-variance tradeoff between the reduction in expressivity and the gain in regularity of the model. (See Sections 3, 4)\n3. We illustrate our approach empirically for image inpainting, traffic flow prediction, and human pose estimation. (See Section 5 for an overview, and Appendix D for the details on how to implement equivariant graph networks with respect to different symmetry groups).\n\n### Related Work\n\n**Graph Neural Networks and Equivariant Networks.** Graph Neural Networks (GNNs) [17, 18, 19] are typically permutation-equivariant (with respect to node relabeling). These include message-passing neural networks (MPNNs) [19, 20, 21], spectral GNNs [22, 23, 24, 18], and subgraph-based GNNs [25, 26, 27, 28, 29, 30]. Permutation equivariance in GNNs can extend to edge types [31] and higher-order tensors representing the graph structure [32, 33]. Equivariant networks generalize symmetries on graphs to other objects, such as sets [34], images [35], shapes [36, 37], point clouds [38, 39, 40, 41, 42], manifolds [43, 44], and physical systems [45, 46] among many others. Notably, many equivariant machine learning problems concern the setting where the domain is fixed, e.g., learning functions on a fixed sphere [47], and thus focus on active symmetries rather than passive symmetries such as the node ordering in standard GNNs. Yet learning on a fixed graph domain arises naturally in many applications such as molecular dynamics modeling [48]. This motivates us to consider active symmetries of GNNs for learning functions on a fixed graph. Our work is closely related to Natural Graph Networks (NGNs) in [49], which use global and local graph isomorphisms to design maximally expressive GNNs for distinguishing _different graphs_. In contrast, we focus on generalization and thus consider symmetry model selection on a _fixed_ graph.\n\n**Generalization of Graph Neural Networks and Equivariant Networks.** Most existing works focus on the graph-level tasks, where in-distribution generalization bounds of GNNs have been derived using statistical learning-theoretic measures such as Rademacher complexity [50] and Vapnik-Chervonenkis (VC) dimension [51, 52], or uniform generalization analysis based on random graph models [53]; Out-of-distribution generalization properties have been investigated using different notions including transferability [54, 55] (also known as size generalization [56]), and extrapolation [57]. In general, imposing symmetry constraints improves the sample complexity and the generalization error [7, 8, 9, 10, 11, 12, 13]. Recently, Petrache and Trivedi [58] investigated approximation-generalization tradeoffs using approximate symmetries for general groups. Their results are based on uniform convergence generalization bounds, which measure the worst-case performance of all functions in a hypothesis class and differ from our non-uniform analysis.\n\n**Approximate Symmetries.** For physical dynamical problems, Wang et al. [45] formalized approximate symmetries by relaxing exact equivariance to allow for a small equivariance error. For reinforcement learning applications, Finzi et al. [59] proposed Residual Pathway Priors to expand network layers into a sum of equivariant layers and non-equivariant layers, and thus relax strict equivariance into approximate equivariance priors. In the context of self-supervised learning, Suau et al. [60], Gupta et al. [61] proposed to learn structural latent representations that satisfy approximate equivariance. Inspired by randomized algorithms, Cotta et al. [62] formalizes probabilistic notions of invariances and universal approximation. In [63], scattering transforms (a specific realization of a CNN) are shown to be approximately invariant to small deformations in the image domain, which can be seen as a form of approximate symmetry. Scattering transforms on graphs are discussed in [64, 65, 66]. Additional discussions on approximate symmetries can be found in [67, Section 6].\n\n## 2 Problem Setup: Learning Equivariant Maps on a Fixed Graph\n\n**Notations.** We let \\(\\mathbb{R}\\), \\(\\mathbb{R}_{+}\\) denote the reals and the nonegative reals, \\(I\\) denote the identity matrix and \\(\\mathbb{1}\\) denote the all-ones matrix. For a matrix \\(Y\\in\\mathbb{R}^{N\\times k}\\), we write the Frobenious norm as \\(\\|Y\\|_{F}\\). We denote by \\(X\\odot Y\\) the element-wise multiplication of matrices \\(X\\) and \\(Y\\). We write \\([N]=\\{1,\\ldots,N\\}\\) and \\(\\mathcal{S}_{N}\\) as the permutation group on the set \\([N]\\). Groups are typically noted by calligraphic letters. Given \\(\\mathcal{H},\\mathcal{K}\\) groups, we denote a semidirect product by \\(\\mathcal{H}\\rtimes\\mathcal{K}\\). For a group \\(\\mathcal{G}\\) with representation \\(\\phi\\), we denote by \\(\\chi_{\\phi|_{\\mathcal{G}}}\\) the corresponding character (see Definition 6 in Appendix A.1). Denoting \\(\\mathcal{H}\\leq\\mathcal{G}\\) means that \\(\\mathcal{H}\\) is a subgroup of \\(\\mathcal{G}\\).\n\n**Equivariance.** We consider a compact group \\(\\mathcal{G}\\) with Haar measure \\(\\lambda\\) (the unique \\(\\mathcal{G}\\)-invariant probability measure on \\(\\mathcal{G}\\)). Let \\(\\mathcal{G}\\) act on spaces \\(\\mathcal{X}\\) and \\(\\mathcal{Y}\\) by representations \\(\\phi\\) and \\(\\psi\\), respectively. We say that a map \\(f:\\mathcal{X}\\rightarrow\\mathcal{Y}\\) is \\(\\mathcal{G}\\)-equivariant if for all \\(g\\in\\mathcal{G}\\), \\(x\\in\\mathcal{X},\\psi(g^{-1})f(\\phi(g)\\,x)=f(x)\\). Given any map \\(f:\\mathcal{X}\\rightarrow\\mathcal{Y}\\), a projection of \\(f\\) onto the space of \\(\\mathcal{G}\\)-equivariant maps can be computed by averaging over orbits with respect to \\(\\lambda\\)\n\n\\[(\\mathcal{Q}_{\\mathcal{G}}f)(x)=\\int_{\\mathcal{G}}\\psi(g^{-1})\\,f\\left(\\phi(g) x\\right)\\,\\mathrm{d}\\lambda(g). \\tag{2}\\]\n\nFor \\(u,v\\in\\mathcal{Y}\\), let \\(\\langle u,v\\rangle\\) be a \\(\\mathcal{G}\\)-invariant inner product, i.e. \\(\\langle\\psi(g)u,\\psi(g)v\\rangle=\\langle u,v\\rangle\\), for all \\(g\\in\\mathcal{G}\\), for all \\(u,v\\in\\mathcal{Y}\\). Given two maps \\(f_{1},f_{2}:\\mathcal{X}\\rightarrow\\mathcal{Y}\\), we define their inner product as \\(\\langle f_{1},f_{2}\\rangle_{\\mu}=\\int_{\\mathcal{X}}\\langle f_{1}(x),f_{2}(x) \\rangle\\,\\mathrm{d}\\mu(x)\\), where \\(\\mu\\) is a \\(\\mathcal{G}\\)-invariant measure on \\(\\mathcal{X}\\). Let \\(V\\) be the space of all (measurable) map \\(f:\\mathcal{X}\\rightarrow\\mathcal{Y}\\) such that \\(\\|f\\|_{\\mu}=\\sqrt{\\langle f,f\\rangle_{\\mu}}<\\infty\\).\n\n**Graphs.** We consider edge-node weighted graphs \\(G=([N],A,b)\\), where \\([N]\\) is a finite set of nodes, \\(A\\in[0,1]^{N\\times N}\\) is the adjacency matrix describing the edge weights, and \\(b=\\{b_{1},\\ldots,b_{N}\\}\\subset\\mathbb{R}\\) are the node weights. An edge weighted graph is a special case of \\(G\\) where all node weights are \\(1\\). A simple graph is a special case of an edge weighted graph, where all edge weights are binary. Let \\(\\mathcal{A}_{G}\\) be the automorphism group of a graph defined as \\(\\mathcal{A}_{G}\\coloneqq\\{\\Pi\\in\\mathcal{S}_{N}:\\Pi\\,A\\,\\Pi^{\\top}=A,\\Pi\\,b=b\\}\\), which characterizes the symmetries of \\(G\\). Hereinafter, \\(\\mathcal{G}\\) is assumed to be a subgroup of \\(\\mathcal{S}_{N}\\).\n\n**Graph Signals and Learning Task.** We consider graph signals supported in the nodes of a fixed graph \\(G\\) and maps between graphs signals. Let \\(\\mathcal{X}=\\mathbb{R}^{N\\times d},\\mathcal{Y}=\\mathbb{R}^{N\\times k}\\) be the input and output graph signal spaces. We denote by \\(f\\) a map between graph signals, \\(f:\\mathcal{X}\\rightarrow\\mathcal{Y}\\). Even though the functions \\(f\\) depend on \\(G\\), we don't explicitly write \\(G\\) as part of the notation of \\(f\\) because \\(G\\) is fixed. Our goal is to learn a target map between graph signals \\(f^{*}:\\mathcal{X}\\rightarrow\\mathcal{Y}\\). To this end, we assume access to a training set \\(\\{(X_{i},Y_{i})\\}\\) where the \\(X_{i}\\) are i.i.d. sampled from an \\(\\mathcal{S}_{N}\\)-invariant distribution \\(\\mu\\) on\\(\\mathcal{X}\\) where \\(\\mathcal{S}_{n}\\) acts on \\(\\mathcal{X}\\) by permuting the rows, and \\(Y_{i}=f^{*}(X_{i})+\\xi_{i}\\) for some noise \\(\\xi_{i}\\). A natural assumption is that \\(f^{*}\\) is approximately equivariant with respect to \\(\\mathcal{A}_{G}\\) in some sense. Our symmetry model selection problem concerns a sequence of hypothesis class \\(\\{\\mathcal{H}_{\\mathcal{G}}\\}\\) indexed by \\(\\mathcal{G}\\), where we choose the best class \\(\\mathcal{H}_{\\mathcal{G}}^{*}\\) such that the estimator \\(\\hat{f}\\in\\mathcal{H}_{\\mathcal{G}}^{*}\\) gives the best generalization performance.\n\n**Equivariant Graph Networks.** We propose to learn the target function \\(f^{*}\\) on the fixed graph using \\(\\mathcal{G}\\)-_equivariant graph networks_ (\\(\\mathcal{G}\\)-Net), which are equivariant to a chosen symmetry group \\(\\mathcal{G}\\) depending on the graph domain. Using standard techniques from representation theory, such as Schur's lemma and projections to isotypic components [68], we can parameterize \\(\\mathcal{G}\\)-Net by interleaving \\(\\mathcal{G}\\)-equivariant linear layers \\(f_{\\mathcal{G}}:\\mathbb{R}^{N\\times d}\\to\\mathbb{R}^{N\\times k}\\) with pointwise nonlinearity, where the weights in the linear map \\(f_{\\mathcal{G}}\\) are constrained in patterns depending on the group structure (also known as parameter sharing or weight tying [69], see Appendix A.1 for technical details). In practice, we can make \\(\\mathcal{G}\\)-Net more flexible by systematically breaking the symmetry, such as incorporating graph convolutions (i.e., \\(Af_{\\mathcal{G}}\\)) and locality constraints (i.e., \\(A\\odot f_{\\mathcal{G}}\\)). Compared to standard GNNs (described in Appendix D.1), \\(\\mathcal{G}\\)-Net uses a more expressive linear map to gather global information (i.e., weights are not shared among all nodes). Importantly, \\(\\mathcal{G}\\)-Net yields a suite of models that allows us to flexibly choose the hypothesis class (and estimator) reflecting the active symmetries in the data, and subsumes standard GNNs that are permutation-equivariant with respect to passive symmetries (but not necessarily equivariant with respect to active symmetries).\n\n**Graphons.** A graphon is a symmetric measurable function \\(W:[0,1]^{2}\\to[0,1]\\). Graphons represent (dense) graph limits where the number of nodes goes to infinity; they can also be viewed as random graph models where the value \\(W(x,y)\\) represents the probability of having an edge between the nodes \\(x\\) and \\(y\\). Let \\(\\mathcal{W}\\) denote the space of all graphons and \\(\\eta\\) denote the Lebesgue measure. Lovasz and Szegedy [70] introduced the cut norm on \\(\\mathcal{W}\\) as\n\n\\[\\|W\\|_{\\square}\\coloneqq\\sup_{\\begin{subarray}{c}S,T\\subset[0,1]\\\\ S,T\\text{measurable}\\end{subarray}}\\left|\\int_{S\\times T}W(u,v)\\,\\mathrm{d}\\eta (u)\\,\\mathrm{d}\\eta(v)\\right|. \\tag{3}\\]\n\nBased on the cut norm, Borgs et al. [71] defined the cut distance between two graphons \\(W,U\\in\\mathcal{W}\\),\n\n\\[\\delta_{\\square}(W,U)\\coloneqq\\inf_{f\\in S_{[0,1]}}\\|W-U^{f}\\|_{\\square}, \\tag{4}\\]\n\nwhere \\(S_{[0,1]}\\) is the set of all measure-preserving bijective measurable maps between \\([0,1]\\) and itself, and \\(U^{f}(x,y)=U(f(x),f(y))\\). Note that the cut distance is \"permutation-invariant,\" where measure preserving bijections are seen as the continuous counterparts of permutations.\n\n## 3 Generalization with Exact Symmetry\n\nIn this section, we assume the target map \\(f^{*}\\) is \\(\\mathcal{A}_{G}\\)-equivariant and study the symmetry model selection problem by comparing the (statistical) risk of different models. Concretely, the risk quantifies how a given model performs on average on any potential input. The smaller the risk, the better the model performs. Thus, we use the _risk gap_ of two functions \\(f,f^{\\prime}\\), defined as\n\n\\[\\Delta(f,f^{\\prime})\\coloneqq\\mathbb{E}\\left[\\|Y-f(X)\\|_{F}^{2}\\right]-\\mathbb{ E}\\left[\\|Y-f^{\\prime}(X)\\|_{F}^{2}\\right], \\tag{5}\\]\n\nas our model selection metric. Following ideas from Elesedy and Zaidi [8], we analyze the risk gap between any function \\(f\\in V\\) and its \\(\\mathcal{G}\\)-equivariant version, for \\(\\mathcal{G}\\subseteq\\mathcal{S}_{N}\\).\n\n**Lemma 1** (Risk Gap).: _Let \\(\\mathcal{X}=\\mathbb{R}^{N\\times d},\\mathcal{Y}=\\mathbb{R}^{N\\times k}\\) be the input and output graph signal spaces on a fixed graph \\(G\\). Let \\(X\\sim\\mu\\) where \\(\\mu\\) is a \\(\\mathcal{S}_{N}\\)-invariant distribution on \\(\\mathcal{X}\\). Let \\(Y=f^{*}(X)+\\xi\\), where \\(\\xi\\in\\mathbb{R}^{N\\times k}\\) is random, independent of \\(X\\) with zero mean and finite variance and \\(f^{*}:\\mathcal{X}\\to\\mathcal{Y}\\) is \\(\\mathcal{A}_{G}\\)-equivariant. Then, for any \\(f\\in V\\) and for any compact group \\(\\mathcal{G}\\subseteq\\mathcal{S}_{N}\\), we can decompose \\(f\\) as_\n\n\\[f=\\bar{f}_{\\mathcal{G}}+f_{\\mathcal{G}}^{\\perp},\\]\n\n_where \\(\\bar{f}_{\\mathcal{G}}=\\mathcal{Q}_{\\mathcal{G}}f,f_{\\mathcal{G}}^{\\perp}=f- \\bar{f}_{\\mathcal{G}}\\). Moreover, the risk gap satisfies_\n\n\\[\\Delta(f,\\bar{f}_{\\mathcal{G}})=\\mathbb{E}\\left[\\|Y-f(X)\\|_{F}^{2}\\right]- \\mathbb{E}\\left[\\|Y-\\bar{f}_{\\mathcal{G}}(X)\\|_{F}^{2}\\right]=\\underbrace{-2 \\langle f^{*},f_{\\mathcal{G}}^{\\perp}\\rangle_{\\mu}}_{\\text{mismatch}}+ \\underbrace{\\left\\|f_{\\mathcal{G}}^{\\perp}\\right\\|_{\\mu}^{2}}_{\\text{constraint}}.\\]Lemma 1, proven in Appendix B.1, adapts ideas from [8] to equate the risk gap to the symmetry \"mismatch\" between the chosen group \\(\\mathcal{G}\\) and the target group \\(\\mathcal{A}_{G}\\), plus the symmetry \"constraint\" captured by the norm of the anti-symmetric part of \\(f\\) with respect to \\(\\mathcal{G}\\). Our symmetry model selection problem aims to find the group \\(\\mathcal{G}\\subseteq\\mathcal{S}_{N}\\) such that the risk gap is maximized. Note that for \\(\\mathcal{G}\\leq\\mathcal{A}_{G}\\), the mismatch term vanishes since \\(f^{*}\\) is \\(\\mathcal{A}_{G}\\)-equivariant, but the constraint term decreases with \\(\\mathcal{G}\\); When \\(\\mathcal{G}=\\mathcal{A}_{G}\\), we recover Lemma 6 in [8]. On the other hand, for \\(\\mathcal{G}>\\mathcal{A}_{G}\\), the mismatch term can be positive, negative or zero (depending on \\(f^{*}\\)) whereas the constraint term increases with \\(\\mathcal{G}\\).\n\n### Linear Regression\n\nIn this section, we focus on the linear regression setting and analyze the risk gap of using the equivariant estimator versus the vanilla estimator. We consider linear estimator \\(\\hat{\\Theta}:\\mathbb{R}^{N\\times d}\\rightarrow\\mathbb{R}^{N\\times k}\\) that predicts \\(\\hat{Y}=f_{\\hat{\\Theta}}(X)=\\hat{\\Theta}^{\\top}X\\). Given a compact group \\(\\mathcal{G}\\) and any linear map \\(\\Theta\\), we obtain its \\(\\mathcal{G}\\)-equivariant version via projection to the \\(\\mathcal{G}\\)-equivariant space (also known as intertwiner average),\n\n\\[\\Psi_{\\mathcal{G}}(\\Theta)=\\int_{\\mathcal{G}}\\phi(g)\\,\\Theta\\,\\psi\\left(g^{-1} \\right)\\mathrm{d}\\lambda(g). \\tag{6}\\]\n\nWe denote \\(\\Psi_{\\mathcal{G}}^{\\perp}(\\Theta)=\\Theta-\\Psi_{\\mathcal{G}}(\\Theta)\\) as the projection of \\(\\hat{\\Theta}\\) to the orthogonal complement of the \\(\\mathcal{G}\\)-equivariant space. Here \\(\\Psi_{\\mathcal{G}}\\) instantiates the orbit-average operator \\(\\mathcal{Q}_{\\mathcal{G}}\\) (eqn. 2) for linear functions.\n\n**Theorem 2** (Bias-Variance-Tradeoff).: _Let \\(\\mathcal{X}=\\mathbb{R}^{N\\times d},\\mathcal{Y}=\\mathbb{R}^{N\\times k}\\) be the graph signals spaces on a fixed graph \\(G\\). Let \\(\\mathcal{S}_{N}\\) act on \\(\\mathcal{X}\\) and \\(\\mathcal{Y}\\) by permuting the rows with representations \\(\\phi\\) and \\(\\psi\\). Let \\(\\mathcal{G}\\) be a subgroup of \\(\\mathcal{S}_{N}\\) acting with restricted representations \\(\\phi|_{\\mathcal{G}}\\) on \\(\\mathcal{X}\\) and \\(\\psi|_{\\mathcal{G}}\\) on \\(\\mathcal{Y}\\). Let \\(X_{[i,j]}\\stackrel{{ i.i.d.}}{{\\sim}}\\mathcal{N}\\left(0,\\sigma_{ \\mathcal{X}}^{2}\\right)\\) and \\(Y=f^{*}(X)+\\xi\\) where \\(f^{*}(x)=\\Theta^{\\top}x\\) is \\(\\mathcal{A}_{G}\\)-equivariant and \\(\\Theta\\in\\mathbb{R}^{Nd\\times Nk}\\). Assume \\(\\xi_{[i,j]}\\) is random, independent of \\(X\\), with mean 0 and \\(\\mathbb{E}\\left[\\xi\\xi^{\\top}\\right]=\\sigma_{\\xi}^{2}I<\\infty\\). Let \\(\\hat{\\Theta}\\) be the least-squares estimate of \\(\\Theta\\) from \\(n\\) i.i.d. examples \\(\\{(X_{i},Y_{i}):i=1,\\ldots,n\\}\\), \\(\\Psi_{\\mathcal{G}}(\\hat{\\Theta})\\) be its equivariant version with respect to \\(\\mathcal{G}\\). Let \\(\\left(\\chi_{\\psi|_{\\mathcal{G}}}\\mid\\chi_{\\phi|_{\\mathcal{G}}}\\right)=\\int_{ \\mathcal{G}}\\chi_{\\psi|_{\\mathcal{G}}}(g)\\chi_{\\phi|_{\\mathcal{G}}}(g)d\\lambda (g)\\) denote the inner product of the characters. If \\(n>Nd+1\\) the risk gap is_\n\n\\[\\mathbb{E}\\left[\\Delta\\left(f_{\\hat{\\Theta}},f_{\\Psi_{\\mathcal{G}}(\\hat{\\Theta })}\\right)\\right]=\\underbrace{-\\sigma_{X}^{2}\\,\\|\\Psi_{\\mathcal{G}}^{\\perp}( \\Theta)\\|_{F}^{2}}_{\\text{bias}}+\\underbrace{\\sigma_{\\xi}^{2}\\frac{N^{2}dk- \\left(\\chi_{\\psi|_{\\mathcal{G}}}\\mid\\chi_{\\phi|_{\\mathcal{G}}}\\right)}{n-Nd-1} }_{\\text{variance}}.\\]\n\nThe bias term depends on the anti-symmetric part of \\(\\Theta\\) with respect to \\(\\mathcal{G}\\), whereas the variance term captures the difference of the dimension between the space of linear maps \\(\\mathbb{R}^{Nd}\\rightarrow\\mathbb{R}^{Nk}\\) (measured by \\(N^{2}dk\\)), and the space of \\(\\mathcal{G}\\)-equivariant linear maps (measured by \\(\\left(\\chi_{\\psi|_{\\mathcal{G}}}\\mid\\chi_{\\phi|_{\\mathcal{G}}}\\right)\\); a proof can be found in [72, Section 2.2]). The bias term reduces to zero for \\(\\mathcal{G}\\leq\\mathcal{A}_{G}\\) and we recover Theorem 1 in [8] when \\(\\mathcal{G}=\\mathcal{A}_{G}\\). Notably, by using a larger group \\(\\mathcal{G}\\), the dimension of the equivariant space measured by \\(\\left(\\chi_{\\psi|_{\\mathcal{G}}}\\mid\\chi_{\\phi|_{\\mathcal{G}}}\\right)\\) is smaller, and thus the variance term increases; meanwhile, the bias term decreases due to extra (symmetry) constraints on the estimator. We remark that \\(\\left(\\chi_{\\psi|_{\\mathcal{G}}}\\mid\\chi_{\\phi|_{\\mathcal{G}}}\\right)\\) depends on \\(d,k\\) as well: For \\(d=k=1\\), \\(\\phi(g)=\\psi(g)\\in\\mathbb{R}^{N\\times N}\\) are standard permutation matrices; For general \\(d>1,k>1\\), \\(\\phi(g)\\in\\mathbb{R}^{Nd\\times Nd},\\psi(g)\\in\\mathbb{R}^{Nk\\times Nk}\\) are block-diagonal matrices, where each block is a permutation matrix.\n\nTo understand the significance of the risk gap, we note that (see Appendix B.1) when \\(n>Nd+1\\), the risk of the least square estimator is\n\n\\[\\mathbb{E}\\left[\\|Y-\\hat{\\Theta}^{\\top}X\\|_{F}^{2}\\right]=\\sigma_{\\xi}^{2}\\frac {Nd}{n-Nd-1}+\\sigma_{\\xi}^{2}. \\tag{7}\\]\n\nThus, when \\(n\\) is small enough so that the risk gap in Theorem 2 is dominated by the variance term, enforcing equivariance gives substantial generalization gain of order \\(\\frac{N^{2}dk-\\left(\\chi_{\\psi|_{\\mathcal{G}}}\\mid\\chi_{\\phi|_{\\mathcal{G}}} \\right)}{n-Nd-1}\\).\n\n**Example 3.1**.: In Appendix C.1 we construct an example with \\(\\mathcal{X}=\\mathbb{R}^{3},\\mathcal{Y}=\\mathbb{R}^{3}\\), and \\(x\\sim\\mathcal{N}(0,\\sigma_{\\mathcal{X}}^{2}I_{d})\\). We consider a target linear function \\(f^{*}\\) that is \\(\\mathcal{S}_{2}\\)-equivariant and _approximately \\(\\mathcal{S}_{3}\\)_-equivariant, and compare the estimators \\(f_{\\psi_{\\mathcal{S}_{2}}(\\hat{\\Theta})}\\) versus \\(f_{\\psi_{\\mathcal{S}_{3}}(\\hat{\\Theta})}\\). When the number of training samples \\(n\\) is small,using a \\(\\mathcal{S}_{3}\\)-equivariant least square estimator yields better test error than the \\(\\mathcal{S}_{2}\\)-equivariant one, as shown in figure inset. The dashed vertical line denotes the theoretical threshold \\(n^{*}\\approx 35\\), before which using \\(\\mathcal{S}_{3}\\) yields better generalization than \\(\\mathcal{S}_{2}\\). This is an example where enforcing more symmetries than the target (symmetry) results on better generalization properties.\n\n## 4 Generalization with Approximate Symmetries\n\nLarge graphs are typically asymmetric. Therefore, the assumption of \\(f^{*}\\) being \\(\\mathcal{A}_{G}\\)-equivariant in Section 3 becomes trivial. Yet, graphon theory asserts that graphs can be seen as living in a \"continuous\" metric space, with the cut distance (Definition 1). Since graphs are continuous entities, the combinatorial notion of exact symmetry of graphs is not appropriate. We hence relax the notion of exact symmetry to a property that is \"continuous\" in the cut distance. The regularity lemma [73, Theorem 5.1] asserts that any large graph can be approximated by a coarser graph in the cut distance. Since smaller graphs tend to exhibit more symmetries than larger ones, we are motivated to consider the symmetries of coarse-grained versions of graphs as their approximate symmetries (Definition 2, 3). We then present the notion of approximately equivariant mappings (Definition 4), which allows us to precisely characterize the bias-variance tradeoff (Corollary 3) in the approximate symmetry setting based on Lemma 1. Effectively, we reduce selecting the symmetry group to choosing the coarsened graph \\(G^{\\prime}\\). Moreover, our symmetry model selection perspective allows for exploring different coarsening procedures and choosing the one that works best for the problem.\n\n**Definition 1** (Induced graphon).: _Let \\(G\\) be a (possibly edge weighted) graph with node set \\([N]\\) and adjacency matrix \\(A=\\{a_{i,j}\\in[0,1]\\}_{i,j=1}^{N}\\). Let \\(\\mathcal{P}_{N}=\\{I_{N}^{n}=(\\frac{n-1}{N},\\frac{n}{N})\\}_{n=1}^{N}\\) be the equipartition of \\([0,1]\\) to \\(N\\) intervals (where formally the last interval is the closed interval \\([\\frac{N-1}{N},1]\\)). We define the graphon \\(W_{G}\\) induced by \\(G\\) as_\n\n\\[W_{G}(x,y)=\\sum_{i,j=1}^{N}a_{i,j}\\mathbb{1}_{I_{N}^{i}}(x)\\mathbb{1}_{I_{N}^{ i}}(y),\\]\n\n_where \\(\\mathbb{1}_{I_{N}^{i}}\\) is the indicator function of the set \\(I_{N}^{i}\\subset[0,1]\\), \\(i=1,\\ldots,N\\)._\n\nWe induce a graphon from an edge-node weighted graph \\(G^{\\prime}=([M],A,b)\\), with rational node weights \\(b_{i}\\), as follows. Let \\(N\\in\\mathbb{N}\\) be a value such that the node weights can be written in the form \\(b=\\{b_{m}=\\frac{q_{m}}{N}\\}_{m=1}^{M},\\) for some \\(q_{m}\\in\\mathbb{N}_{\\geq 0}\\), \\(m=1,\\ldots,M\\). We _blow-up_\\(G^{\\prime}\\) to an edge weighted graph \\(\\overline{G}_{N}\\) of \\(\\sum q_{m}\\) nodes by splitting each node \\(m\\) of \\(G^{\\prime}\\) into \\(q_{m}\\) nodes \\(\\{n_{m}^{j}\\}_{j=1}^{q_{m}}\\) of weight \\(1\\), and defining the adjacency matrix \\(\\overline{A}_{N}\\) with entries \\(\\overline{a}_{n_{m}^{j},n_{m^{\\prime}}^{j}}=a_{m,m^{\\prime}}\\). Note that for any two choices \\(N_{1},N_{2}\\in\\mathbb{N}\\) in the above construction, \\(\\delta_{\\square}(W_{\\overline{G}_{N_{1}}},W_{\\overline{G}_{N_{2}}})=0\\), where the infimum in the definition of \\(\\delta_{\\square}\\) is realized on some permutation of intervals in \\([0,1]\\), as explained below. We hence define the induced graphon of the edge-node weighted graph \\(G^{\\prime}\\) by \\(W_{G^{\\prime}}:=W_{\\overline{G}_{N}}\\) for some \\(N\\), where, in fact, each choice of \\(N\\) gives a representative of an equivalence class of piecewise constant graphons with zero \\(\\delta_{\\square}\\) distance.\n\n**Definition 2**.: _Let \\(G^{\\prime}\\) be an edge-node weighted graph with \\(M\\) nodes and node weights \\(b=\\{b_{m}=\\frac{q_{m}}{N^{\\prime}}\\}_{m=1}^{M},\\) satisfying \\(\\{q_{m}\\}_{m}\\subset\\mathbb{N}_{\\geq 0}\\) and \\(\\sum q_{m}=N\\), and let \\(G\\) be an edge weighted graph with nodes. We say that \\(G^{\\prime}\\)_ coarsens \\(G\\)_up to error \\(\\epsilon\\) if_\n\n\\[\\delta_{\\square}(W_{G},W_{G^{\\prime}})<\\epsilon.\\]\n\nSuppose that \\(G^{\\prime}\\) coarsens \\(G\\). This implies the existence of an assignment of nodes of \\(G\\) to nodes of \\(G^{\\prime}\\). Namely, the supremum underlying the definition of \\(\\delta_{\\square}\\) over the measure preserving bijections \\(\\phi:[0,1]\\to[0,1]\\) is realized by a permutation \\(\\Pi:[N]\\to[N]\\) applied on the intervals \\(\\{I_{N}^{i}\\}_{i=1}^{N}\\). With this permutation, the assignment \\(C_{G\\to G^{\\prime}}\\) defined by \\([N]\\ni\\Pi(n_{m}^{j})\\mapsto m\\in[M]\\), for every \\(m=1,\\ldots,M\\) and \\(j=1,\\ldots,q_{m}\\), is called a _cluster assignment_ of \\(G\\) to \\(G^{\\prime}\\).\n\nLet \\(G^{\\prime}\\), with \\(M\\) nodes, be a edge-node weighted graph that coarsens the simple graph \\(G\\) of \\(N\\) nodes. Let \\(C_{G\\to G^{\\prime}}\\) be a cluster assignment of \\(G\\) to \\(G^{\\prime}\\). Let \\(\\mathcal{A}_{G^{\\prime}}\\) be the automorphism group of \\(G^{\\prime}\\). Note that two nodes can belong to the same orbit of \\(\\mathcal{A}_{G^{\\prime}}\\) only if they have the same node weight. Hence, for every two nodes \\(m,m^{\\prime}\\) in the same orbit of \\(\\mathcal{A}_{G^{\\prime}}\\) (i.e., equivalent clusters), there exists a bijection \\(\\psi_{m,m^{\\prime}}:C_{G\\to G^{\\prime}}^{-1}\\{m\\}\\to C_{G\\to G^{\\prime}}^{-1} \\{m^{\\prime}\\}\\). We choose a set of such bijections, such that for every \\(m,m^{\\prime}\\) on the same orbit of \\(\\mathcal{A}_{G^{\\prime}}\\)\n\n\\[\\psi_{m,m^{\\prime}}=\\psi_{m^{\\prime},m}^{-1}.\\]\n\nWe moreover choose \\(\\psi_{m,m}\\) as the identity mapping for each \\(m=1,\\ldots,M\\). We identify each element \\(g\\) of \\(\\mathcal{A}_{G^{\\prime}}\\) with an element \\(\\overline{g}\\) in a permutation group \\(\\overline{\\mathcal{A}}_{G^{\\prime}}\\) of \\([N]\\), called the _blown-up symmetry group_, as follows. Given \\(g\\in\\mathcal{A}_{G^{\\prime}}\\), for every \\(m\\in[M]\\) and every \\(n\\in C_{G\\to G^{\\prime}}^{-1}\\{m\\}\\), define\n\n\\[\\overline{g}n\\coloneqq\\psi_{m,gm}n.\\]\n\n**Definition 3**.: _Let \\(G^{\\prime}\\), with \\(M\\) nodes, be a edge-node weighted graph that coarsen the (simple or weighted) graph \\(G\\) of \\(N\\) nodes. Let \\(C_{G\\to G^{\\prime}}\\) be a cluster assignment of \\(G\\) to \\(G^{\\prime}\\) with cluster sizes \\(c_{1},\\ldots,c_{M}\\). Let \\(\\mathcal{A}_{G^{\\prime}}\\) be the automorphism group of \\(G^{\\prime}\\), and \\(\\overline{\\mathcal{A}}_{G^{\\prime}}\\) be the blown-up symmetry group of \\([N]\\). For every \\(m=1,\\ldots M\\), let \\(\\mathcal{S}_{c_{m}}\\) be the symmetry group of \\(C_{G\\to G^{\\prime}}^{-1}\\{m\\}\\). We call the group of permutations_\n\n\\[\\mathcal{G}_{G\\to G^{\\prime}}\\coloneqq\\left(\\mathcal{S}_{c_{1}}\\times \\mathcal{S}_{c_{2}}\\ldots\\times\\mathcal{S}_{c_{M}}\\right)\\rtimes\\overline{ \\mathcal{A}}_{G^{\\prime}}\\subseteq\\mathcal{S}_{N}\\]\n\n_the symmetry group of \\(G\\) induced by the coarsening \\(G^{\\prime}\\). We call any element of \\(\\mathcal{G}_{G\\to G^{\\prime}}\\) an approximate symmetry of \\(G\\)._\n\nSpecifically, every element \\(s\\in\\mathcal{G}_{G\\to G^{\\prime}}\\) can be written in coordinates by\n\n\\[s=s_{1}s_{2}\\ldots s_{M}a\\sim(s_{1},\\ldots,s_{M},a),\\]\n\nfor a unique choice of \\(s_{j}\\in\\mathcal{S}_{c_{j}}\\), \\(j=1,\\ldots,M\\), and \\(a\\in\\overline{\\mathcal{A}}_{G^{\\prime}}\\) (see Appendix B.2 for details).\n\nIn words, we consider the symmetry of the graph \\(G\\) of \\(N\\) nodes not by its own automorphism group, but via the symmetry of its coarsened graph \\(G^{\\prime}\\). The nodes in the same orbit in \\(G\\) are either in the same cluster of \\(G^{\\prime}\\) (i.e., from the same coarsened node), or in the equivalent clusters of \\(G^{\\prime}\\) (i.e., they belong to the same orbit of the coarsened graph and share the same cluster size). See Figure 1 (and Figure 9 in Appendix A.2) for examples.\n\n**Definition 4**.: _Let \\(G\\) be a graph with \\(N\\) nodes, and \\(\\mathcal{X}=\\mathbb{R}^{N\\times d}\\) be the space of graph signals. Let \\(X\\sim\\mu\\) where \\(\\mu\\) is a \\(\\mathcal{S}_{N}\\)-invariant measure on \\(\\mathbb{R}^{N\\times d}\\). We call \\(f^{*}:\\mathbb{R}^{N\\times d}\\to\\mathbb{R}^{N\\times k}\\) an approximately equivariant mapping if there exists a function \\(\\kappa:\\mathbb{R}_{+}\\to\\mathbb{R}_{+}\\) satisfying \\(\\lim_{\\epsilon\\to 0}\\kappa(\\epsilon)=0\\) (called the equivariance rate), such that for every \\(\\epsilon>0\\) and every edge-node weigted graph \\(G^{\\prime}\\) that coarsen \\(G\\) up to error \\(\\epsilon\\),_\n\n\\[\\|f^{*}-f^{*}_{\\mathcal{G}_{G\\to G^{\\prime}}}\\|_{\\mu}\\leq\\kappa(\\epsilon),\\]\n\n_where \\(f^{*}_{\\mathcal{G}_{G\\to G^{\\prime}}}=\\mathcal{Q}_{\\mathcal{G}_{G\\to G^{ \\prime}}}(f)\\) is the intertwining projection of \\(f^{*}\\) with respect to \\(\\mathcal{G}_{G\\to G^{\\prime}}\\)._\n\n**Example 4.1**.: Here we give an example of an approximately equivariant mapping in a natural setting. Suppose that \\(G\\) is a random geometric graph, namely, a graph that was sampled from a metric-probability space \\(\\mathcal{M}\\) by randomly and independently sampling nodes \\(\\{x_{n}\\}_{n=1}^{N}\\subset\\mathcal{M}\\). Hence, \\(G\\) can be viewed as a discretization of \\(\\mathcal{M}\\), and the graphs that coarsen \\(G\\) can be seen as coarser discretizations of \\(\\mathcal{M}\\). Therefore, the approximate symmetries are _local deformations_ of \\(G\\), namely, permutations that swap nodes if they are close in the metric of \\(\\mathcal{M}\\). This now gives an interpretation for approximately equivariance mappings of geometric graphs: these are mappings that are stable to local deformations. In Appendix C.2 we develop this example in detail.\n\nWe are ready to present the bias-variance tradeoff in the setting with approximate symmetries. The proofs are deferred to Appendix B.2.\n\n**Corollary 3** (Risk Gap via Graph Coarsening).: _Let \\(\\mathcal{X}=\\mathbb{R}^{N\\times d},\\mathcal{Y}=\\mathbb{R}^{N\\times k}\\) be the input and output graph signal spaces on a fixed graph \\(G\\). Let \\(X\\sim\\mu\\) where \\(\\mu\\) is a \\(\\mathcal{S}_{N}\\)-invariant distribution on \\(\\mathcal{X}\\). Let \\(Y=f^{*}(X)+\\xi\\), where \\(\\xi\\in\\mathbb{R}^{N\\times k}\\) is random, independent of \\(X\\) with zero mean and finite variance, and \\(f^{*}:\\mathbb{R}^{N\\times d}\\rightarrow\\mathbb{R}^{N\\times k}\\) be an approximately equivariant mapping with equivariance rate \\(\\kappa\\). Then, for any \\(G^{\\prime}\\) that coarsens \\(G\\) up to error \\(\\epsilon\\), for any \\(f\\in V\\), we have_\n\n\\[\\Delta(f,\\bar{f}_{\\mathcal{G}_{G\\to G^{\\prime}}})=\\underbrace{-2 \\langle f^{*},f^{\\perp}_{\\mathcal{G}_{G\\to G^{\\prime}}}\\rangle_{\\mu}}_{ minatch}+\\underbrace{\\left\\|f^{\\perp}_{\\mathcal{G}_{G\\to G^{\\prime}}} \\right\\|^{2}_{\\mu}}_{\\text{constraint}}\\geq-2\\kappa(\\epsilon)\\left\\|f^{\\perp}_ {\\mathcal{G}_{G\\to G^{\\prime}}}\\right\\|_{\\mu}+\\left\\|f^{\\perp}_{ \\mathcal{G}_{G\\to G^{\\prime}}}\\right\\|^{2}_{\\mu}\\]\n\nNotably, Corollary 3 illustrates the tradeoff explicitly in the form of choosing the coarsened graph \\(G^{\\prime}\\): If we choose \\(G^{\\prime}\\) close to \\(G\\) such that the coarsening error \\(\\epsilon\\) and \\(\\|f^{\\perp}_{\\mathcal{G}_{G\\to G^{\\prime}}}\\|_{\\mu}\\) are small, then the mismatch term is close to zero; meanwhile the constraint term is also small since \\(\\mathcal{A}_{\\mathcal{G}}\\) is typically trivial. On the other hand, if we choose \\(G^{\\prime}\\) far from \\(G\\) that yields large coarsening error \\(\\epsilon\\), then the constraint term \\(\\|f^{\\perp}_{\\mathcal{G}_{G\\to G^{\\prime}}}\\|^{2}_{\\mu}\\) also increases.\n\n**Corollary 4** (Bias-Variance-Tradeoff via Graph Coarsening).: _Consider the same linear regression setting in Theorem 2, except now \\(f^{*}\\) is an approximately equivariant mapping with equivariance rate \\(\\kappa\\), and \\(\\mathcal{G}=\\mathcal{G}_{G\\to G^{\\prime}}\\) is controlled by \\(G^{\\prime}\\) that coarsens \\(G\\) up to error \\(\\epsilon\\). Denote the canonical permutation representations of \\(\\mathcal{G}_{G\\to G^{\\prime}}\\) on \\(\\mathcal{X},\\mathcal{Y}\\) as \\(\\phi^{\\prime},\\psi^{\\prime}\\), respectively. Let \\((\\chi_{\\psi^{\\prime}}\\mid\\chi_{\\phi^{\\prime}})=\\int_{\\mathcal{G}_{G\\to G^{ \\prime}}}\\chi_{\\psi^{\\prime}}(g)\\mathrm{d}\\lambda(g)\\) denote the inner product of the characters. If \\(n>Nd+1\\) the risk gap is bounded by_\n\n\\[\\mathbb{E}\\left[\\Delta\\left(f_{\\Theta},f_{\\Psi_{\\mathcal{G}_{G\\to G^{ \\prime}}}(\\Theta)}\\right)\\right]\\geq-2\\kappa(\\epsilon)\\sqrt{\\sigma_{\\xi}^{2} \\frac{N^{2}dk-(\\chi_{\\psi^{\\prime}}\\mid\\chi_{\\psi^{\\prime}})}{n-Nd-1}}+\\sigma_ {\\xi}^{2}\\frac{N^{2}dk-(\\chi_{\\psi^{\\prime}}\\mid\\chi_{\\psi^{\\prime}})}{n-Nd-1}.\\]\n\n## 5 Experiments\n\nWe illustrate our theory in three real-world tasks for learning on a fixed graph: image inpainting, traffic flow prediction, and human pose estimation1. For image inpainting, we demonstrate the bias-variance tradeoff via graph coarsening using \\(\\mathcal{G}\\)-Net; For the other two applications, we show that the tradeoff not only emerges from \\(\\mathcal{G}\\)-Net that enforces strict equivariance, but also \\(\\mathcal{G}\\)-Net augmented with symmetry-breaking modules, allowing us to _recover_ standard GNN architectures as a special case. Concretely, we consider the following variants (more details in Appendix D.4.2):\n\nFootnote 1: The source code is available at [https://github.com/nhuang37/Approx_Equivariant_Graph_Nets](https://github.com/nhuang37/Approx_Equivariant_Graph_Nets).\n\n1. \\(\\mathcal{G}\\)-Net with strict equivariance using equivariant linear map \\(f_{\\mathcal{G}}\\).\n2. \\(\\mathcal{G}\\)-Net augmented with graph convolution \\(Af_{\\mathcal{G}}(x)\\), denoted as \\(\\mathcal{G}\\)-Net(gc).\n3. \\(\\mathcal{G}\\)-Net augmented with graph convolution and learnable edge weights: \\(\\mathcal{G}\\)-Net(gc+ew).\n4. \\(\\mathcal{G}\\)-Net augmented with graph locality constraints \\((A\\odot f_{\\mathcal{G}})(x)\\) and learnable edge weights, denoted as \\(\\mathcal{G}\\)-Net(pt+ew).\n\n\\(\\mathcal{G}\\)-Net serves as a baseline to validate our generalization analysis for equivariant estimators (see Section 3). Yet in practice, we observe that augmenting \\(\\mathcal{G}\\)-Net with symmetry breaking can further improve performance, thereby justifying the analysis for approximate symmetries (see Section 4). In particular, \\(\\mathcal{G}\\)-Net(gc) and \\(\\mathcal{G}\\)-Net(gc+ew) are motivated by graph convolutions used in standard GNNs; \\(\\mathcal{G}\\)-Net(pt+ew) is inspired by the concept of locality in CNNs, where \\(A\\odot f_{\\mathcal{G}}\\) effectively restricts the receptive field to the \\(1\\)-hop neighrborhood induced by the graph \\(A\\).\n\n### Application: Image Inpainting\n\nWe consider a \\(28\\times 28\\) grid graph as the fixed domain, with grey-scale images as the graph signals. The learning task is to reconstruct the original images given _masked_ images as inputs (i.e., image inpainting). We use subsets of MNIST [74] and FashionMNIST [75], each comprising 100 training samples and 1000 test samples. The input and output graph signals are \\((m_{i}\\odot x_{i},x_{i})\\), where \\(x_{i}\\in\\mathbb{R}^{28\\times 28}\\equiv\\mathbb{R}^{784}\\) denotes the image signals and \\(m_{i}\\) denotes a random mask (size \\(14\\times 14\\) for MNIST and \\(20\\times 20\\) for FashionMNIST). We investigate the symmetry model selection problem by clustering the grid into \\(M\\) patches with size \\(d\\times d\\), where \\(d\\in\\{28,14,7,4,2,1\\}\\). Here \\(d=28\\) means one cluster (with \\(\\mathcal{S}_{N}\\) symmetry); \\(d=1\\) is \\(784\\) singleton clusters with no symmetry (trivial).\n\nWe consider \\(\\mathcal{G}_{G\\to G^{\\prime}}\\)-equivariant networks \\(\\mathcal{G}\\)-Net with ReLU nonlinearity. We parameterize the equivariant linear layer \\(f:\\mathbb{R}^{N}\\to\\mathbb{R}^{N}\\) with respect to \\(\\mathcal{G}_{G\\to G^{\\prime}}=(\\mathcal{S}_{c_{1}}\\times\\ldots\\times\\mathcal{ S}_{c_{M}})\\rtimes\\overline{\\mathcal{A}}_{G^{\\prime}}\\) using the following block-matrix form (assuming the nodes are ordered by their cluster assignment), with \\(f_{kl}\\) denoting block matrices, and \\(a_{k},b_{k},e_{kl}\\) representing scalars:\n\n\\[f=\\begin{bmatrix}f_{11}&\\cdots&f_{1M}\\\\ &\\cdots&f_{MM}\\end{bmatrix},\\,f_{kk}=a_{k}I+b_{k}\\mathds{1}\\mathds{1}^{\\top}, \\,f_{kl}=e_{kl}\\mathds{1}\\mathds{1}^{\\top}\\text{ for }k\\neq l. \\tag{8}\\]\n\nThe coarsened graph symmetry \\(\\mathcal{A}_{G^{\\prime}}\\) induces constraints on \\(a_{k},b_{k},e_{kl}\\). If \\(\\mathcal{A}_{G^{\\prime}}\\) is trivial, then these scalars are unconstrained. In the experiment, we consider a reflection symmetry on the coarsened grid graph, i.e., \\(\\mathcal{A}_{G^{\\prime}}=\\mathcal{S}_{2}\\) which acts by reflecting the left (coarsened) patches to the right (coarsened) patches. Suppose the reflected patch pairs are ordered consecutively, then \\(a_{k}=a_{k+1},b_{k}=b_{k+1}\\) for \\(k\\in\\{1,3,\\ldots,M-1\\}\\), and \\(e_{kl}=e_{k+1,l-1}\\) for \\(k\\in\\{1,3,\\ldots,M-1\\},l\\in\\{2,4,\\ldots,M\\}\\) (see Figure inset for an illustration). In practice, we extend the parameterization to \\(f:\\mathbb{R}^{N\\times d}\\to\\mathbb{R}^{N\\times k}\\). More details can be found in Appendix D.2.\n\nFigure 2 shows the empirical risk first decreases and then increases as the group decreases, illustrating the bias-variance tradeoff from our theory. Figure 2 (left) compares a \\(2\\)-layer \\(\\mathcal{G}\\)-Net with a \\(1\\)-layer linear \\(\\mathcal{G}\\)-Net, demonstrating that the tradeoff occurs in both linear and nonlinear models. Figure 2 (right) shows that using reflection symmetry of the coarsened graph outperforms the trivial symmetry baseline, highlighting the utility of modeling coarsened graph symmetries.\n\n### Application: Traffic Flow Prediction\n\nThe traffic flow prediction problem can be formulated as follows: Let \\(X^{(t)}\\in\\mathbb{R}^{n\\times d}\\) represent the traffic graph signal (e.g., speed or volume) observed at time \\(t\\). The goal is to learn a function \\(h(\\cdot)\\) that maps \\(T^{\\prime}\\) historical traffic signals to \\(T\\) future graph signals, assuming the fixed graph domain \\(G\\):\n\n\\[\\left[X^{\\left(t-T^{\\prime}+1\\right)},\\ldots,X^{(t)};G\\right]\\xrightarrow{h( \\cdot)}\\left[X^{(t+1)},\\ldots,X^{(t+T)}\\right].\\]\n\nThe traffic graph is typically large and asymmetric. Therefore we leverage our approximate symmetries to study the symmetry model selection problem (Section 4). We use the METR-LA dataset which represents traffic volume of highways in Los Angeles (see figure inset). We use both the original graph \\(G\\) from [76], and its sparsified version \\(G_{s}\\) which is more faithful to the road geometry. In \\(G_{s}\\), the \\((i,j)\\)-edge exists if and only if nodes \\(i,j\\) lie on the same highway. We first validate our Definition 4 in such dataset (see Appendix D.3.1). In terms of the models, we consider a simplified version of DCRNN model proposed in [76], where the spatial module is modelled by a standard GNN, and the temporal module is modelled via an encoder-decoder architecture for sequence-to-sequence modelling. We follow the same temporal module and extending its GNN module using \\(\\mathcal{G}\\)-Net(gc), which recovers DCRNN when choosing \\(\\mathcal{S}_{N}\\). We then consider different choices of the coarsened graph \\(G^{\\prime}\\) (to induce approximate symmetries). As shown in Table 3, using \\(2\\) clusters as approximate symmetries yields better generalization error than using \\(9\\) clusters, or the full permutation group \\(\\mathcal{S}_{N}\\).\n\nFigure 2: Bias-variance tradeoff via graph coarsening. Left:2-layer \\(\\mathcal{G}\\)-Net(blue) and \\(1\\)-layer linear \\(\\mathcal{G}\\)-equivariant functions (orange), assuming the coarsened graph is asymmetric; Right: 2-layer \\(\\mathcal{G}\\)-Net with both trivial and non-trivial coarsened graph symmetry. See Table 7 for more numerical details.\n\n### Application: Human Pose Estimation\n\nWe consider the simple (loopy) graph \\(G\\) with adjacency matrix \\(A\\in\\{0,1\\}^{16\\times 16}\\) representing 16 human joints, illustrated on the right. The input graph signals \\(X\\in\\mathbb{R}^{16\\times 2}\\) describe the joint 2D spatial location. The goal is to learn a map to reconstruct the 3D pose \\(Y\\in\\mathbb{R}^{16\\times 3}\\). We use the standard benchmark dataset Human3.6M [77] and follow the evaluation protocol in [78]. The generalization performance is measured by mean per joint position error (MPJPE) and MPJPE after alignment (P-MPJPE). We implement a \\(4\\)-Layer \\(\\mathcal{G}\\)-Net(gc+ew), which recovers the same model architecture in SemGCN [78] when choosing \\(\\mathcal{G}=\\mathcal{S}_{16}\\) (further details are provided in Appendix D.4.1).\n\nWe consider the following symmetry groups: the permutation group \\(\\mathcal{S}_{16}\\), the graph automorphism \\(\\mathcal{A}_{G}\\), and the trivial symmetry. Note that the human skeleton graph has \\(\\mathcal{A}_{G}=(\\mathcal{S}_{2})^{2}\\) (corresponding to the arm flip and leg flip). Additionally, we investigate an approximate symmetry called Relax-\\(\\mathcal{S}_{16}\\); This relaxes the global weight sharing in \\(\\mathcal{S}_{16}\\) (that only learns \\(2\\) scalars per equivariant linear layer \\(f_{\\mathcal{S}_{16}}:\\mathbb{R}^{N}\\rightarrow\\mathbb{R}^{N}\\), where \\(f_{\\mathcal{S}_{16}}[i,i]=a,f_{\\mathcal{S}_{16}}[i,j]=b\\) for \\(i\\neq j\\)) to local weight sharing (that learns \\(2\\times 16=32\\) scalars, where \\(f_{\\text{Relax-}\\mathcal{S}_{16}}[i,i]=a_{i},f_{\\text{Relax-}\\mathcal{S}_{16 }}[i,j]=b_{i}\\) for \\(i\\neq j\\)). Table 4 shows that the best performance is achieved at the hypothesis class induced from Relax-\\(\\mathcal{S}_{16}\\), which is smaller than \\(\\mathcal{S}_{16}\\) but differs from \\(\\mathcal{A}_{G}\\). Furthermore, using \\(\\mathcal{G}\\)-Net(gc+ew) with Relax-\\(\\mathcal{S}_{16}\\) gives a substantial improvement over \\(\\mathcal{S}_{16}\\) (representing SemGCN in [78]). This demonstrates the utility of enforcing active symmetries in GNNs that results in more expressive models. More details and additional ablation studies can be found in Appendix D.4.\n\n## 6 Discussion\n\nIn this paper, we focus on learning tasks where the graph is fixed, and the dataset consists of different signals on the graph. We developed an approach for designing GNNs based on active symmetries and approximate symmetries induced by the symmetries of the graph and its coarse-grained versions. A layer of an approximately equivariant graph network uses a linear map that is equivariant to the chosen symmetry group; the graph is not used as an input but rather induces a hypothesis class. In practice, we further break the symmetry by incorporating the graph in the model computation, thus combining symmetric and asymmetric components.\n\nWe theoretically show a bias-variance tradeoff between the loss of expressivity due to imposing symmetries, and the gain in regularity, for settings where the target map is assumed to be (approximately) equivariant. For simplicity, the theoretical analysis focuses on the equivariant models without symmetry breaking; Theoretically analyzing the combination of symmetric and asymmetric components in machine learning models is an interesting open problem. The bias-variance formula is computed only for a simple linear regression model with white noise and in the underparamterized setting; Extending it to more realistic models and overparameterized settings is a promising direction.\n\nAs a proof of concept, our approximately equivariant graph networks only consider symmetries of the fixed graph. An interesting future direction is to extend our approach to also account for (approximate) symmetries in node features and labels, using suitably generalized cut distance (e.g., graphon-signal cut distance in [79]; see Appendix C.2 for an overview). Our network architecture consists only of linear layers and pointwise nonlinearity. Another promising direction is to incorporate pooling layers (e.g., [80; 22]) and investigate them through the lens of approximate symmetries. Finally, extending our analysis to general groups is a natural next step. Our techniques are based on compact groups with orthogonal representation. While we believe that the orthogonality requirement can be lifted straightforwardly, relaxing the compactness requirement appears to be more challenging (see related discussion in [46]).\n\n\\begin{table}\n\\begin{tabular}{l c c c} \\hline \\hline \\(\\mathcal{G}\\)-Net(gc) & \\(\\mathcal{S}_{N}\\) & \\(\\mathcal{S}_{i_{1}}\\times\\mathcal{S}_{n_{2}}\\) & \\(\\mathcal{S}_{i_{2}}\\times\\ldots\\times\\mathcal{S}_{n_{N}}\\) \\\\ \\hline Graph \\(G\\) & \\(3.173\\pm 0.013\\) & \\(\\mathbf{3.150\\pm 0.008}\\) & \\(3.204\\pm 0.006\\) \\\\ Graph \\(G\\) & \\(3.106\\pm 0.013\\) & \\(\\mathbf{3.092\\pm 0.008}\\) & \\(3.174\\pm 0.013\\) \\\\ \\hline \\hline \\end{tabular}\n\\end{table}\nTable 3: Traffic forecasting with different choices of graph clustering. Table shows Mean Absolute Error (MAE) across \\(3\\) runs.\n\n## Acknowledgments\n\nThis research project was started at the BIRS 2022 Workshop \"Deep Exploration of non-Euclidean Data with Geometric and Topological Representation Learning\" held at the UBC Okanagan campus in Kelowna, B.C. The authors thank Ben Blum-Smith (JHU), David Hogg (NYU), Erik Thiede (Cornell), Wenda Zhou (Open AI), Carey Priebe (JHU), and Rene Vidal (UPenn) for valuable discussions, and Hong Ye Tan (Cambridge) for pointing out typos in an earlier version of the manuscript. The authors also thank the anonymous NeurIPS reviewers and area chair for helpful feedback. Ningyuan Huang is partially supported by the MINDS Data Science Fellowship from Johns Hopkins University. Ron Levie is partially funded by ISF (Israel Science Foundation) grant #1937/23. Soledad Villar is partially funded by the NSF-Simons Research Collaboration on the Mathematical and Scientific Foundations of Deep Learning (MoDL) (NSF DMS 2031985), NSF CISE 2212457, ONR N00014-22-1-2126 and an Amazon AI2AI Faculty Research Award.\n\n## References\n\n* [1] M. M. Bronstein, J. Bruna, Y. LeCun, A. Szlam, and P. Vandergheynst (2017) Geometric deep learning: going beyond euclidean data. IEEE Signal Processing Magazine34 (4), pp. 18-42. Cited by: SS1.\n* [2] Fernando Gama, Joan Bruna, and Alejandro Ribeiro (2020) Stability properties of graph neural networks. IEEE Transactions on Signal Processing68, pp. 5680-5695. Cited by: SS1.\n* [3]Michael M. Bronstein, Joan Bruna, Taco Cohen, and Petar Velickovic (2021) Geometric deep learning: grids, groups, graphs, geodesics, and gauges. External Links: 2101.01108 Cited by: SS1.\n* [4]Carlo Rovelli and Marcus Gaul (2000) Loop quantum gravity and the meaning of diffeomorphism invariance. In Towards Quantum Gravity: Proceeding of the XXXV International Winter School on Theoretical Physics Held in Polanica, Poland, 2-11 February 1999, pp. 277-324. Cited by: SS1.\n* [5]S. Villar, D. W. Hogg, W. Yao, G. A. Kevrekidis, and B. Scholkopf (2023) The passive symmetries of machine learning. arXiv preprint arXiv:2301.13724. Cited by: SS1.\n* [6]M. Mohri, A. Rostamizadeh, and A. Talwalkar (2018) Foundations of machine learning. MIT press. Cited by: SS1.\n* [7]Alberto Bietti, Luca Venturi, and Joan Bruna (2021) On the sample complexity of learning under invariance and geometric stability. External Links: 2102.00178 Cited by: SS1.\n* [8]B. Elesedy and S. Zaidi (2021) Provably strict generalisation benefit for equivariant models. In International Conference on Machine Learning, pp. 2959-2969. Cited by: SS1.\n* [9]S. Mei, T. Misiakiewicz, and A. Montanari (2021) Learning with invariances in random features and kernel models. arXiv preprint arXiv:2102.13219. Cited by: SS1.\n* [10]C. Lyle, M. van der Wilk, M. Kwiatkowska, Y. Gal, and B. Bloem-Reddy (2020) On the benefits of invariance in neural networks. arXiv preprint arXiv:2005.00178. Cited by: SS1.\n* [11]B. Elesedy (2022) Group symmetry in pac learning. In ICLR 2022 Workshop on Geometrical and Topological Representation Learning, Cited by: SS1.\n* [12]B. Elesedy (2021) Provably strict generalisation benefit for invariance in kernel methods. Advances in Neural Information Processing Systems34, pp. 17273-17283. Cited by: SS1.\n* [13]A. Behboodi, G. Cesa, and T. Cohen (2022) A pac-bayesian generalization bound for equivariant networks. arXiv preprint arXiv:2210.13150. Cited by: SS1.\n* [14]G. Benton, M. Finzi, P. Izmailov, and A. G. Wilson (2020) Learning invariances in neural networks from training data. Advances in neural information processing systems33, pp. 17605-17616. Cited by: SS1.\n* [15]J. Cahill, D. G. Mixon, and H. Parshall (2023) Lie pca: density estimation for symmetric manifolds. Applied and Computational Harmonic Analysis65, pp. 279-295. Cited by: SS1.\n* [16]V. Portilheiro (2022) A tradeoff between universality of equivariant models and learnability of symmetries. arXiv preprint arXiv:2210.09444. Cited by: SS1.\n* [17]F. Scarselli, M. Gori, A. C. Tsoi, M. Hagenbuchner, and G. Monfardini (2009) The graph neural network model. IEEE Transactions on Neural Networks20 (1), pp. 61-80. Cited by: SS1.\n\n* [18] J. Bruna, W. Zaremba, A. Szlam, and Y. LeCun. Spectral networks and deep locally connected networks on graphs. In _International Conference on Learning Representation_, 2014.\n* [19] Justin Gilmer, Samuel S. Schoenholz, Patrick F. Riley, Oriol Vinyals, and George E. Dahl. Neural message passing for quantum chemistry. In _International Conference on Machine Learning_, pages 1263-1272, 2017.\n* [20] Thomas N. Kipf and Max Welling. Semi-supervised classification with graph convolutional networks. In _International Conference on Learning Representations_, 2017.\n* [21] Petar Velickovic, Guillem Cucurull, Arantxa Casanova, Adriana Romero, Pietro Lio, and Yoshua Bengio. Graph attention networks. In _International Conference on Learning Representations_, 2018.\n* [22] Michael Defferrard, Xavier Bresson, and Pierre Vandergheynst. Convolutional neural networks on graphs with fast localized spectral filtering. In _Advances in Neural Information Processing Systems_, pages 3837-3845, 2016.\n* [23] F. Gama, A. G. Marques, G. Leus, and A. Ribeiro. Convolutional neural network architectures for signals supported on graphs. _IEEE Transactions on Signal Processing_, 67(4):1034-1049, 2019.\n* [24] R. Levie, F. Monti, X. Bresson, and M. M. Bronstein. Cayleynets: Graph convolutional neural networks with complex rational spectral filters. _IEEE Transactions on Signal Processing_, 67(1):97-109, 2019.\n* [25] Zhengdao Chen, Lei Chen, Soledad Villar, and Joan Bruna. Can graph neural networks count substructures? _Advances in neural information processing systems_, 33:10383-10395, 2020.\n* [26] Erik Henning Thiede, Wenda Zhou, and Risi Kondor. Autobahn: Automorphism-based graph neural nets, 2021. URL [https://arxiv.org/abs/2103.01710](https://arxiv.org/abs/2103.01710).\n* [27] Beatrice Bevilacqua, Fabrizio Frasca, Derek Lim, Balasubramaniam Srinivasan, Chen Cai, Gopinath Balamurugan, Michael M Bronstein, and Haggai Maron. Equivariant subgraph aggregation networks. _arXiv preprint arXiv:2110.02910_, 2021.\n* [28] Giorgos Bouritsas, Fabrizio Frasca, Stefanos P Zafeiriou, and Michael Bronstein. Improving graph neural network expressivity via subgraph isomorphism counting. _IEEE Transactions on Pattern Analysis and Machine Intelligence_, 2022.\n* [29] Fabrizio Frasca, Beatrice Bevilacqua, Michael M Bronstein, and Haggai Maron. Understanding and extending subgraph gnns by rethinking their symmetries. _arXiv preprint arXiv:2206.11140_, 2022.\n* [30] Bohang Zhang, Guhao Feng, Yiheng Du, Di He, and Liwei Wang. A complete expressiveness hierarchy for subgraph gnns via subgraph weisfeiler-lehman tests. _CoRR_, abs/2302.07090, 2023.\n* [31] Jianfei Gao, Yangze Zhou, and Bruno Ribeiro. Double permutation equivariance for knowledge graph completion. _arXiv preprint arXiv:2302.01313_, 2023.\n* [32] Haggai Maron, Heli Ben-Hamu, Nadav Shamir, and Yaron Lipman. Invariant and equivariant graph networks. _arXiv preprint arXiv:1812.09902_, 2018.\n* [33] Christopher Morris, Martin Ritzert, Matthias Fey, William L. Hamilton, Jan Eric Lenssen, Gaurav Rattan, and Martin Grohe. Weisfeiler and leman go neural: Higher-order graph neural networks. In _AAAI Conference on Artificial Intelligence_, pages 4602-4609, 2019.\n* [34] Manzil Zaheer, Satwik Kottur, Siamak Ravanbakhsh, Barnabas Poczos, Russ R Salakhutdinov, and Alexander J Smola. Deep sets. _Advances in neural information processing systems_, 30, 2017.\n* [35] Taco Cohen and Max Welling. Group equivariant convolutional networks. In _International conference on machine learning_, pages 2990-2999. PMLR, 2016.\n* [36] Yashil Sukurdeep et al. _Elastic shape analysis of geometric objects with complex structures and partial correspondences_. PhD thesis, Johns Hopkins University, 2023.\n* [37] Jameson Cahill, Joseph W Iverson, Dustin G Mixon, and Daniel Packer. Group-invariant max filtering. _arXiv preprint arXiv:2205.14039_, 2022.\n\n* Thomas et al. [2018] Nathaniel Thomas, Tess Smidt, Steven Kearnes, Lusann Yang, Li Li, Kai Kohlhoff, and Patrick Riley. Tensor field networks: Rotation-and translation-equivariant neural networks for 3d point clouds. _arXiv:1802.08219_, 2018.\n* Fuchs et al. [2020] Fabian Fuchs, Daniel Worrall, Volker Fischer, and Max Welling. Se (3)-transformers: 3d roto-translation equivariant attention networks. _Neural Information Processing Systems_, 33, 2020.\n* Geiger and Smidt [2022] Mario Geiger and Tess Smidt. e3nn: Euclidean neural networks. _arXiv preprint arXiv:2207.09453_, 2022.\n* Villar et al. [2021] Soledad Villar, David W Hogg, Kate Storey-Fisher, Weichi Yao, and Ben Blum-Smith. Scalars are universal: Equivariant machine learning, structured like classical physics. _Advances in Neural Information Processing Systems_, 34:28848-28863, 2021.\n* Finkelshtein et al. [2022] Ben Finkelshtein, Chaim Baskin, Haggai Maron, and Nadav Dym. A simple and universal rotation equivariant point-cloud network. In _Topological, Algebraic and Geometric Learning Workshops 2022_, pages 107-115. PMLR, 2022.\n* Kondor et al. [2018] Risi Kondor, Zhen Lin, and Shubhendu Trivedi. Clebsch-gordan nets: a fully fourier space spherical convolutional neural network. _Advances in Neural Information Processing Systems_, 31:10117-10126, 2018.\n* Weiler et al. [2021] Maurice Weiler, Patrick Forre, Erik Verlinde, and Max Welling. Coordinate independent convolutional networks-isometry and gauge equivariant convolutions on riemannian manifolds. _arXiv preprint arXiv:2106.06020_, 2021.\n* Wang et al. [2022] Rui Wang, Robin Walters, and Rose Yu. Approximately equivariant networks for imperfectly symmetric dynamics. In Kamalika Chaudhuri, Stefanie Jegelka, Le Song, Csaba Szepesvari, Gang Niu, and Sivan Sabato, editors, _Proceedings of the 39th International Conference on Machine Learning_, volume 162 of _Proceedings of Machine Learning Research_, pages 23078-23091. PMLR, 17-23 Jul 2022. URL [https://proceedings.mlr.press/v162/wang22aa.html](https://proceedings.mlr.press/v162/wang22aa.html).\n* Villar et al. [2023] Soledad Villar, Weichi Yao, David W Hogg, Ben Blum-Smith, and Bianca Dumitrascu. Dimensionless machine learning: Imposing exact units equivariance. _Journal of Machine Learning Research_, 24(109):1-32, 2023.\n* Cohen et al. [2018] Taco S. Cohen, Mario Geiger, Jonas Koehler, and Max Welling. Spherical cnns, 2018.\n* Han et al. [2022] Jiaqi Han, Wenbing Huang, Tingyang Xu, and Yu Rong. Equivariant graph hierarchy-based neural networks. _Advances in Neural Information Processing Systems_, 35:9176-9187, 2022.\n* de Haan et al. [2020] Pim de Haan, Taco S Cohen, and Max Welling. Natural graph networks. _Advances in Neural Information Processing Systems_, 33:3636-3646, 2020.\n* Garg et al. [2020] Vikas Garg, Stefanie Jegelka, and Tommi Jaakkola. Generalization and representational limits of graph neural networks. In Hal Daume III and Aarti Singh, editors, _Proceedings of the 37th International Conference on Machine Learning_, volume 119 of _Proceedings of Machine Learning Research_, pages 3419-3430. PMLR, 13-18 Jul 2020. URL [https://proceedings.mlr.press/v119/garg20c.html](https://proceedings.mlr.press/v119/garg20c.html).\n* Esser et al. [2021] Pascal Esser, Leena Chennuru Vankadara, and Debarghya Ghoshdastidar. Learning theory can (sometimes) explain generalisation in graph neural networks. _Advances in Neural Information Processing Systems_, 34:27043-27056, 2021.\n* Morris et al. [2023] Christopher Morris, Floris Geerts, Jan Tonshoff, and Martin Grohe. Wl meet vc, 2023.\n* Maskey et al. [2022] Sohir Maskey, Ron Levie, Yunseok Lee, and Gitta Kutyniok. Generalization analysis of message passing neural networks on large random graphs. In _Advances in Neural Information Processing Systems_, 2022.\n* Levie et al. [2021] Ron Levie, Wei Huang, Lorenzo Bucci, Michael Bronstein, and Gitta Kutyniok. Transferability of spectral graph convolutional neural networks. _Journal of Machine Learning Research_, 22(272):1-59, 2021.\n* Ruiz et al. [2020] Luana Ruiz, Luiz Chamon, and Alejandro Ribeiro. Graphon neural networks and the transferability of graph neural networks. _Advances in Neural Information Processing Systems_, 33:1702-1712, 2020.\n\n* [56] Beatrice Bevilacqua, Yangze Zhou, and Bruno Ribeiro. Size-invariant graph representations for graph classification extrapolations. In _International Conference on Machine Learning_, pages 837-851. PMLR, 2021.\n* [57] Keyulu Xu, Mozhi Zhang, Jingling Li, Simon S Du, Ken-ichi Kawarabayashi, and Stefanie Jegelka. How neural networks extrapolate: From feedforward to graph neural networks. _arXiv preprint arXiv:2009.11848_, 2020.\n* [58] Mircea Petrache and Shubhendu Trivedi. Approximation-generalization trade-offs under (approximate) group equivariance, 2023.\n* [59] Marc Finzi, Gregory Benton, and Andrew G Wilson. Residual pathway priors for soft equivariance constraints. _Advances in Neural Information Processing Systems_, 34:30037-30049, 2021.\n* [60] Xavier Suau, Federico Danieli, T. Anderson Keller, Arno Blaas, Chen Huang, Jason Ramapuram, Dan Busbridge, and Luca Zappella. Duet: 2d structured and approximately equivariant representations. In _International Conference on Machine Learning_. PMLR, 2023.\n* [61] Sharut Gupta, Joshua Robinson, Derek Lim, Soledad Villar, and Stefanie Jegelka. Structuring representation geometry with rotationally equivariant contrastive learning. _arXiv preprint arXiv:2306.13924_, 2023.\n* [62] Leonardo Cotta, Gal Yehuda, Assaf Schuster, and Chris J Maddison. Probabilistic invariant learning with randomized linear classifiers. _arXiv preprint arXiv:2308.04412_, 2023.\n* [63] Stephane Mallat. Group invariant scattering. _Communications on Pure and Applied Mathematics_, 65(10):1331-1398, 2012. doi: [https://doi.org/10.1002/cpa.21413](https://doi.org/10.1002/cpa.21413).\n* [64] Fernando Gama, Alejandro Ribeiro, and Joan Bruna. Diffusion scattering transforms on graphs. _arXiv preprint arXiv:1806.08829_, 2018.\n* [65] Michael Perlmutter, Feng Gao, Guy Wolf, and Matthew Hirn. Understanding graph neural networks with asymmetric geometric scattering transforms. _arXiv preprint arXiv:1911.06253_, 2019.\n* [66] Dongmian Zou and Gilad Lerman. Graph convolutional neural networks via scattering. _Applied and Computational Harmonic Analysis_, 49(3):1046-1074, 2020.\n* [67] Alexander Bogatskiy, Sanmay Ganguly, Thomas Kipf, Risi Kondor, David W Miller, Daniel Murnane, Jan T Offermann, Mariel Pettee, Phiala Shanahan, Chase Shimmin, et al. Symmetry group equivariant architectures for physics. _arXiv preprint arXiv:2203.06153_, 2022.\n* [68] William Fulton and Joe Harris. _Representation theory: a first course_, volume 129. Springer Science & Business Media, 2013.\n* [69] Siamak Ravanbakhsh, Jeff Schneider, and Barnabas Poczos. Equivariance through parameter-sharing. In _International conference on machine learning_, pages 2892-2901. PMLR, 2017.\n* [70] Laszlo Lovasz and Balazs Szegedy. Limits of dense graph sequences. _Journal of Combinatorial Theory, Series B_, 96(6):933-957, 2006.\n* [71] Christian Borgs, Jennifer T Chayes, Laszlo Lovasz, Vera T Sos, and Katalin Vesztergombi. Convergent sequences of dense graphs i: Subgraph frequencies, metric properties and testing. _Advances in Mathematics_, 219(6):1801-1851, 2008.\n* [72] Mark Reeder. Notes on representations of finite groups, 2014.\n* [73] Laszlo Lovasz and Balazs Szegedy. Szemeredi's lemma for the analyst. _GAFA Geometric And Functional Analysis_, 17(1):252-270, 2007.\n* [74] Yann LeCun. The mnist database of handwritten digits. _http://yann. lecun. com/exdb/mnist/_, 1998.\n* [75] Han Xiao, Kashif Rasul, and Roland Vollgraf. Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. _arXiv preprint arXiv:1708.07747_, 2017.\n* [76] Yaguang Li, Rose Yu, Cyrus Shahabi, and Yan Liu. Diffusion convolutional recurrent neural network: Data-driven traffic forecasting. _arXiv preprint arXiv:1707.01926_, 2017.\n* [77] Catalin Ionescu, Dragos Papava, Vlad Olaru, and Cristian Sminchisescu. Human3. 6m: Large scale datasets and predictive methods for 3d human sensing in natural environments. _IEEE transactions on pattern analysis and machine intelligence_, 36(7):1325-1339, 2013.\n\n* Zhao et al. [2019] Long Zhao, Xi Peng, Yu Tian, Mubbasir Kapadia, and Dimitris N Metaxas. Semantic graph convolutional networks for 3d human pose regression. In _Proceedings of the IEEE/CVF conference on computer vision and pattern recognition_, pages 3425-3435, 2019.\n* Levie [2023] Ron Levie. A graphon-signal analysis of graph neural networks, 2023.\n* Parada-Mayorga et al. [2022] Alejandro Parada-Mayorga, Zhiyang Wang, and Alejandro Ribeiro. Graphon pooling for reducing dimensionality of signals and convolutional operators on graphs, 2022.\n* Spielman [2018] Daniel A. Spielman. Testing isomorphism of graphs with distinct eigenvalues, 2018.\n* Sagan [2001] Bruce Sagan. _The symmetric group: representations, combinatorial algorithms, and symmetric functions_, volume 203. Springer Science & Business Media, 2001.\n* Serre et al. [1977] Jean-Pierre Serre et al. _Linear representations of finite groups_, volume 42. Springer, 1977.\n* Hastie et al. [2022] Trevor Hastie, Andrea Montanari, Saharon Rosset, and Ryan J Tibshirani. Surprises in high-dimensional ridgeless least squares interpolation. _The Annals of Statistics_, 50(2):949-986, 2022.\n* Huang et al. [2022] Ningyuan Huang, David W. Hogg, and Soledad Villar. Dimensionality reduction, regularization, and generalization in overparameterized regressions. _SIAM Journal on Mathematics of Data Science_, 4(1):126-152, feb 2022. doi: 10.1137/20m1387821. URL [https://doi.org/10.1137%2F20m1387821](https://doi.org/10.1137%2F20m1387821).\n* Gupta [1968] Somesh Das Gupta. Some aspects of discrimination function coefficients. _Sankhya: The Indian Journal of Statistics, Series A_, pages 387-400, 1968.\n* Jagadish et al. [2014] Hosagrahar V Jagadish, Johannes Gehrke, Alexandros Labrinidis, Yannis Papakonstantinou, Jignesh M Patel, Raghu Ramakrishnan, and Cyrus Shahabi. Big data and its technical challenges. _Communications of the ACM_, 57(7):86-94, 2014.\n* Pavllo et al. [2019] Dario Pavllo, Christoph Feichtenhofer, David Grangier, and Michael Auli. 3d human pose estimation in video with temporal convolutions and semi-supervised training. In _Proceedings of the IEEE/CVF conference on computer vision and pattern recognition_, pages 7753-7762, 2019.\n\nParameterization of equivariant linear Maps\n\nWe consider a group \\(\\mathcal{G}\\) acting on spaces \\(\\mathcal{X}\\) and \\(\\mathcal{Y}\\) via representations \\(\\phi\\) and \\(\\psi\\), respectively. Our goal is to find the equivariant linear maps \\(f:\\mathcal{X}\\to\\mathcal{Y}\\) such that \\(f(\\phi(g)x)=\\psi(g)f(x)\\) for all \\(g\\in\\mathcal{G}\\) and \\(x\\in\\mathcal{X}\\). The standard way to do this, used extensively in the equivariant machine learning literature (e.g. [40, 43]), is to decompose \\(\\phi\\) and \\(\\psi\\) in irreducibles and use Schur's lemma.\n\nIn a nutshell, a group representation \\(\\varphi\\) is an homomorphism \\(\\mathcal{G}\\to\\mathrm{GL}(V)\\), where \\(\\mathrm{GL}(V)\\) denotes the General Linear group of the vector space \\(V\\) (sometimes mathematicians say that \\(V\\) is a representation of \\(\\mathcal{G}\\), but we need to know the homomorphism \\(\\varphi\\) too). One way to interpret the group homomorphism (i.e. \\(\\varphi(gh)=\\varphi(g)\\circ\\varphi(h)\\)) is that the group multiplication corresponds to the composition of linear invertible maps (i.e. matrix multiplication). A linear subspace \\(W\\) of \\(V\\) is said to be a subrepresentation of \\(\\varphi\\) if \\(\\varphi(\\mathcal{G})(W)\\subset W\\). An irreducible representation is one that only has itself and the trivial subspace as subrepresentations.\n\nSchur's lemma states that if \\(V\\), \\(W\\) are vector spaces over \\(\\mathbb{C}\\) and \\(\\varphi_{V}\\), \\(\\varphi_{W}\\) are irreducible representations, then either (1) \\(\\varphi_{V}\\) and \\(\\varphi_{W}\\) are not isomorphic as representations (and the only equivariant linear map between \\(V\\), \\(W\\) is the zero map), or (2) \\(\\varphi_{V}\\) and \\(\\varphi_{W}\\) are isomorphic and the only non-trivial equivariant maps are of the form \\(\\lambda\\,I\\) where \\(\\lambda\\in\\mathbb{C}\\) and \\(I\\) is the identity (See Chapter 1 of [68]).\n\nGiven \\(\\mathcal{G}\\) acting on spaces \\(\\mathcal{X}\\) and \\(\\mathcal{Y}\\) via representations \\(\\phi\\) and \\(\\psi\\), respectively, one can decompose \\(\\phi\\) and \\(\\psi\\) in irreducibles over \\(\\mathbb{C}\\)\n\n\\[\\phi=\\oplus_{k=1}^{\\ell}a_{k}\\mathcal{T}_{k}\\quad\\psi=\\oplus_{k=1}^{\\ell}b_{k} \\mathcal{T}_{k}\\]\n\n(this notation assumes the same irreducibles appear in both decompositions, which can be done if we allow some of the \\(a_{k}\\) and \\(b_{k}\\) to be zero). Then one can parameterize the equivariant linear maps by having one complex parameter per irreducible that appears in both decompositions.\n\nThese ideas can be applied to real spaces too. By Maschke's theorem we can decompose the representation in irreducibles over \\(\\mathbb{R}\\). Then we can check further how to decompose these irreducibles over \\(\\mathbb{C}\\), and apply Schur's lemma. We have 3 cases for the decomposition:\n\n1. The irreducible over \\(\\mathbb{R}\\) is also irreducible over \\(\\mathbb{C}\\). In this case we directly apply Schur's lemma.\n2. The irreducible over \\(\\mathbb{R}\\) decomposes in two different irreducibles over \\(\\mathbb{C}\\). In this case we can send each \\(\\mathbb{C}\\)-irreducible to their isomorphic counterpart.\n3. The irreducibles over \\(\\mathbb{R}\\) decompose in two copies of the same irreducible over \\(\\mathbb{C}\\). In this case we can send each irreducible to any isomorphic copy independently.\n\nTherefore, finding the equivariant linear maps reduces to decomposing the corresponding representations in irreducibles. In the next sections we explain in detail how to do this for the specific problems described in this paper. The appendix is organized as follows: We first show how to parameterize equivariant linear layers for abelian group using isotypical decomposition (Section A.1), and then discuss the case for the symmetry group induced by graph coarsening, which further considers parameterizing permutation-equivariant linear maps to model the within-cluster symmetries (Section A.2).\n\n### Equivariant Linear Maps via Isotypical Decomposition\n\nIn this section, we assume that the graph adjacency matrix \\(A\\) has distinct eigenvalues \\(\\lambda_{1}>\\lambda_{2}>\\ldots>\\lambda_{n}\\). Then \\(\\mathcal{A}_{G}\\) is an abelian group [81, Lemma 3.8.1]. Under this assumption, we present the construction of \\(\\mathcal{A}_{G}\\)-equivariant linear maps using isotypical decomposition (i.e. decomposition into isomorphism classes of irreducible representations) and group characters. We remark that such construction extends to non-abelian groups and refer the interested reader to [82], but we omit it here for the ease of exposition.\n\nWe consider the simplest setting where \\(f:\\mathbb{R}^{N}\\to\\mathbb{R}^{N}\\) is a linear function that maps graph signals. Let \\(x\\in\\mathbb{R}^{N}\\) be the node features, then \\(\\mathcal{A}_{G}\\)-equivariance requires\n\n\\[f(g\\,x)=g\\,f(x)\\quad\\text{for all}\\,g\\in\\mathcal{A}_{G}. \\tag{9}\\]\n\nTo construct equivariant linear functions \\(f\\), our roadmap is outlined as follows:1. Decompose the vector space \\(\\mathcal{X}=\\mathbb{R}^{N}\\) into a sum of components such that different components cannot be mapped to each other equivariantly (also known as the isotypic decomposition);\n2. Given \\(\\mathcal{X}=\\oplus_{i}\\mathcal{X}_{i}\\) an isotypic representation, we then parameterize \\(f\\) by linear maps at each \\(\\mathcal{X}_{i}\\) such that for all \\(i,f(\\mathcal{X}_{i})\\subseteq\\mathcal{X}_{i}\\).\n\nTo this end, we need the following definitions.\n\n**Definition 5**.: _(\\(\\mathcal{G}\\)-module, [82, Defn 1.3.1]) Let \\(\\mathcal{X}\\) be a vector space and \\(\\mathcal{G}\\) be a group. We say the vector space \\(\\mathcal{X}\\) is a \\(\\mathcal{G}\\)-module or \\(\\mathcal{X}\\) carries a representation of \\(\\mathcal{G}\\) if there is a group homomorphism \\(\\rho:\\mathcal{G}\\to GL(\\mathcal{X})\\), where \\(GL\\) denotes the General Linear group. Equivalently, if the following holds:_\n\n1. \\(gv\\in\\mathcal{X}\\)_,_\n2. \\(g(cv+dw)=c(gv)+d(gw)\\)_,_\n3. \\((gh)v=g(hv)\\)_,_\n4. \\(ev=v\\)__\n\n_for all \\(g,h\\in\\mathcal{G};v,w\\in\\mathcal{X}\\) and scalars \\(c,d\\in\\mathbb{C}\\) (\\(e\\in\\mathcal{G}\\) denotes the identity element)._\n\nIn what follows, we consider \\(\\mathcal{X}=\\mathbb{R}^{N}\\) carries a representation of \\(G\\).\n\n**Definition 6**.: _(Group characters) Given a group \\(\\mathcal{G}\\) and a representation \\(\\phi:G\\to GL(V)\\), the group character is a function \\(\\chi:\\mathcal{G}\\to\\mathbb{R}\\) (or \\(\\mathbb{C}\\)) defined as \\(\\chi(g):=\\operatorname{tr}\\phi(g)\\)._\n\n**Definition 7**.: _(Group orbits) Let \\(\\mathcal{X}\\) be a vector space and \\(\\mathcal{G}\\) be a group. The group orbit of an element \\(x\\in\\mathcal{X}\\) is \\(O(x):=\\{gx:g\\in\\mathcal{G}\\}\\)._\n\nLet \\(g_{1},\\ldots,g_{s}\\) be the generators of \\(\\mathcal{A}_{G}\\subset(\\mathcal{S}_{2})^{n}\\), or simply \\(\\mathcal{A}_{G}\\equiv(\\mathcal{S}_{2})^{k}\\) for some \\(k\\leq n\\). Since \\(\\mathcal{A}_{G}\\) is abelian, any irreducible representation is \\(1\\)-dimensional [68, p.8]. In other words, the irreducible representations of an abelian group are homomorphisms\n\n\\[\\rho:\\mathcal{A}_{G}\\to\\mathbb{C}. \\tag{10}\\]\n\nSince all the elements of the group \\(\\mathcal{A}_{G}=(\\mathcal{S}_{2})^{k}\\) is of order \\(1\\) or \\(2\\), the homomorphisms are \\(\\rho:\\mathcal{A}_{G}\\to\\{\\pm 1\\}\\subset\\mathbb{R}\\). By Definition 6, the irreducible characters (i.e., characters of irreducible matrix representation) are also homomorphisms \\(\\rho:\\mathcal{A}_{G}\\to\\{\\pm 1\\}\\). In other words, \\(\\chi(g)\\in\\{\\pm 1\\}\\) for all \\(g\\in\\mathcal{A}_{G}\\). Then we can write down the \\(2^{k}\\times 2^{k}\\) character table, where the rows are the characters \\(\\chi\\), and the columns are the group elements \\(g\\in\\mathcal{A}_{G}\\) (see Table 1 as an example). Now, define the projection onto the isotypic component of the representation \\(X\\) as\n\n\\[P_{\\chi}:=\\frac{\\deg(X)}{|\\mathcal{A}_{G}|}\\sum_{g\\in\\mathcal{A}_{G}}\\overline {\\chi(g)}\\,g=\\frac{1}{|\\mathcal{A}_{G}|}\\sum_{g\\in\\mathcal{A}_{G}}\\chi(g)\\,g, \\tag{11}\\]\n\nwhere the second equality uses the fact that \\(\\mathcal{A}_{G}\\) is abelian.\n\nIntuitively, applying \\(P_{\\chi}\\) on \\(\\mathcal{X}=\\operatorname{span}(\\{e_{1},\\ldots,e_{N}\\})\\) picks out all \\(v\\in\\mathcal{X}\\) that stays in the same subspace defined by the group character \\(\\chi\\). (Note that for the \\((\\mathcal{S}_{2})^{k}\\) case \\(\\chi^{-1}(g)=\\chi(g)\\) since \\(\\chi(g)\\in\\{\\pm 1\\}\\)).\n\nAlgorithm 1 presents the construction of equivariant linear map \\(f\\) with respect to an abelian group. Such construction can parameterize all abelian \\(\\mathcal{A}_{G}\\)-equivariant linear maps, as shown in Lemma 5.\n\n**Lemma 5**.: \\(f\\) _is linear, equivariant with respect to the abelian group \\(\\mathcal{A}_{G}\\) if and only if \\(f\\) can be written as (13) in Algorithm 1._\n\nProof.: By construction in Algorithm 1, \\(f\\) is linear and equivariant. To show the converse, we first recall some useful facts and notations. Since \\(\\mathcal{A}_{G}\\) is abelian with all irreducible representations being one-dimensional, for isotypic components \\(\\mathcal{X}_{\\chi_{1}}\\not\\cong\\mathcal{X}_{\\chi_{2}}\\), we have\n\n\\[g\\,v_{1}=\\lambda_{1}(g)\\,v_{1},\\quad\\text{for all}\\,g\\in\\mathcal{G},v_{1}\\in \\mathcal{X}_{\\chi_{1}}, \\tag{14}\\]\n\n\\[g\\,v_{2}=\\lambda_{2}(g)\\,v_{2},\\quad\\text{for all}\\,g\\in\\mathcal{G},v_{2}\\in \\mathcal{X}_{\\chi_{2}}, \\tag{15}\\]\n\nwhere there exists some \\(g\\in\\mathcal{G}\\) such that \\(\\lambda_{1}(g)\\neq\\lambda_{2}(g)\\). To show \\(f\\) being linear and equivariant implies for all \\(v\\in\\mathcal{X}_{\\chi}\\), \\(f(v)\\in\\mathcal{X}_{\\chi}\\), we prove by contradiction. Without loss of generality, suppose\n\n\\[f(v_{\\chi_{1}})=\\alpha_{1}v_{\\chi_{1}}+\\alpha_{2}v_{\\chi_{2}}, \\tag{16}\\]for some scalars \\(\\alpha_{1},\\alpha_{2}\\) and \\(v_{\\chi_{1}}\\in\\mathcal{X}_{\\chi_{1}},v_{\\chi_{2}}\\in\\mathcal{X}_{\\chi_{2}}\\). Then by (14), for all \\(g\\in\\mathcal{G}\\),\n\n\\[f(g\\,v_{\\chi_{1}})=f(\\lambda_{1}(g)\\,v_{\\chi_{1}})=\\lambda_{1}(g)f(v_{\\chi_{1}} )=\\lambda_{1}(g)\\alpha_{1}v_{\\chi_{1}}+\\lambda_{1}(g)\\alpha_{2}v_{\\chi_{2}}. \\tag{17}\\]\n\nNow, since \\(f\\) is equivariant, for all \\(g\\in\\mathcal{G}\\),\n\n\\[f(g\\,v_{\\chi_{1}})=gf(v_{\\chi_{1}})=g(\\alpha_{1}v_{\\chi_{1}}+\\alpha_{2}v_{\\chi _{2}})=\\lambda_{1}(g)\\alpha_{1}v_{\\chi_{1}}+\\lambda_{2}(g)\\alpha_{2}v_{\\chi_{ 2}}. \\tag{18}\\]\n\nBut there exists some \\(g^{\\prime}\\in\\mathcal{G}\\) such that \\(\\lambda_{1}(g^{\\prime})\\neq\\lambda_{2}(g^{\\prime})\\), which leads to \\(f(g^{\\prime}v_{\\chi_{1}})\\neq f(g^{\\prime}v_{\\chi_{1}})\\), a contradiction. One can easily extend the proof strategy to the general case for \\(f(v_{\\chi_{1}})=\\sum_{i=1}^{1}v_{\\chi_{i}}\\). \n\n**Example A.1**.: Consider the path graph on \\(4\\) nodes (i.e., \\(P_{4}\\)). We have \\(\\mathrm{aut}(P_{4})=\\{e,(14)(23)\\}\\cong Z_{2}\\).\n\nSteps \\(1\\): Note that \\(Z_{2}\\) is abelian and thus all irreducible characters \\(\\chi(g)\\in\\{\\pm 1\\}\\), for all \\(g\\in Z_{2}\\). The character table is shown in Table 1.\n\nStep \\(2\\): using (11) we have\n\n\\[P_{\\chi_{e}}[e_{1};e_{2};e_{3};e_{4}]=\\frac{1}{2}\\begin{bmatrix}1 &0&0&1\\\\ 0&1&1&0\\\\ 1&0&0&1\\end{bmatrix}\\text{ which yields basis }\\mathcal{B}(P_{\\chi_{e}})=[e_{1}+e_{4};e_{2}+e _{3}].\\] \\[P_{\\chi_{2}}[e_{1};e_{2};e_{3};e_{4}]=\\frac{1}{2}\\begin{bmatrix}1 &0&0&-1\\\\ 0&1&-1&0\\\\ 0&-1&1&0\\\\ -1&0&0&1\\end{bmatrix}\\text{ which yields basis }\\mathcal{B}(P_{\\chi_{2}})=[e_{1}-e_{4};e_{2}-e _{3}].\\]\n\nStep \\(3\\): Parameterize \\(f:\\mathbb{R}^{4}\\to\\mathbb{R}^{4}\\) by \\(f:\\mathcal{B}(P_{\\chi_{e}})\\to\\mathcal{B}(P_{\\chi_{e}})\\) and \\(f:\\mathcal{B}(P_{\\chi_{2}})\\to\\mathcal{B}(P_{\\chi_{2}})\\). For all \\(v\\in\\mathbb{R}^{4}\\), write \\(v=c_{1}(e_{1}+e_{4})+c_{2}(e_{2}+e_{3})+c_{3}(e_{1}-e_{4})+c_{4}(e_{2}-e_{3})\\), then\n\n\\[f(v)=\\begin{bmatrix}\\alpha_{1}&\\alpha_{2}\\\\ \\alpha_{3}&\\alpha_{4}\\end{bmatrix}\\begin{bmatrix}c_{1}\\\\ c_{2}\\end{bmatrix}+\\begin{bmatrix}\\alpha_{5}&\\alpha_{6}\\\\ \\alpha_{7}&\\alpha_{8}\\end{bmatrix}\\begin{bmatrix}c_{3}\\\\ c_{4}\\end{bmatrix}, \\tag{19}\\]\n\nwhere \\(\\alpha_{1},\\ldots,\\alpha_{8}\\) are (learnable) real scalars. Now \\(f\\) is linear, equivariant by construction.\n\n\\begin{table}\n\\begin{tabular}{l c c} \\hline \\hline  & \\(e\\) & \\(\\sigma\\) \\\\ \\hline \\(\\chi_{e}\\) & \\(1\\) & \\(1\\) \\\\ \\(\\chi_{2}\\) & \\(1\\) & \\(-1\\) \\\\ \\hline \\hline \\end{tabular}\n\\end{table}\nTable 1: Character table for \\(\\mathrm{aut}(P_{4})\\cong Z_{2}\\)\n\n### Equivariant Linear Map for Symmetries Induced by Graph Coarsening\n\nIn this section, we provide additional details of constructing equivariant linear maps for using the symmetry group induced by graph coarsening (Definition 3). Recall the symmetry group with \\(M\\) clusters of \\(G\\) (with the associated coarsened graph \\(G^{\\prime}\\)) is given by\n\n\\[\\mathcal{G}_{G\\to G^{\\prime}}:=\\left(\\mathcal{S}_{1}\\times\\mathcal{S}_{2}\\ldots \\times\\mathcal{S}_{M}\\right)\\rtimes\\overline{\\mathcal{A}}_{G^{\\prime}}\\subset \\mathcal{S}_{N}.\\]\n\nWe first assume that \\(\\overline{\\mathcal{A}}_{G^{\\prime}}\\) is trivial and show how to parameterize equivariant functions with respect to products of permutations. Then we discuss more general cases, for instance if \\(\\overline{\\mathcal{A}}_{G^{\\prime}}\\) is abelian. For the ease of exposition, we consider \\(X\\in\\mathbb{R}^{N},Y\\in\\mathbb{R}^{N}\\).\n\nSuppose \\(\\overline{\\mathcal{A}}_{G^{\\prime}}\\) is trivial, we claim that a linear function \\(f:\\mathbb{R}^{N}\\to\\mathbb{R}^{N}\\) is equivariant with respect to \\(\\mathcal{G}_{G\\to G^{\\prime}}\\) if and only if it admits the following block-matrix form:\n\n\\[f=\\begin{bmatrix}f_{11}&f_{12}&\\cdots&f_{1M}\\\\ f_{21}&f_{22}&\\cdots&f_{2M}\\\\ &&\\ddots&\\\\ f_{M1}&f_{M2}&\\cdots&f_{MM}\\end{bmatrix},\\,f_{kk}=a_{k}I_{c_{k}}+b_{k}\\mathds{1 }_{c_{k}}\\mathds{1}_{c_{k}}^{\\top},\\,f_{kl}=e_{kl}\\mathds{1}_{c_{k}}\\mathds{1} _{c_{l}}^{\\top}\\text{ for }k\\neq l, \\tag{20}\\]\n\nwhere \\(f_{kl}\\) are block matrices, \\(I_{k}\\) is a size-\\(k\\) identity matrix, \\(\\mathds{1}_{k}\\) is a length-\\(k\\) vector of all ones, and \\(a_{k},b_{k},e_{kl}\\) are scalars. The subscript \\(c_{k}\\) denotes the size of the \\(k\\)-th cluster. Figure 5 illustrates the block structure of \\(f\\).\n\nWe provide a proof sketch of our claim. Since \\(\\overline{\\mathcal{A}}_{G^{\\prime}}\\) is trivial by assumption, it remains to show any function that is equivariant to the product of permutations \\(\\left(\\mathcal{S}_{1}\\times\\mathcal{S}_{2}\\ldots\\times\\mathcal{S}_{M}\\right)\\) admits the form in eqn. 20. We justify as follows:\n\n1. (within-cluster) \\(f_{kk}\\) is a linear permutation-equivariant function if and only if its diagonal elements are the same and its off-diagonal elements are the same ([34, Lemma 3.]);\n2. (across-cluster) \\(f_{kl}\\) for \\(k\\neq l\\) is a constant matrix since nodes within a cluster are indistinguishable; moreover, \\(e_{kl}\\) are unconstrained since the coarsened symmetry \\(\\overline{\\mathcal{A}}_{G^{\\prime}}\\) is trivial.\n\nAs an example, we illustrate the equivariant linear layer for two-cluster graph coarsening. Without loss of generality, assume that the node signals \\(X\\) are ordered according to the cluster assignment (e.g., \\(X_{[1:|V_{1}|]}\\) are node features for the first cluster, etc). Let \\(X_{(1)},X_{(2)}\\) denote the node features for the first and second cluster, \\(W^{s}_{(1)},W^{s}_{(2)}\\) denote the weights on the block diagonal for self-feature transformation, \\(W^{n}_{(1)},W^{n}_{(2)}\\) denote the weights on the block diagonal for within-cluster neighbors, and \\(W^{n}_{(12)},W^{n}_{(21)}\\) denote the weights off the block diagonal for across-cluster neighbors. Let \\(I\\) denote the identity matrix, \\(\\mathbf{1}_{(1)},\\mathbf{1}_{(2)}\\) denote the all-ones matrices with the same size as the corresponding\n\nFigure 5: The block structure of equivariant linear function \\(f:\\mathbb{R}^{n}\\to\\mathbb{R}^{n}\\) with respect to \\(\\mathcal{G}_{G\\to G^{\\prime}}\\) (where \\(G,G^{\\prime}\\) are asymmetric): Each diagonal block \\(f_{kk}\\) is diagonally constant and off-diagonally constant; Each off-diagonal block \\(f_{kl}\\) is a constant matrix.\n\ncluster, and \\(\\mathbf{1}_{(12)},\\mathbf{1}_{(21)}\\) denote the all-ones matrices mapping across clusters. Recall \\(\\odot\\) denotes the element-wise multiplication of two matrices. Then the equivariant linear layer is parameterized as\n\n\\[I\\odot\\begin{bmatrix}X_{(1)}W_{(2)}^{s}\\\\ X_{(2)}W_{(2)}^{s}\\end{bmatrix}+\\Big{(}\\begin{bmatrix}\\mathbf{1}_{(1)}&0\\\\ 0&\\mathbf{1}_{(2)}\\end{bmatrix}-I\\Big{)}\\odot\\begin{bmatrix}X_{(1)}W_{(1)}^{n} \\\\ X_{(2)}W_{(2)}^{n}\\end{bmatrix}+\\begin{bmatrix}0&\\mathbf{1}_{(12)}\\\\ \\mathbf{1}_{(21)}&0\\end{bmatrix}\\odot\\begin{bmatrix}X_{(1)}W_{(12)}^{n}\\\\ X_{(2)}W_{(21)}^{n}\\end{bmatrix}. \\tag{21}\\]\n\nFor cases where \\(\\overline{\\mathcal{A}}_{G^{\\prime}}\\) is nontrivial, if \\(\\overline{\\mathcal{A}}_{G^{\\prime}}\\) is abelian, we can use a construction by Serre ([83] Section 8.2). Observe that the symmetry of \\(\\overline{\\mathcal{A}}_{G^{\\prime}}\\) effectively constrains the patterns of \\(e_{kl}\\) in equation 20. We provide an example where \\(\\overline{\\mathcal{A}}_{G^{\\prime}}=\\mathcal{S}_{2}\\) in our image inpainting experiment (see Section 5.1), and defer the investigation of general cases for future work.\n\n## Appendix B Proofs of Our Theoretical Results\n\n### Proofs of Generalization with Exact Symmetries\n\n**Lemma 1** (Risk Gap).: _Let \\(\\mathcal{X}=\\mathbb{R}^{N\\times d},\\mathcal{Y}=\\mathbb{R}^{N\\times k}\\) be the input and output graph signal spaces on a fixed graph \\(G\\). Let \\(X\\sim\\mu\\) where \\(\\mu\\) is a \\(\\mathcal{S}_{N}\\)-invariant distribution on \\(\\mathcal{X}\\). Let \\(Y=f^{*}(X)+\\xi\\), where \\(\\xi\\in\\mathbb{R}^{N\\times k}\\) is random, independent of \\(X\\) with zero mean and finite variance and \\(f^{*}:\\mathcal{X}\\rightarrow\\mathcal{Y}\\) is \\(\\mathcal{A}_{G}\\)-equivariant. Then, for any \\(f\\in V\\) and for any compact group \\(\\mathcal{G}\\subseteq\\mathcal{S}_{N}\\), we can decompose \\(f\\) as_\n\n\\[f=\\bar{f}_{\\mathcal{G}}+f_{\\mathcal{G}}^{\\perp},\\]\n\n_where \\(\\bar{f}_{\\mathcal{G}}=\\mathcal{Q}_{\\mathcal{G}}f,f_{\\mathcal{G}}^{\\perp}=f- \\bar{f}_{\\mathcal{G}}\\). Moreover, the risk gap satisfies_\n\nLemma 1 is a straightforward extension of Lemma 6 in [8], which makes use of Lemma 1 in [8].\n\n**Lemma 1 in [8]**.: _Let \\(U\\) be any subspace of \\(V\\) that is closed under \\(\\mathcal{Q}\\). Define the subspaces \\(S\\) and \\(A\\) of, respectively, the \\(\\mathcal{G}\\)-symmetric and \\(\\mathcal{G}\\)-anti-symmetric functions in \\(U:S=\\{f\\in U:f\\) is \\(\\mathcal{G}\\)-equivariant \\(\\}\\) and \\(A=\\{f\\in U:\\mathcal{Q}f=0\\}\\). Then \\(U\\) admits admits an orthogonal decomposition into symmetric and anti-symmetric parts_\n\n\\[U=S\\oplus A\\]\n\nProof of Lemma 1.: The first part of Lemma 1\\(f=\\bar{f}_{\\mathcal{G}}+f_{\\mathcal{G}}^{\\perp}\\) follows from Lemma 1 in [8]. For the second part, by the assumption that the noise \\(\\xi\\) is independent of \\(X\\) with zero mean and finite variance, we can simplify the risk gap as\n\n\\[\\Delta(f,\\bar{f}_{\\mathcal{G}}) \\coloneqq\\mathbb{E}\\left[\\|Y-f(X)\\|_{F}^{2}\\right]-\\mathbb{E} \\left[\\|Y-\\bar{f}_{\\mathcal{G}}(X)\\|_{F}^{2}\\right]\\] \\[=\\mathbb{E}\\left[\\|f^{*}(X)-f(X)\\|_{F}^{2}\\right]-\\mathbb{E} \\left[\\|f^{*}(X)-\\bar{f}_{\\mathcal{G}}(X)\\|_{F}^{2}\\right]. \\tag{22}\\]\n\nSubstituting \\(f=\\bar{f}_{\\mathcal{G}}+f_{\\mathcal{G}}^{\\perp}\\) yields\n\n\\[\\mathbb{E}\\left[\\|f^{*}(X)-\\bar{f}_{\\mathcal{G}}(X)-f_{\\mathcal{ G}}^{\\perp}(X)\\|_{F}^{2}\\right]-\\mathbb{E}\\left[\\|f^{*}(X)-\\bar{f}_{\\mathcal{G}}(X )\\|_{F}^{2}\\right]\\] \\[= -2(f^{*}(X)-\\bar{f}_{\\mathcal{G}}(X),f_{\\mathcal{G}}^{\\perp}(X))_ {\\mu}+\\mathbb{E}\\left[\\|f_{\\mathcal{G}}^{\\perp}(X)\\|_{F}^{2}\\right]\\] \\[= -2(f^{*},f_{\\mathcal{G}}^{\\perp})_{\\mu}+\\left\\|f_{\\mathcal{G}}^{ \\perp}\\right\\|_{\\mu}^{2}. \\tag{23}\\]\n\nWe remark that Lemma 6 in [8] assumes that \\(f^{*}\\) is \\(\\mathcal{G}\\)-equivariant, so the first term in (23) vanishes. We are motivated from the symmetry model selection problem, and thereby relax the assumption of the chosen symmetry group \\(\\mathcal{G}\\) can differ from the target symmetry group \\(\\mathcal{A}_{\\mathcal{G}}\\).\n\n**Theorem 2** (Bias-Variance-Tradeoff).: _Let \\(\\mathcal{X}=\\mathbb{R}^{N\\times d},\\mathcal{Y}=\\mathbb{R}^{N\\times k}\\) be the graph signals spaces on a fixed graph \\(G\\). Let \\(\\mathcal{S}_{N}\\) act on \\(\\mathcal{X}\\) and \\(\\mathcal{Y}\\) by permuting the rows with representations \\(\\phi\\) and \\(\\psi\\). Let \\(\\mathcal{G}\\) be a subgroup of \\(\\mathcal{S}_{N}\\) acting with restricted representations \\(\\phi|_{\\mathcal{G}}\\) on \\(\\mathcal{X}\\) and \\(\\psi|_{\\mathcal{G}}\\) on \\(\\mathcal{Y}\\).__Let \\(X_{[i,j]}\\) be \\(\\mathcal{N}\\left(0,\\sigma_{X}^{2}\\right)\\) and \\(Y=f^{*}(X)+\\xi\\) where \\(f^{*}(x)=\\Theta^{\\top}x\\) is \\(\\mathcal{A}_{G}\\)-equivariant and \\(\\Theta\\in\\mathbb{R}^{Nd\\times Nk}\\). Assume \\(\\xi_{[i,j]}\\) is random, independent of \\(X\\), with mean 0 and \\(\\mathbb{E}\\left[\\xi\\xi^{\\top}\\right]=\\sigma_{\\xi}^{2}I<\\infty\\). Let \\(\\hat{\\Theta}\\) be the least-squares estimate of \\(\\Theta\\) from \\(n\\) i.i.d. examples \\(\\{(X_{i},Y_{i}):i=1,\\ldots,n\\}\\), \\(\\Psi_{\\mathcal{G}}(\\hat{\\Theta})\\) be its equivariant version with respect to \\(\\mathcal{G}\\). Let \\(\\left(\\chi_{\\psi|_{\\mathcal{G}}}\\mid\\chi_{\\phi|_{\\mathcal{G}}}\\right)=\\int_{ \\mathcal{G}}\\chi_{\\psi|_{\\mathcal{G}}}(g)\\chi_{\\phi|_{\\mathcal{G}}}(g)\\mathrm{ d}\\lambda(g)\\) denote the inner product of the characters. If \\(n>Nd+1\\) the risk gap is_\n\n\\[\\mathbb{E}\\left[\\Delta\\left(f_{\\hat{\\Theta}},f_{\\Psi_{\\mathcal{G}}(\\hat{\\Theta })}\\right)\\right]=\\underbrace{-\\sigma_{X}^{2}\\,\\|\\Psi_{\\mathcal{G}}^{\\perp}( \\Theta)\\|_{F}^{2}}_{\\text{bias}}+\\underbrace{\\sigma_{\\xi}^{2}\\frac{N^{2}dk- \\left(\\chi_{\\psi|_{\\mathcal{G}}}\\mid\\chi_{\\phi|_{\\mathcal{G}}}\\right)}{n-Nd-1} }_{\\text{variance}}.\\]\n\nTheorem 2 presents the risk gap in expectation, which follows from Lemma 1, by taking \\(f\\) as the least-squares estimator and using assumptions in the linear regression setting. To this end, we denote \\(\\mathbf{X}\\in\\mathbb{R}^{n\\times Nd},\\mathbf{Y}\\in\\mathbb{R}^{n\\times Nk},\\mathbf{\\xi}\\in \\mathbb{R}^{n\\times Nk}\\) as the training data arranged in matrix form, where \\(\\mathbf{Y}=f^{*}(\\mathbf{X})+\\mathbf{\\xi}\\). Recall that the least-squares estimator of \\(\\Theta\\) in the classic regime (\\(n>Nd\\)) is given by\n\n\\[\\hat{\\Theta}:=(\\mathbf{X}^{\\top}\\mathbf{X})^{\\dagger}\\mathbf{X}^{\\top}\\mathbf{Y}\\overset{\\text {a.e.}}{=}\\Theta+(\\mathbf{X}^{\\top}\\mathbf{X})^{-1}\\mathbf{X}^{\\top}\\mathbf{\\xi}, \\tag{24}\\]\n\nwhile its equivariant map is\n\n\\[\\Psi_{\\mathcal{G}}(\\hat{\\Theta})=\\int_{\\mathcal{G}}\\phi(g)\\,\\hat{\\Theta}\\, \\psi\\left(g^{-1}\\right)\\mathrm{d}\\lambda(g). \\tag{25}\\]\n\nOur proof makes use of the following results in [8], which we restate adapted versions here for our setting.\n\n**Proposition 11 in [8].**_Let \\(V=\\{f_{W}:f_{W}(x)=W^{\\top}x,W\\in\\mathbb{R}^{d\\times k},x\\in\\mathbb{R}^{d}\\}\\) denote the space of linear functions. Let \\(X\\sim\\mu\\) with \\(\\mathbb{E}[XX^{\\top}]=\\Sigma\\). For any linear functions \\(f_{W_{1}},f_{W_{2}}\\in V\\), the inner product on \\(V\\) satisfies_\n\n\\[\\langle f_{W_{1}},f_{W_{2}}\\rangle_{\\mu}=\\operatorname{Tr}(W_{1}^{\\top}\\Sigma W _{2}). \\tag{26}\\]\n\n**Theorem 13 in [8]** (Simplified, Adapted).: _Consider the same setting as Theorem 2. For \\(n>Nd+1\\),_\n\n\\[\\sigma_{X}^{2}\\mathbb{E}\\left[\\left\\|\\Psi_{\\mathcal{G}}^{\\perp}\\left(\\left(\\bm {X}^{\\top}\\mathbf{X}\\right)^{+}\\mathbf{X}^{\\top}\\mathbf{\\xi}\\right)\\right\\|_{F}^{2}\\right] =\\sigma_{\\xi}^{2}\\frac{N^{2}dk-\\left(\\chi_{\\psi|_{\\mathcal{G}}}\\mid\\chi_{\\phi| _{\\mathcal{G}}}\\right)}{n-Nd-1}.\\]\n\nProof of Theorem 2.: We first plug in the least-squares expressions \\(\\hat{\\Theta},\\Psi_{\\mathcal{G}}(\\hat{\\Theta})\\) to Lemma 1 and treat the mismatch term and constraint term separately; We complete the proof by collecting common terms together.\n\nFor the mismatch term, our goal is to compute\n\n\\[-2\\,\\mathbb{E}\\left[\\langle\\Theta,\\hat{\\Theta}-\\Psi_{\\mathcal{G}}(\\hat{\\Theta} )\\rangle_{\\mu}\\right], \\tag{27}\\]\n\nwhere the expectation is taken over the test point \\(X\\) and the training data \\(\\mathbf{X},\\mathbf{\\xi}\\).\n\nTo that end, we write\n\n\\[\\left(\\hat{\\Theta}-\\Psi_{\\mathcal{G}}(\\hat{\\Theta})\\right)x\\overset{a.e.}{=} \\Theta^{\\top}x+\\mathbf{\\xi}^{\\top}\\mathbf{X}(\\mathbf{X}^{\\top}\\mathbf{X})^{-1}x-\\int_{ \\mathcal{G}}\\psi(g^{-1})\\left(\\Theta^{\\top}+\\mathbf{\\xi}^{\\top}\\mathbf{X}(\\mathbf{X}^{ \\top}\\mathbf{X})^{-1}\\right)\\phi(g)x\\,\\mathrm{d}\\lambda(g). \\tag{28}\\]\n\nTaking expectation yields\n\n\\[\\mathbb{E}_{X,\\mathbf{X},\\mathbf{\\xi}}\\left[\\langle\\Theta,\\hat{\\Theta}- \\Psi_{\\mathcal{G}}(\\hat{\\Theta})\\rangle_{\\mu}\\right] =\\|\\Theta\\|_{\\mu}^{2}+\\mathbb{E}_{X,\\mathbf{X},\\mathbf{\\xi}}\\left[\\langle \\Theta^{\\top}\\mathbf{X},\\mathbf{\\xi}^{\\top}\\mathbf{X}(\\mathbf{X}^{\\top}\\mathbf{X})^{-1}x\\rangle\\right]\\] \\[-\\mathbb{E}_{X,\\mathbf{X},\\mathbf{\\xi}}\\left[\\langle\\Theta^{\\top}x,\\int_{ \\mathcal{G}}\\psi(g^{-1})\\left(\\Theta^{\\top}+\\mathbf{\\xi}^{\\top}\\mathbf{X}(\\mathbf{X}^{ \\top}\\mathbf{X})^{-1}\\right)\\phi(g)x\\,\\mathrm{d}\\lambda(g)\\rangle\\right]. \\tag{29}\\]\n\nNote that \\(\\mathbf{\\xi}\\) is independent with \\(\\mathbf{X}\\) and mean 0, so the second term in (29) vanishes. Similarly, the part \\(\\mathbb{E}_{X,\\mathbf{X},\\mathbf{\\xi}}\\int_{\\mathcal{G}}\\psi(g^{-1})\\left(\\mathbf{\\xi}^{ \\top}\\mathbf{X}(\\mathbf{X}^{\\top}\\mathbf{X})^{-1}\\right)\\phi(g)x\\,\\mathrm{d}\\lambda(g)\\) also vanishes (by first taking conditional expectation of \\(\\mathbf{\\xi}\\) conditioned on \\(\\mathbf{X}\\)). Thus, we arrive at\n\n\\[\\mathbb{E}\\left[\\left\\langle\\Theta,\\hat{\\Theta}-\\Psi_{\\mathcal{G}}( \\hat{\\Theta})\\right\\rangle_{\\mu}\\right] =\\left\\|\\Theta\\right\\|_{\\mu}^{2}-\\mathbb{E}_{x}\\left[\\left\\langle \\Theta^{\\top}x,\\int_{\\mathcal{G}}\\psi(g^{-1})\\,\\Theta^{\\top}\\,\\phi(g)x\\,\\mathrm{ d}\\lambda(g)\\right\\rangle\\right]\\] \\[=\\left\\|\\Theta\\right\\|_{\\mu}^{2}-\\left\\langle\\Theta,\\Psi_{ \\mathcal{G}}(\\Theta)\\right\\rangle_{\\mu}\\] \\[=\\left\\|\\Psi_{\\mathcal{G}}^{\\perp}(\\Theta)\\right\\|_{\\mu}^{2}\\] \\[=-2\\,\\sigma_{X}^{2}\\|\\Psi_{\\mathcal{G}}^{\\perp}(\\Theta)\\|_{F}^{2}, \\tag{30}\\]\n\nwhere the last equality follows from Proposition 11 in [8] with the assumption that \\(\\Sigma=\\sigma_{X}^{2}\\). This finishes the computation for the mismatch term.\n\nNow for the constraint term, we have\n\n\\[\\|f_{\\mathcal{G}}^{\\perp}\\|_{\\mu}^{2} =\\|\\Psi_{\\mathcal{G}}^{\\perp}(\\hat{\\Theta})\\|_{\\mu}^{2} \\tag{31}\\] \\[=\\sigma_{X}^{2}\\,\\mathbb{E}_{\\mathbf{X},\\mathbf{\\xi}}\\|\\Psi_{\\mathcal{G} }^{\\perp}\\left(\\Theta+(\\mathbf{X}^{\\top}\\mathbf{X})^{-1}\\mathbf{X}^{\\top}\\mathbf{\\xi}\\right)\\| ^{2}\\] (32) \\[=\\sigma_{X}^{2}\\|\\Psi_{\\mathcal{G}}^{\\perp}(\\Theta)\\|_{F}^{2}+ \\sigma_{X}^{2}\\,\\mathbb{E}_{\\mathbf{X},\\mathbf{\\xi}}\\|\\Psi_{\\mathcal{G}}^{\\perp}\\left( (\\mathbf{X}^{\\top}\\mathbf{X})^{-1}\\mathbf{X}^{\\top}\\mathbf{\\xi}\\right)\\|^{2}, \\tag{33}\\]\n\nwhere the last equality follows from linearity of expectation, \\(\\mathbb{E}[\\mathbf{\\xi}]=0\\) and \\(\\mathbf{\\xi}\\) independent of \\(x\\).\n\nCombining the mismatch term in (30) with the constraint term in (33), the risk gap becomes\n\n\\[\\mathbb{E}\\left[\\Delta\\left(f_{\\hat{\\Theta}},f_{\\Psi_{\\mathcal{G}}(\\hat{\\Theta })}\\right)\\right]=-\\sigma_{X}^{2}\\|\\,\\Psi_{\\mathcal{G}_{L}}^{\\perp}(\\Theta)\\| ^{2}+\\sigma_{X}^{2}\\,\\mathbb{E}_{\\mathbf{X},\\mathbf{\\xi}}\\|\\Psi_{\\mathcal{G}_{L}}^{ \\perp}\\left((\\mathbf{X}^{\\top}\\mathbf{X})^{-1}\\mathbf{X}^{\\top}\\mathbf{\\xi}\\right)\\|^{2}, \\tag{34}\\]\n\nApplying Theorem 13 in [8], the second term in (34) reduces to\n\n\\[\\sigma_{X}^{2}\\,\\mathbb{E}_{\\mathbf{X},\\mathbf{\\xi}}\\|\\Psi_{\\mathcal{G}_{L}}^{\\perp} \\left((\\mathbf{X}^{\\top}\\mathbf{X})^{-1}\\mathbf{X}^{\\top}\\mathbf{\\xi}\\right)\\|^{2}=\\sigma_{ \\xi}^{2}\\frac{N^{2}dk-\\left(\\chi_{\\psi|\\sigma}\\mid\\chi_{\\phi|\\sigma}\\right)}{ n-Nd-1}, \\tag{35}\\]\n\nfrom which the theorem follows immediately.\n\nFinally, we state a well-known result for the risk of (Ordinary) Least-Squares Estimator (see [84, 85] and references therein).\n\n**Lemma 6** (Risk of Least-Squares Estimator).: _Consider the same set-up as Theorem 2. For \\(n>Nd+1\\),_\n\n\\[\\mathbb{E}\\left[\\|Y-\\hat{\\Theta}^{\\top}X\\|_{F}^{2}\\right]=\\sigma_{\\xi}^{2} \\frac{Nd}{n-Nd-1}+\\sigma_{\\xi}^{2}.\\]\n\nProof of Lemma 6.: Recall \\(X,Y\\) denote the test sample. We denote the risk of the least-squares estimator _conditional on the training data_\\(\\mathbf{X}\\in\\mathbb{R}^{n\\times Nd}\\) as \\(\\mathcal{R}(\\hat{\\Theta}\\mid\\mathbf{X})\\), which has the following bias-variance decomposition:\n\n\\[\\mathcal{R}(\\hat{\\Theta}\\mid\\mathbf{X}) =\\mathbb{E}\\left[\\|Y-\\hat{\\Theta}^{\\top}X\\|_{F}^{2}\\mid\\mathbf{X}\\right] \\tag{36}\\] \\[=\\mathbb{E}\\left[\\|\\Theta^{\\top}X+\\xi-\\hat{\\Theta}^{\\top}X\\|_{F} ^{2}\\mid\\mathbf{X}\\right]\\] (37) \\[=\\mathbb{E}\\left[\\|(\\Theta-\\hat{\\Theta})^{\\top}X\\|_{F}^{2}\\mid \\mathbf{X}\\right]+\\sigma_{\\xi}^{2}, \\tag{38}\\]\n\nwhere the last equality follows from \\(\\xi\\) being zero mean and independent with \\(X\\). The second term \\(\\sigma_{\\xi}^{2}\\) is also known as _irreducible error_. We decompose the first term into\n\n\\[\\mathbb{E}\\left[\\|(\\Theta-\\hat{\\Theta})^{\\top}X\\|_{F}^{2}\\mid\\mathbf{X}\\right]= \\mathbb{E}\\left[\\|(\\Theta-\\mathbb{E}[\\hat{\\Theta}])^{\\top}X\\|_{F}^{2}+\\|( \\mathbb{E}[\\hat{\\Theta}]-\\hat{\\Theta})^{\\top}X\\|_{F}^{2}\\mid\\mathbf{X}\\right]. \\tag{39}\\]\n\nRecall that \\(\\hat{\\Theta}\\stackrel{{ a.e.}}{{=}}(\\mathbf{X}^{\\top}\\mathbf{X})^{-1}\\mathbf{X} ^{\\top}\\mathbf{Y}=(\\mathbf{X}^{\\top}\\mathbf{X})^{-1}\\mathbf{X}^{\\top}(\\mathbf{X}\\Theta+\\xi)=\\Theta +(\\mathbf{X}^{\\top}\\mathbf{X})^{-1}\\mathbf{X}^{\\top}\\xi\\). Thus \\(\\mathbb{E}[\\hat{\\Theta}]=\\Theta\\) and (39) simplifies to \\(\\mathbb{E}\\left[\\|(\\mathbb{E}[\\hat{\\Theta}]-\\hat{\\Theta})^{\\top}X\\|_{F}^{2} \\mid\\mathbf{X}\\right]\\).\n\nWe finish computing the risk by taking expectation over \\(\\mathbf{X}\\), and using \\(\\mathbb{E}[\\hat{\\Theta}]-\\hat{\\Theta}=(\\mathbf{X}^{\\top}\\mathbf{X})^{-1}\\mathbf{X}^{\\top}\\xi\\),\n\n\\[\\mathbb{E}\\left[\\left\\|Y-\\hat{\\Theta}^{\\top}X\\right\\|_{F}^{2}\\right] =\\mathbb{E}\\left[\\mathcal{R}(\\hat{\\Theta}\\mid\\mathbf{X})\\right] \\tag{40}\\] \\[=\\mathbb{E}_{\\mathbf{X}}\\left[\\mathbb{E}_{X,\\xi}\\left[\\left\\|( \\mathbb{E}[\\hat{\\Theta}]-\\hat{\\Theta})^{\\top}X\\right\\|_{F}^{2}\\mid\\mathbf{X} \\right]\\right]+\\sigma_{\\xi}^{2}\\] (41) \\[=\\mathbb{E}\\left[\\left\\|\\left((\\mathbf{X}^{\\top}\\mathbf{X})^{-1}\\mathbf{X}^{ \\top}\\xi\\right)^{\\top}X\\right\\|_{F}^{2}\\right]+\\sigma_{\\xi}^{2}\\] (42) \\[=\\sigma_{\\xi}^{2}\\operatorname{tr}\\left(\\mathbb{E}[(\\mathbf{X}^{\\top} \\mathbf{X})^{-1}]\\,\\sigma_{X}^{2}I\\right)+\\sigma_{\\xi}^{2}. \\tag{43}\\]\n\nBy [86, Lemma 2.3], for \\(n>Nd+1\\), \\(\\mathbb{E}[(\\mathbf{X}^{\\top}\\mathbf{X})^{-1}]=\\frac{Nd}{n-Nd-1}I\\). Putting this in (43) completes the proof. \n\n### Proofs of Generalization with Approximate Symmetries\n\nIn Definition 3, we construct the _symmetry group of \\(G\\) induced by the coarsening \\(G^{\\prime}\\)_ via a semidirect product,\n\n\\[\\mathcal{G}_{G\\to G^{\\prime}}=\\left(\\mathcal{S}_{c_{1}}\\times\\mathcal{S}_{c_{2 }}\\ldots\\times\\mathcal{S}_{c_{M}}\\right)\\rtimes\\overline{\\mathcal{A}}_{G^{ \\prime}}.\\]\n\nWe explain the construction in more details here. We first recall the definition of semidirect product. Given two groups \\(\\mathcal{G}_{1},\\mathcal{G}_{2}\\) and a group homomorphism \\(\\varphi:\\mathcal{G}_{2}\\to\\operatorname{aut}(\\mathcal{G}_{1})\\), we can construct a new group \\(\\mathcal{G}_{1}\\rtimes_{\\varphi}\\mathcal{G}_{2}\\), called the semidirect product of \\(\\mathcal{G}_{1},\\mathcal{G}_{2}\\) with respect to \\(\\varphi\\) as follows:\n\n1. The underlying set is the Cartesian product \\(\\mathcal{G}_{1}\\times\\mathcal{G}_{2}\\);\n2. The group operation \\(\\circ\\) is determined by the homomorphism \\(\\varphi\\), such that \\[\\circ:(\\mathcal{G}_{1}\\rtimes_{\\varphi}\\mathcal{G}_{2})\\times( \\mathcal{G}_{1}\\rtimes_{\\varphi}\\mathcal{G}_{2}) \\to(\\mathcal{G}_{1}\\rtimes_{\\varphi}\\mathcal{G}_{2})\\] \\[(g_{1},g_{2})\\circ(g_{1}^{\\prime},g_{2}^{\\prime}) =(g_{1}\\,\\varphi_{g_{2}}(g_{1}^{\\prime}),g_{2}g_{2}^{\\prime}), \\quad g_{1},g_{1}^{\\prime}\\in\\mathcal{G}_{1};g_{2},g_{2}^{\\prime}\\in\\mathcal{G }_{2},\\]\n\nTake \\(\\mathcal{G}_{1}=\\left(\\mathcal{S}_{c_{1}}\\times\\mathcal{S}_{c_{2}}\\ldots\\times \\mathcal{S}_{c_{M}}\\right)\\), \\(\\mathcal{G}_{2}=\\overline{\\mathcal{A}}_{G^{\\prime}}\\). Note that \\(\\mathcal{G}_{1}\\) is a normal subgroup in \\(\\mathcal{G}_{G\\to G^{\\prime}}\\); Namely, for all \\(s\\in\\mathcal{G}_{G\\to G^{\\prime}},g_{1}\\in\\mathcal{G}_{1}\\), we have \\(sg_{1}s^{-1}\\in\\mathcal{G}_{1}\\). Thus, the map \\(g_{1}\\mapsto sg_{1}s^{-1}\\) is an automorphism of \\(\\mathcal{G}_{1}\\). In particular, the homomorphism \\(\\varphi_{g_{2}}(g_{1})=g_{2}\\,g_{1}\\,g_{2}^{-1}\\) for \\(g_{2}\\in\\mathcal{G}_{2}\\) describes the action of \\(\\mathcal{G}_{2}\\) on \\(\\mathcal{G}_{1}\\) by conjugation 2. In the context of a graph \\(G\\) with \\(N\\) nodes and its coarsening \\(G^{\\prime}\\) with \\(M\\) clusters, the homomorphism \\(\\varphi\\) describes how across-cluster permutations in \\(G^{\\prime}\\) act on the original graph \\(G\\). Note that in the special case where \\(\\mathcal{A}_{G^{\\prime}}\\) acts transitively on the coarsened nodes, we recover the wreath product -- a special kind of semidirect product. Our construction of \\(\\mathcal{G}_{G\\to G^{\\prime}}\\) can be seen as a natural generalization of the wreath product.\n\nFootnote 2: In our context, it is sensible to write \\(\\varphi_{g_{2}}(g_{1})=g_{2}\\,g_{1}\\,g_{2}^{-1}\\) given that \\(\\mathcal{G}_{1},\\mathcal{G}_{2}\\) are originally inside a common group \\(\\mathcal{S}_{N}\\). Yet semidirect product applies to two arbitrary groups — not necessarily inside a common group initially, where \\(\\varphi_{g_{2}}(g_{1})\\) is an abstraction of \\(g_{2}\\,g_{1}\\,g_{2}^{-1}\\).\n\n**Corollary 3** (Risk Gap via Graph Coarsening).: _Let \\(\\mathcal{X}=\\mathbb{R}^{N\\times d},\\mathcal{Y}=\\mathbb{R}^{N\\times k}\\) be the input and output graph signal spaces on a fixed graph \\(G\\). Let \\(X\\sim\\mu\\) where \\(\\mu\\) is a \\(\\mathcal{S}_{N}\\)-invariant distribution on \\(\\mathcal{X}\\). Let \\(Y=f^{*}(X)+\\xi\\), where \\(\\xi\\in\\mathbb{R}^{N\\times k}\\) is random, independent of \\(X\\) with zero mean and finite variance, and \\(f^{*}:\\mathbb{R}^{N\\times d}\\to\\mathbb{R}^{N\\times k}\\) be an approximately equivariant mapping with equivariance rate \\(\\kappa\\). Then, for any \\(G^{\\prime}\\) that coarsens \\(G\\) up to error \\(\\epsilon\\), for any \\(f\\in V\\), we have_\n\n\\[\\Delta(f,\\bar{f}_{\\mathcal{G}_{G\\to G^{\\prime}}})=\\underbrace{-2(f^{*},f^{ \\perp}_{\\mathcal{G}_{G\\to G^{\\prime}}})_{\\mu}}_{\\text{mismatch}}+\\underbrace{ \\left\\|f^{\\perp}_{\\mathcal{G}_{G\\to G^{\\prime}}}\\right\\|_{\\mu}^{2}}_{\\text{ constraint}}\\geq-2\\kappa(\\epsilon)\\left\\|f^{\\perp}_{\\mathcal{G}_{G\\to G^{\\prime}}} \\right\\|_{\\mu}+\\left\\|f^{\\perp}_{\\mathcal{G}_{G\\to G^{\\prime}}}\\right\\|_{\\mu}^{2}\\]Proof of Corollary 3.: We start by simplifying the mismatch term in Lemma 1,\n\n\\[-2\\mathbb{E}\\left[\\langle f^{*}(x),f^{\\perp}_{\\mathcal{G}_{G\\to G^{\\prime}}}(x) \\rangle\\right] =-2\\mathbb{E}\\left[\\langle f^{*}(x)-f^{*}_{\\mathcal{G}_{G\\to G^{ \\prime}}}(x)+f^{*}_{\\mathcal{G}_{G\\to G^{\\prime}}}(x),f^{\\perp}_{\\mathcal{G}_{ G\\to G^{\\prime}}}(x)\\rangle\\right]\\] \\[=-2\\mathbb{E}\\left[\\langle\\underbrace{f^{*}(x)-f^{*}_{\\mathcal{G}_ {G\\to G^{\\prime}}}(x)}_{\\mathcal{G}_{L}\\text{-anti-symmetric part of }f^{*}}\\cdot \\underbrace{f^{\\perp}_{\\mathcal{G}_{L}\\to G^{\\prime}}(x)}_{\\mathcal{G}_{L}\\text {-anti-symmetric part of }f}\\rangle\\right]\\] \\[\\geq-2\\left\\|f^{*}-f^{*}_{\\mathcal{G}_{G\\to G^{\\prime}}}\\right\\|_{\\mu} \\left\\|f^{\\perp}_{\\mathcal{G}_{G\\to G^{\\prime}}}\\right\\|_{\\mu}\\quad\\text{( By Cauchy Schwarz Ineq.)}\\] \\[\\geq-2\\left\\langle\\epsilon\\right\\rangle\\left\\|f^{\\perp}_{\\mathcal{ G}_{G\\to G^{\\prime}}}\\right\\|_{\\mu}.\\quad\\text{( By Definition \\ref{def:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq: eq:eqTherefore, the variance term evaluates to\n\n\\[\\sigma_{\\xi}^{2}\\frac{N^{2}-\\left(\\chi_{\\psi|_{\\mathcal{G}}}\\mid\\chi_{\\psi|_{ \\mathcal{G}}}\\right)}{n-N-1}=\\sigma_{\\xi}^{2}\\frac{7}{n-4}. \\tag{49}\\]\n\nPutting (47) and (49) together yields the generalization gap of for the least square estimator \\(f_{\\Theta}\\) compared to its \\(\\mathcal{S}_{3}\\)-equivariant version \\(f_{\\Psi_{\\mathcal{S}_{3}}(\\hat{\\Theta})}\\).\n\nAs a comparison, when choosing the symmetry group of the target function \\(\\mathcal{G}=\\mathcal{S}_{2}\\), the bias vanishes and note that \\(\\left(\\chi_{\\psi_{\\mathcal{S}_{2}}}\\mid\\chi_{\\phi_{\\mathcal{S}_{2}}}\\right)= \\frac{1}{2}(3^{2}+1^{2})=5\\), so generalization gap is\n\n\\[\\mathbb{E}\\left[\\Delta\\left(f_{\\hat{\\Theta}},f_{\\Psi_{\\mathcal{S}_{2}}(\\hat{ \\Theta})}\\right)\\right]=\\sigma_{\\xi}^{2}\\frac{4}{n-4}. \\tag{50}\\]\n\nWe see that choosing \\(\\mathcal{G}=\\mathcal{S}_{3}\\) is better if \\(a\\approx e,b\\approx c\\approx d\\) (i.e., \\(f^{*}\\) is approximately \\(\\mathcal{S}_{3}\\)-invariant) and the training sample size \\(n\\) small, whereas \\(\\mathcal{S}_{2}\\) is better vice versa. This analysis illustrates the advantage of choosing a (suitably) larger symmetry group to induce a smaller hypothesis class when learning with limited data, and introduce useful inductive bias when the target function is approximately symmetric with respect to a larger group. We further illustrate our theoretical analysis via simulations, with details and results shown in Figure 6.\n\n### Example: Approximately Equivariant Mapping on a Geometric Graph\n\nIn this section, we illustrate a construction of an approximately equivariant mapping. We focus on a version of Definition 3 that does not take to account the symmetries of \\(G^{\\prime}\\). Namely, we consider a definition of the approximate symmetries as\n\n\\[\\mathcal{G}_{G\\to G^{\\prime}}:\\;=\\;\\mathcal{S}_{c_{1}}\\times\\mathcal{S}_{c_{2}} \\ldots\\times\\mathcal{S}_{c_{M}}\\subset\\mathcal{S}_{N}.\\]\n\nEquivalently, we restrict the analysis to coarsening graphs \\(G^{\\prime}\\) that are asymmetric.\n\n**Background from graphon-signal analysis.** To support our construction, we cite some definitions and results from [79].\n\n**Definition 8**.: _Let \\(r>0\\). The graphon-signal space with signals bounded by \\(r\\) is \\(\\mathcal{WL}_{r}:=\\mathcal{W}\\times L_{r}^{\\infty}[0,1]\\), where \\(L_{r}^{\\infty}[0,1]\\) is the ball of radius \\(r\\) in \\(L^{\\infty}[0,1]\\). The distance in \\(\\mathcal{WL}_{r}\\) is defined for \\((W,s),(V,g)\\in\\mathcal{WL}_{r}\\) by_\n\n\\[d_{\\square}\\big{(}(W,s),(V,g)\\big{)}:=\\left\\|(W,s)-(V,g)\\right\\|_{\\square}:= \\left\\|W-V\\right\\|_{\\square}+\\left\\|s-g\\right\\|_{1}.\\]\n\nFigure 6: Choosing the symmetry group corresponding to the target function usually yields the best generalization ((a), (b), (d)), but not always: when the number of training data \\(n\\) is small and the target function \\(f\\) is approximately equivariant with respect to a larger group, choosing the larger symmetry group could yield further generalization gain, as shown in (c) empirically. The dashed gray vertical line highlights the theoretical threshold \\(n^{*}\\approx 35\\), before which using \\(\\mathcal{S}_{3}\\) yields better generalization than \\(\\mathcal{S}_{2}\\), validating our theoretical analysis. We set \\(\\sigma_{X}^{2}=1,\\sigma_{\\xi}^{2}=\\frac{1}{64}\\), conduct \\(10\\) random runs and compute the generalization error based on \\(300\\) test points. We obtain the estimators via stochastic gradient descent, and enforce the symmetry via tying weights. The titles of each subplot indicate the symmetry of the target function, and display the target function values.\n\n_Moreover,_\n\n\\[\\delta_{\\square}\\big{(}(W,s),(V,g)\\big{)}=\\inf_{\\phi}d_{\\square}\\big{(}(W,s),(V^{ \\phi},g^{\\phi})\\big{)},\\]\n\n_where \\(g^{\\phi}(x)=g(\\phi(x))\\) and \\(\\phi\\) is a measure preserving bijection._\n\nAny graph-signal induces a graphon signal in the natural way, as in Definition 1. The cut norm and distance between two graph-signals is defined to be the cut norm and distance between the two induced graphon-signal respectively. Similarly, the \\(L_{1}\\) distance between a signal \\(q\\) on a graph and a signal \\(s\\) on \\([0,1]\\) is defined to be the \\(L_{1}\\) distance between the induced signal from \\(q\\) and \\(s\\). The supremum in the definition of cut distance between two induced graphon-signals is realized by some measure preserving bijection.\n\n**Sampling graphon-signals.** The following construction is from [79, Section 3.4]. Let \\(\\Lambda=(\\lambda_{1},\\ldots\\lambda_{N})\\in[0,1]^{N}\\) be \\(N\\) independent uniform random samples from \\([0,1]\\), and \\((W,s)\\in\\mathcal{WL}_{r}\\). We define the _random weighted graph_\\(W(\\Lambda)\\) as the weighted graph with \\(N\\) nodes and edge weight \\(w_{i,j}=W(\\lambda_{i},\\lambda_{j})\\) between node \\(i\\) and node \\(j\\). We similarly define the _random sampled signal_\\(s(\\Lambda)\\) with value \\(s_{i}=s(\\lambda_{i})\\) at each node \\(i\\). Note that \\(W(\\Lambda)\\) and \\(s(\\Lambda)\\) share the sample points \\(\\Lambda\\). We then define a random simple graph as follows. We treat each \\(w_{i,j}=W(\\lambda_{i},\\lambda_{j})\\) as the parameter of a Bernoulli variable \\(e_{i,j}\\), where \\(\\mathbb{P}(e_{i,j}=1)=w_{i,j}\\) and \\(\\mathbb{P}(e_{i,j}=0)=1-w_{i,j}\\). We define the _random simple graph_\\(\\mathbb{G}(W,\\Lambda)\\) as the simple graph with an edge between each node \\(i\\) and node \\(j\\) if and only if \\(e_{i,j}=1\\). The following theorem is [79, Theorem 3.6]\n\n**Theorem 3.6 from [79]** (Sampling lemma for graphon-signals).: _Let \\(r>1\\). There exists a constant \\(N_{0}>0\\) that depends on \\(r\\), such that for every \\(N\\geq N_{0}\\), every \\((W,s)\\in\\mathcal{WL}_{r}\\), and for \\(\\Lambda=(\\lambda_{1},\\ldots\\lambda_{N})\\in[0,1]^{N}\\) independent uniform random samples from \\([0,1]\\), we have_\n\n\\[\\mathbb{E}\\bigg{(}\\delta_{\\square}\\Big{(}\\big{(}W,s\\big{)},\\big{(}\\mathbb{G}(W,\\Lambda),s(\\Lambda)\\big{)}\\Big{)}\\bigg{)}<\\frac{15}{\\sqrt{\\log(N)}}. \\tag{51}\\]\n\nBy Markov's inequality and (51), for any \\(0<p<1\\), there is an event of probability \\(1-p\\) (regarding the choice of \\(\\Lambda\\)) in which\n\n\\[\\delta_{\\square}\\Big{(}\\big{(}W,s\\big{)},\\big{(}\\mathbb{G}(W,\\Lambda),s( \\Lambda)\\big{)}\\Big{)}<\\frac{15}{p\\sqrt{\\log(N)}}. \\tag{52}\\]\n\n**Stability to deformations of mappings on geometric graphs.** Let \\(\\mathcal{M}\\) be a metric space with an atomless standard probability measure defined over the Borel sets (up to completion of the measure). Such a probability space is equivalent to the standard probabiltiy space \\([0,1]\\) with Lebesgue measure. Namely, there are co-null sets \\(A\\subset\\mathcal{M}\\) and \\(B\\subset[0,1]\\), and a measure preserving bijection \\(\\phi:A\\to B\\). Hence, graphon analysis applied as-is when replacing the domain \\([0,1]\\) with \\(\\mathcal{M}\\). Suppose that we are interested in a target function \\(f_{\\mathcal{M}}:L^{1}(\\mathcal{M})\\to L^{1}(\\mathcal{M})\\) that is stable to deformations in the following sense.\n\n**Definition 9**.: _Let \\(\\epsilon>0\\). A measurable bijection \\(\\nu:\\mathcal{M}\\to\\mathcal{M}\\) is called a deformation up to \\(\\epsilon\\), if there exists an event \\(B_{\\epsilon}\\subset\\mathcal{M}\\) with probability greater than \\(1-\\epsilon\\) such that for every \\(x\\in B_{\\epsilon}\\)_\n\n\\[d_{\\mathcal{M}}\\big{(}\\nu(x),x\\big{)}<\\epsilon.\\]\n\n_The mapping \\(f_{\\mathcal{M}}:L^{1}(\\mathcal{M})\\to L^{1}(\\mathcal{M})\\) is called stable to deformations with stability constant \\(C\\), if for any deformation \\(\\nu\\) up to \\(\\epsilon\\), and every \\(s\\in L^{1}(\\mathcal{M})\\), we have_\n\n\\[\\|f_{\\mathcal{M}}(s)-f_{\\mathcal{M}}(s\\circ\\nu)\\circ\\nu^{-1}\\|_{1}<C\\epsilon.\\]\n\nSuppose that we observe a discretized version of the domain \\(\\mathcal{M}\\), defined as follows. There is a graphon \\(W:\\mathcal{M}^{2}\\to[0,1]\\) defined as\n\n\\[W(x,y)=r\\big{(}d(x,y)\\big{)}, \\tag{53}\\]\n\nwhere \\(r:\\mathbb{R}_{+}\\to[0,1]\\) is a decreasing function with support \\([0,\\rho]\\). Instead of observing \\(W\\), we observe a graph \\(G=\\mathbb{G}(W,\\Lambda)\\) with node set \\([N]\\), sampled from \\(W\\) on the random independent points \\(\\Lambda=\\{\\lambda_{n}\\}_{n=1}^{N}\\subset\\mathcal{M}\\) as above. Suppose moreover that any graph signal is sampled from a signal in \\(L^{1}(\\mathcal{M})\\), on the same random points \\(\\Lambda\\), as above. Suppose that the target \\(f_{\\mathcal{M}}\\) on the continuous domain is well approximated by some mapping \\(f^{*}:L^{1}[N]\\to L^{1}[N]\\) on the discrete domain in the following sense. For every \\(s\\in L^{1}(\\mathcal{M})\\), let \\(s_{G}\\) be the graph signal sampled on the random samples \\(\\{\\lambda_{n}\\}_{n}\\). Then there is an event of high probability such that\n\n\\[\\|f^{*}s_{G}-\\{\\big{(}f_{\\mathcal{M}}(s)\\big{)}(x_{n})\\}_{n}\\|_{1}<e\\]\n\nfor some small \\(e\\). We hence consider \\(f^{*}\\) as the target mapping of the learning problem. One example of such a scenario is when there exists some Lipschitz continuous mapping \\(\\Theta:\\mathcal{WL}_{r}\\to\\mathcal{WL}_{r}\\) with Lipschitz constant \\(L\\), such that \\(f_{\\mathcal{M}}=\\Theta(W,\\cdot)\\) and \\(f^{*}=\\Theta(G,\\cdot)\\). Indeed, by (52), for some \\(p\\) as small as we like, there is an event of probability \\(1-p\\) in which, up to a measure preserving bijection,\n\n\\[\\|f_{\\mathcal{M}}s-f^{*}s_{G}\\|_{1} \\leq\\delta_{\\square}\\Big{(}\\big{(}W,f_{\\mathcal{M}}s\\big{)},\\big{(} G,f^{*}s_{G}\\big{)}\\Big{)}\\] \\[\\leq L\\delta_{\\square}\\Big{(}\\big{(}W,s\\big{)},\\big{(}G,s_{G} \\big{)}\\Big{)}<\\frac{15L}{p\\sqrt{\\log(N)}}=e. \\tag{54}\\]\n\nA concrete example is when \\(\\Theta\\) is a message passing neural network (MPNN) with Lipschitz continuous message and update functions, and normalized sum aggregation [79, Theorem 4.1].\n\nLet \\(G^{\\prime}\\) be a graph that coarsens \\(G\\) up to error \\(\\epsilon\\). In the same event as above, by (52), up to a measure preserving bijection,\n\n\\[\\delta_{\\square}(W_{G^{\\prime}},W)\\leq\\delta_{\\square}(W_{G^{\\prime}},W_{G})+ \\delta_{\\square}(W_{G},W)\\leq\\epsilon+e=u. \\tag{55}\\]\n\nWe next show an approximation property that we state here informally: Since \\(W(x,y)\\approx 0\\) for \\(x\\) away from \\(y\\), we must have \\(W_{G^{\\prime}}(x,y)\\approx 0\\) as well for a set of high measure. Otherwise, \\(\\delta_{\\square}(W_{G^{\\prime}},W)\\) cannot be small. By this, any approximate symmetry of \\(G\\) is a small deformation, and, hence, \\(f^{*}\\) is an approximately equivariant mapping.\n\n**Equivariant mappings on geometric graphs.** In the following, we construct a scenario in which \\(f^{*}\\) can be shown to be approximately equivariant in a restricted sense. For simplicity, we assume \\(f^{*}(s_{G})\\in L^{2}[0,1]\\), and restrict to the case \\(r=\\mathbb{1}_{[0,\\rho]}\\) in the geometric graphon \\(W\\) of (53). Denote the induced graphon \\(W_{G^{\\prime}}=T\\). Given \\(h>0\\), define the \\(h\\)_-diagonal_\n\n\\[d_{h}=\\{(x,y)\\in\\mathcal{M}^{2}\\mid d_{\\mathcal{M}}(x,y)\\leq h\\}.\\]\n\nIn the following, all distances are assumed to be up to the best measure preserving bijection.\n\nIf there is a domain \\(S^{\\prime}\\times T^{\\prime}\\in\\mathcal{M}^{2}\\) outside the \\(\\rho\\)-diagonal in which \\(T(x,y)>c\\) for some \\(c>0\\), by reverse triangle inequality, we must have\n\n\\[\\|W-T\\|_{\\square}\\geq\\int_{S^{\\prime}}\\int_{T^{\\prime}}T(x,y)dydx=c\\mu(S^{ \\prime})\\mu(T^{\\prime}).\\]\n\nHence, since by (55), \\(\\|W-T\\|_{\\square}<u\\), for every \\(S^{\\prime}\\times T^{\\prime}\\) that does not intersect \\(d_{\\rho}\\), we must have\n\n\\[\\int_{S^{\\prime}}\\int_{T^{\\prime}}T(x,y)dydx\\leq u.\\]\n\nIn other words, for any two sets \\(S,T\\) with distance more than \\(\\rho\\) (\\(\\inf_{s\\in S,t\\in T}d_{\\mu}(s,t)>\\rho\\)), we have\n\n\\[\\int_{S}\\int_{T}T(x,y)dydx\\leq u.\\]\n\nThis formalizes the statement \"\\(W_{G^{\\prime}}(x,y)\\approx 0\\) for \\(x\\) away from \\(y\\)\" from above.\n\nNext, we develop the analysis for the special case \\(\\mathcal{M}=[0,1]\\) with the standard metric and Lebesgue probability measure. We note that the analysis can be extended to \\(\\mathcal{M}=[0,1]^{D}\\) for a general dimension \\(D\\in\\mathbb{N}\\). For every \\(z\\in[0,1]\\), we have\n\n\\[\\int_{[z+\\rho/\\sqrt{2},1]}\\int_{[0,z-\\rho/\\sqrt{2}]}T(x,y)dydx\\leq u,\\]\n\nand\n\n\\[\\int_{[0,z-\\rho/\\sqrt{2}]}\\int_{[z+\\rho/\\sqrt{2},1]}T(x,y)dydx\\leq u.\\]Let \\(\\nu>0\\). We take a grid \\(\\{x_{j}\\}\\in[0,1]\\) of spacing \\(\\sqrt{2}\\nu\\). The sets\n\n\\[\\bigcup_{j}[x_{j}+\\rho/\\sqrt{2},1]\\times[0,x_{j}-\\rho/\\sqrt{2}]\\,\\quad\\bigcup_{j}[0,x_{j}-\\rho/\\sqrt{2}]\\times[x_{j}+\\rho/\\sqrt{2},1]\\]\n\ncover \\(d_{\\nu}^{c}\\) (where \\(d_{\\nu}^{c}\\) is the complement of \\(d_{\\nu}\\)). Hence,\n\n\\[\\iint_{d_{\\nu}^{c}}T(x,y)dydx\\leq\\sum_{j=1}^{1/\\sqrt{2}\\nu}\\int_{[x_{j}+\\rho/ \\sqrt{2},1]}\\int_{[0,x_{j}-\\rho/\\sqrt{2}]}T(x,y)dydx\\]\n\n\\[+\\sum_{j=1}^{1/\\sqrt{2}\\nu}\\int_{[0,x_{j}-\\rho/\\sqrt{2}]}\\int_{[x_{j}+\\rho/\\sqrt {2},1]}T(x,y)dydx\\]\n\n\\[\\leq\\frac{2}{\\sqrt{2}\\nu}u.\\]\n\nWe take \\(\\frac{2}{\\sqrt{2}\\nu}u=t\\), for \\(u\\ll t\\ll 1\\), namely, \\(\\nu=\\sqrt{2}\\frac{u}{t}\\). For example, we may take \\(t=\\sqrt{2}u^{1/3}\\), and \\(\\nu=u^{2/3}\\), assuming that \\(\\rho<u^{1/3}\\). Hence, we have\n\n\\[\\iint_{d_{u}^{c^{2}/3}}T(x,y)\\leq\\sqrt{2}u^{1/3}.\\]\n\nTo conclude, the probability of having an edge between nodes \\(\\lambda_{i}\\) and \\(\\lambda_{j}\\) in \\(\\overline{G^{\\prime}}_{N}\\) which are further away than \\(u^{2/3}\\), namely, \\(d_{\\mathcal{M}}(\\lambda_{i},\\lambda_{j})>u^{2/3}\\), is less than \\(\\sqrt{2}u^{1/3}\\).\n\nSuppose that \\(G^{\\prime}\\) is asymmetric. This means that symmetries of \\(\\mathcal{G}_{G\\to G^{\\prime}}\\) can only permute between nodes that have an edge between them in the blown-up graph \\(\\overline{G^{\\prime}}_{N}\\). The probability of having an edge between nodes further away than \\(u^{2/3}\\) is less than \\(\\sqrt{2}u^{1/3}\\). Hence, a symmetry in \\(\\mathcal{G}_{G\\to G^{\\prime}}\\) can be seen as a small deformation, where for each node \\(\\lambda_{i}\\) and a random uniform \\(g\\in\\mathcal{G}_{G\\to G^{\\prime}}\\), the probability that \\(\\lambda_{i}\\) it is mapped by \\(g\\) to a node of distance less than \\(u^{2/3}\\) is more than \\(1-\\sqrt{2}u^{1/3}\\).\n\nAny symmetry \\(g\\) in \\(\\mathcal{G}_{G\\to G^{\\prime}}\\) induces a measure preserving bijection \\(\\nu\\) in \\(\\mathcal{M}=[0,1]\\), by permuting the intervals of the partition \\(\\mathcal{P}_{N}\\) of Definition 1. As a result, the set of points that are mapped further away than \\(u^{2/3}\\) under \\(\\nu\\) has probability upper bounded by \\(\\sqrt{2}u^{1/3}\\), and symmetries in \\(\\mathcal{G}_{G\\to G^{\\prime}}\\) can be seen as a small deformation \\(\\nu\\) according to Definition 9 (in high probability). This means that, for any \\(g\\in\\mathcal{G}_{G\\to G^{\\prime}}\\),\n\n\\[\\|f_{\\mathcal{M}}(s)-f_{\\mathcal{M}}(s\\circ g)\\circ g^{-1}\\|_{1}<C\\sqrt{2}u^{1 /3},\\]\n\nso by the triangle inequality, combining with equation 54, we have\n\n\\[\\|f^{*}(s_{G})-g^{-1}f^{*}(gs_{G})\\|_{1}<2e+C\\sqrt{2}u^{1/3}=\\epsilon^{\\prime}. \\tag{56}\\]\n\nEquation (56) leads to\n\n\\[\\|f^{*}(s_{G})-\\mathcal{Q}_{\\mathcal{G}_{G\\to G^{\\prime}}}(f^{*})(s_{G})\\|_{1} =\\|f^{*}(s_{G})-\\frac{1}{|\\mathcal{G}_{G\\to G^{\\prime}}|}\\sum_{g\\in \\mathcal{G}_{G\\to G^{\\prime}}}g^{-1}f^{*}(gs_{G})\\|_{1} \\tag{57}\\] \\[\\leq\\frac{1}{|\\mathcal{G}_{G\\to G^{\\prime}}|}\\sum_{g\\in\\mathcal{G}_{ G\\to G^{\\prime}}}\\|f^{*}(s_{G})-g^{-1}f^{*}(gs_{G})\\|_{1}<\\epsilon^{\\prime}. \\tag{58}\\]\n\nSince for any \\(q\\in L^{2}[0,1]\\cap L^{\\infty}[0,1]\\) we have \\(\\|q\\|_{2}^{2}\\leq\\|q\\|_{\\infty}\\|q\\|_{1}\\), we can bound\n\n\\[\\|f^{*}(s_{G})-\\mathcal{Q}_{\\mathcal{G}_{G\\to G^{\\prime}}}(f^{*})(s_{G})\\|_{2} <\\sqrt{\\|f^{*}(s_{G})-\\mathcal{Q}_{\\mathcal{G}_{G\\to G^{\\prime}}}(f^{*})(s_{G })\\|_{\\infty}}\\sqrt{\\epsilon^{\\prime}}\\]\n\n\\[<\\sqrt{\\|f^{*}(s_{G})\\|_{\\infty}}+\\|\\mathcal{Q}_{\\mathcal{G}_{G\\to G^{\\prime}} }(f^{*})(s_{G})\\|_{\\infty}\\sqrt{\\epsilon^{\\prime}}\\]\n\n\\[<\\sqrt{2\\|f^{*}(s_{G})\\|_{\\infty}}\\sqrt{\\epsilon^{\\prime}}, \\tag{59}\\]\n\nwhere the last inequality follows from \\(\\|\\mathcal{Q}_{\\mathcal{G}_{G\\to G^{\\prime}}}(f^{*})(s_{G})\\|_{\\infty}\\leq\\|f^{ *}(s_{G})\\|_{\\infty}\\).\n\nDenote \\(\\left\\lVert f^{*}\\right\\rVert_{\\infty}:=\\left\\lVert f^{*}(s_{G})\\right\\rVert_{ \\infty}d\\mu(s_{G})\\), and suppose that \\(\\left\\lVert f^{*}\\right\\rVert_{\\infty}\\) is finite. Hence, if \\(\\mu\\) is a probability measure, we have\n\n\\[\\left\\lVert f^{*}-\\mathcal{Q}_{\\mathcal{G}_{G\\to G^{\\prime}}}(f^{*}) \\right\\rVert_{\\mu}<\\sqrt{2\\left\\lVert f^{*}\\right\\rVert_{\\infty}}\\sqrt{ \\epsilon^{\\prime}}.\\]\n\nThis shows an example of approximately equivariant mapping based on random geometric graph, where the approximation rate is also a function of the size of the graph \\(N\\), and goes to zero as \\(N\\to\\infty\\) and \\(\\epsilon\\to 0\\).\n\nIn future work, we will extend this example to more general metric space \\(\\mathcal{M}\\) and to non-trivial symmetry groups \\(\\overline{\\mathcal{A}}_{G^{\\prime}}\\). Intuitively, most random geometric graphs are \"close to asymmetric\". This means that for \"most\" \\(G^{\\prime}\\), the symmetries of \\(\\overline{\\mathcal{A}}_{G^{\\prime}}\\) can only permute between nodes connected by an edge, and so are the symmetries of \\(\\mathcal{G}_{G\\to G^{\\prime}}\\). For this, we need to extend Definition 9 by treating \\(G^{\\prime}\\) probabilistically.\n\n## Appendix D Experiment Details\n\nIn this section, we provide additional details of our experiments. We first give a brief introduction of standard graph neural networks (Section D.1), followed by in-depth explanations of our applications in image inpainting (Section D.2), traffic flow prediction (Section D.3), and human pose estimation (Section D.4). All experiments were conducted on a server with 256 GB RAM and 4 NVIDIA RTX A5000 GPU cards.\n\n### Graph Neural Networks (GNNs)\n\nWe consider standard message-passing graph neural networks (MPNNs) [19; 20; 21] defined as follows. A \\(L\\)-layer MPNN maps input \\(X\\in\\mathbb{R}^{N\\times d}\\) to output \\(Y\\in\\mathbb{R}^{N\\times k}\\) following an iterative scheme: At initialization, \\(\\mathbf{h}^{(0)}=X\\); At each iteration \\(l\\), the embedding for node \\(i\\) is updated to\n\n\\[\\mathbf{h}^{(l)}_{i}=\\phi\\left(\\mathbf{h}^{(l-1)}_{i},\\sum_{j\\in\\mathcal{N}(i) }\\psi\\left(\\mathbf{h}^{(l-1)}_{i},\\mathbf{h}^{(l-1)}_{j},A_{[i,j]}\\right)\\right), \\tag{60}\\]\n\nwhere \\(\\phi,\\psi\\) are the update and message functions, \\(\\mathcal{N}(i)\\) denotes the neighbors of node \\(i\\), and \\(A_{[i,j]}\\) represents the \\((i,j)\\)-edge weight. MPNNs typically have two key design features: (1) \\(\\phi,\\psi\\) are _shared_ across all nodes in the graph, typically chosen to be a linear transformation or a multi-layer perceptions (MLPs), known as _global weight sharing_; (2) the graph \\(A\\) is used for (spatial) convolution.\n\n### Application: Image Inpainting\n\nWe provide additional details of the data, model, and optimization procedure used in the experiment.\n\n**Data.** We consider the datasets MNIST [74] and FashionMNIST [75]. For each dataset, we take \\(100\\) training samples and \\(1000\\) test samples via stratified random sampling. The input and output graph signals are \\((m_{i}\\odot x_{i},x_{i})\\) (\\(\\odot\\) is entrywise multiplication). Here \\(x_{i}\\in\\mathbb{R}^{28\\times 28}\\equiv\\mathbb{R}^{784}\\) denotes the image signals and \\(m_{i}\\) denotes a random mask (size \\(14\\times 14\\) for MNIST and \\(20\\times 20\\) for FashionMNIST). For experiments with reflection symmetry on the coarsened graph, we further transform each image in the FashionMNIST subset using horizontal flip with probability \\(0.5\\) (FashionMNIST+hflip). We remark that it is possible to model the dihedral group \\(\\mathcal{D}_{4}\\) on the square as the coarsened graph symmetry, yet we only consider the reflection symmetry \\(\\mathcal{S}_{2}\\) to simplify the parameterization (since it is Abelian while \\(\\mathcal{D}_{4}\\) is not) while capturing the most relevant symmetries.\n\n**Model.** We consider a \\(2\\)-layer \\(\\mathcal{G}_{G\\to G^{\\prime}}\\)-equivariant networks \\(\\mathcal{G}\\)-Net, a composition of \\(f_{\\text{out}}\\circ\\mathtt{ReLU}\\circ f_{\\text{in}}\\), where \\(f_{\\text{in}},f_{\\text{out}}\\) denote the input/output equivariant linear layers. The input and output feature dimension is \\(1\\) (since the signals are grayscale images), and the hidden dimension is set as \\(28\\). For comparison, we also consider a simple \\(1\\)-layer \\(\\mathcal{G}_{G\\to G^{\\prime}}\\)-equivariant networks \\(\\mathcal{G}\\)-Net.\n\nWe train the models with ADAM (learning rate \\(0.01\\), no weight decay, at most \\(1000\\) epochs). We report the best test accuracy at the model checkpoint selected by the best validation accuracy (with a \\(80/20\\) training-validation split).\n\nWe supplement Figure 2 with Table 7 for further numerical details.\n\n### Application: Traffic Flow Prediction\n\n**Data.** The METR-LA traffic dataset, [76], contains traffic information collected from 207 sensors in the highway of Los Angeles County from Mar 1st 2012 to Jun 30th 2012 [87]. We use the same traffic data normalization and \\(70/10/20\\) train/validation/test data split as [76]. We consider two different traffic graphs constructed from the pairwise road network distance matrix: (1) the sensor graph \\(G\\) introduced in [76] based on applying a thresholded Gaussian kernel (degree distribution in Figure 7(e)); (2) the sparser graph \\(G_{s}\\) based on applying the binary mask where the \\((i,j)\\) entry is nonzero if and only if nodes \\(i,j\\) lie on the same highway (degree distribution in Figure 7(d)). We construct the second variant to more faithfully model the geometry of the highway, illustrated in Figure 7(a).\n\n**Graph coarsening.** We choose \\(2\\) clusters based on highway intersection and flow direction, indicated by colors (Figure 7(b)), and \\(9\\) clusters based on highway labels (Figure 7(c)).\n\n**Model.** We use a standard baseline, DCRNN proposed in [76]. DCRNN is built on a core recurrent module, DCGRU cell, which iterates as follows: Let \\(x_{i,t},h_{i,t}\\) denote the \\(i\\)-th node feature and hidden state vector at time \\(t\\); Let \\(X_{t},R_{t},H_{t-1}\\) be the matrices of stacking feature vectors \\(x_{i,t},r_{i,t},h_{i,t-1}\\) as rows.\n\n\\[z_{i,t} =\\sigma_{g}\\left(W_{z}\\,x_{i,t}+U_{z}\\,h_{i,t-1}+b_{z}\\right) \\tag{61}\\] \\[r_{i,t} =\\sigma_{g}\\left(W_{r}\\,x_{t}+U_{r}\\,h_{t-1}+b_{r}\\right)\\] (62) \\[\\hat{h}_{i,t} =\\phi_{h}\\left([A\\,X\\,W_{h}]_{[i,:]}^{\\top}+[A\\,(R_{t}\\odot H_{t-1 })\\,U_{h}]_{[i,:]}^{\\top}+b_{h}\\right)\\] (63) \\[h_{i,t} =z_{t}\\odot h_{t-1}+(1-z_{t})\\odot\\hat{h}_{t}, \\tag{64}\\]\n\nwhere \\(W_{z},U_{z},b_{z},U_{r},W_{r},b_{r},W_{h},U_{h},b_{h}\\) are learnable weights and biases, \\(\\sigma_{g}\\) is the sigmoid function, \\(\\phi_{g}\\) is the hyperbolic tangent, and \\(h_{i,0}=0\\) for all \\(i\\) at initialization. The crucial different from a vanilla GRU lies in eqn (63) where graph convolution replaces matrix multiplication.\n\nWe then modify the graph convolution in (63) from global weight sharing to tying weights among clusters of nodes, similar to the implementation in Appendix D.4 for Relax-\\(\\mathcal{S}_{16}\\). For example, in the\n\n\\begin{table}\n\\begin{tabular}{l l l l l l l} \\hline \\hline\n**MSE** (\\(\\times 1e^{-2}\\)) & \\(\\mathcal{S}_{28}\\) & \\((\\mathcal{S}_{10})^{3}\\) & \\((\\mathcal{S}_{7})^{16}\\) & \\((\\mathcal{S}_{7})^{16}\\) & \\((\\mathcal{S}_{8})^{16}\\) & \\((\\mathcal{S}_{9})^{16}\\) & \\((\\mathcal{S}_{10})^{16}\\) & \\((\\mathcal{S}_{11})^{16}\\) \\\\ \\hline MNIST & \\(41.56\\pm 0.16\\) & \\(40.53\\pm 0.26\\) & \\(36.06\\pm 0.24\\) & \\(34.68\\pm 0.5\\) & \\(\\mathbf{33.67\\pm 0.07}\\) & \\(33.92\\pm 0.04\\) \\\\ Fashion & \\(23.48\\pm 0.14\\) & \\(22.26\\pm 0.02\\) & \\(16.94\\pm 0.08\\) & \\(15.16\\pm 0.1\\) & \\(\\mathbf{14.47\\pm 0.11}\\) & \\(14.75\\pm 0.11\\) \\\\ \\hline \\hline \\end{tabular}\n\\end{table}\nTable 7: Image inpainting using \\(\\mathcal{G}\\)-Net with different levels of coarsening. Table shows mean squared error (MSE) across \\(3\\) runs on the test set, supplementing Figure 2 (Left, blue curves).\n\nFigure 8: METR-LA traffic graph: visualization, clustering, and degree distributioncase of two clusters (orbits), we change \\(XW_{h}\\) to\n\n\\[\\mathtt{swap}\\left(\\mathtt{concat}[X_{c_{1}}W_{h,c_{1}};X_{c_{2}}W_{h,c_{2}}] \\right), \\tag{65}\\]\n\nwhere \\(X_{c_{i}}\\) denotes the submatrix of \\(X\\) including the rows of nodes from cluster \\(i\\) only, and \\(W_{h,c_{1}},W_{h,c_{2}}\\) are two learnable matrices. In words, we perform cluster-specific linear transformation, combine the transformed features, and reorder the rows (i.e., \\(\\mathtt{swap}\\)) to ensure compatibility with the graph convolution.\n\n**Experiment Set-up.** For our experiments, we use DCRNN model with \\(1\\) RNN layer and \\(1\\) diffusion step. We choose \\(T^{\\prime}=3\\) (i.e., \\(3\\) historical graph signals) and \\(T=3\\) (i.e., predict the next \\(3\\) period graph signals). We train all variants for \\(30\\) epochs using ADAM optimizer with learning rate \\(0.01\\). We report the test set performance selected by the best validation set performance.\n\n#### d.3.1 Assumption Validation: Approximate Equivariant Map\n\nBefore applying our construction of approximate symmetries, we validate the assumption of the target function \\(f^{*}\\) being an approximately equivariant mapping using a trained DCRNN model as a proxy. We proceed as follows:\n\n**Data.** We use the validation set of METR-LA (traffic graph signals in LA), which has \\(207\\) nodes and consists of \\(14,040\\) input and output signals. Each input \\(X\\in\\mathbb{R}^{207\\times 2}\\) represents the traffic volume and speed in the past at the \\(207\\) stations, and output \\(Y\\in\\mathbb{R}^{207}\\) representing future traffic volume.\n\n**Model.** We use a trained DCRNN model on our faithful graph, with input being \\(3\\) historical signals \\(\\mathbf{X}=(X_{T-3},X_{T-2},X_{T-1})\\in\\mathbb{R}^{3\\times 207\\times 2}\\) to predict the future signals \\(\\mathbf{Y}=(X_{T},X_{T+1},X_{T+2})\\in\\mathbb{R}^{3\\times 207}\\). We denote this model as \\(f\\). It gives reasonable performance with Mean Absolute Error \\(\\approx 3\\), and serves as a good proxy for the target (unknown) function \\(f^{*}\\).\n\n**Neighbors.** We take our faithful traffic graph that originally has \\(397\\) non-loop edges, and only consider a subset of \\(260\\) edges by thresholding the distance values to eliminate geometrically far-away nodes. This defines our \\(260\\) neighboring node pairs.\n\n**Equivariance error.** For each node pair \\((i,j)\\), we swap their input signals by interchanging the \\((i,j)\\)-th slices in the node dimension of the tensor \\(\\mathbf{X}\\), denoted as \\(\\mathbf{X}_{(i,j)}\\), and check if the swapped output \\(\\hat{\\mathbf{Y}}_{(i,j)}=f(\\mathbf{X}_{(i,j)})\\) is close to the original output \\(\\hat{\\mathbf{Y}}=f(\\mathbf{X})\\) with \\((i,j)\\)-th slices swapped. We measure \"closeness\" via the relative equivariant error at the node pair. Concretely, let \\(\\mathbf{X}[i,j]\\) denote the tensor slices at the \\((i,j)\\) node pair, and \\(\\mathbf{X}[j,i]\\) being the swapped version by interchanging \\((i,j)\\)-th slices. The relative different is computed as\n\n\\[\\big{|}\\hat{\\mathbf{Y}}_{(i,j)}[j,i]-\\hat{\\mathbf{Y}}[i,j]\\big{|}/\\hat{\\mathbf{Y}}[i,j],\\]\n\nwhere \\(/\\) denotes element-wise division. We then compute the mean relative equivariance error over all instances in the validation set, which equals to 5.17%. This gives concrete justification to enforce approximate equivariance in the traffic flow prediction problems.\n\n### Application: Human Pose Estimation\n\n#### d.4.1 Equivariant Layer for Human Skeleton Graph\n\nWe apply the constructions in Section A.1 to our human skeleton graph. We first show how to parameterize all linear \\(\\mathcal{A}_{G}\\)-equivariant functions. Observe that \\(\\mathcal{A}_{G}\\cong(\\mathcal{S}_{2})^{2}=\\{e,a,l,al\\}\\), where the nontrivial actions correspond to the \\(\\mathbf{arm}\\) flip with respect to the spine, the leg flip with respect to the spine, and their composition. To fix ideas, we first treat both input and output graph signals as vectors, and construct \\(\\mathcal{A}_{G}\\)-equivariant linear maps \\(f:\\mathbb{R}^{16}\\rightarrow\\mathbb{R}^{16}\\).\n\nStep \\(1\\): Obtain the character table for \\((\\mathcal{S}_{2})^{2}\\)Step 2: Construct the basis for isotypic decomposition. Here we choose to index the leg joint pairs as \\((1,4),(2,5),(3,6)\\), arm joint pairs as \\((10,13),(11,14),(12,15)\\), and spline joints \\(0,7,8,9\\).\n\n\\[B =[\\mathcal{B}(P_{\\chi_{e}});\\mathcal{B}(P_{\\chi_{2}});\\mathcal{B}( P_{\\chi_{3}});\\mathcal{B}(P_{\\chi_{4}})]\\text{ where }\\] \\[\\mathcal{B}(P_{\\chi_{e}}) =[(e_{1}+e_{4})/\\sqrt{2};\\ldots;(e_{12}+e_{15})/\\sqrt{2};e_{0};e_ {7};e_{8};e_{9}]\\in\\mathbb{R}^{16\\times 10}.\\] \\[\\mathcal{B}(P_{\\chi_{2}}) =[(e_{1}-e_{4})/\\sqrt{2};(e_{2}-e_{5})/\\sqrt{2};(e_{3}-e_{6})/ \\sqrt{2}]\\in\\mathbb{R}^{16\\times 3};\\] \\[\\mathcal{B}(P_{\\chi_{3}}) =[(e_{10}-e_{13})/\\sqrt{2};(e_{11}-e_{14})/\\sqrt{2};(e_{12}-e_{15 })/\\sqrt{2}]\\in\\mathbb{R}^{16\\times 3};\\] \\[\\mathcal{B}(P_{\\chi_{4}}) =\\emptyset \\tag{66}\\]\n\nStep 3: Parameterize \\(f:\\mathbb{R}^{16}\\rightarrow\\mathbb{R}^{16}\\) by \\(f:\\mathcal{B}(P_{\\chi_{e}})\\rightarrow\\mathcal{B}(P_{\\chi_{e}})\\) and \\(f:\\mathcal{B}(P_{\\chi_{2}})\\rightarrow\\mathcal{B}(P_{\\chi_{2}})\\), i.e. for all \\(v\\in\\mathbb{R}^{16}\\), let \\(v=\\mathcal{B}(P_{\\chi_{e}})\\,\\mathbf{c_{e}}+\\mathcal{B}(P_{\\chi_{2}})\\,\\mathbf{c_{2}}+ \\mathcal{B}(P_{\\chi_{3}})\\,\\mathbf{c_{3}}\\), then\n\n\\[f(v)=W_{e}\\,\\mathbf{c_{e}}+W_{2}\\,\\mathbf{c_{2}}+W_{3}\\,\\mathbf{c_{3}}, \\tag{67}\\]\n\nwhere \\(W_{e}\\in\\mathbb{R}^{10\\times 10},W_{2}\\in\\mathbb{R}^{3\\times 3},W_{3}\\in\\mathbb{R }^{3\\times 3}\\) are (learnable) weight matrices. Now \\(f\\) expresses all linear, equivariant maps w.r.t \\((\\mathcal{S}_{2})^{2}\\).\n\nThe following calculation based on \\(f:\\mathbb{R}^{16}\\rightarrow\\mathbb{R}^{16}\\) shows how much degree of freedom (measured by learnable parameters) is gained by relaxing the symmetry from global (group \\(\\mathcal{S}_{16}\\)), exact \\(\\mathcal{A}_{G}\\cong(\\mathcal{S}_{2})^{2}\\), to trivial group (i.e., no symmetry).\n\n\\[f_{\\mathcal{S}_{16}} =w\\,\\mathbf{1}_{16}+w^{\\prime}(\\mathbf{1}-\\mathbf{1}_{16}),\\quad (\\text{2 parameters}); \\tag{68}\\] \\[f_{\\mathcal{A}_{G}} =W_{e}\\oplus W_{2}\\oplus W_{3},\\quad(118\\text{ parameters on the isotypic components});\\] (69) \\[f_{\\text{friv.}} =W,\\quad(256\\text{ parameters}). \\tag{70}\\]\n\nTo parameterize equivariant linear function \\(f:\\mathbb{R}^{16\\times d}\\rightarrow\\mathbb{R}^{16\\times d^{\\prime}}\\), we proceed by decoupling the input space into \\(\\mathbb{R}^{10\\times d},\\mathbb{R}^{3\\times d},\\mathbb{R}^{3\\times d}\\) and the output space into \\(\\mathbb{R}^{10\\times d^{\\prime}},\\mathbb{R}^{3\\times d^{\\prime}},\\mathbb{R}^{ 3\\times d^{\\prime}}\\). Now the learnable weight matrices for multidimensional input/output become \\(W_{e}\\in\\mathbb{R}^{10d\\times 10d^{\\prime}},W_{2}\\in\\mathbb{R}^{3d\\times 3d^{ \\prime}},W_{3}\\in\\mathbb{R}^{3d\\times 3d^{\\prime}}\\). The construction is summarized in Algorithm 2.\n\n```\n0: The basis \\(B\\in\\mathbb{R}^{16\\times 16}\\) in (66) for isotypic decomposition of \\(\\mathcal{A}_{G}=(\\mathcal{S}_{2})^{2}\\), input \\(h^{(l)}\\in\\mathbb{R}^{16\\times d}\\). Initialize: The learnable weights \\(W_{e}^{(l)}\\in\\mathbb{R}^{10d^{\\prime}\\times 10d};W_{2}^{(l)},W_{3}^{(l)}\\in \\mathbb{R}^{3d^{\\prime}\\times 3d};M^{(l)}\\in\\mathbb{R}^{16\\times 16}\\).\n1. Project \\(h^{(l)}\\) to the isotypic component: \\(z^{(l)}=B^{\\top}h^{(l)}\\);\n2. Perform block-wise linear transformation: * \\(z_{e}=W_{e}\\ \\text{flatten}(z_{[:,10]}^{(l)})\\) * \\(z_{2}=W_{2}\\ \\text{flatten}(z_{[:,10:13]}^{(l)})\\) * \\(z_{3}=W_{3}\\ \\text{flatten}(z_{[:,13:]}^{(l)})\\) * \\(z^{(l+1)}=\\text{concat}[z_{e};\\,z_{2};\\,z_{3}]\\in\\mathbb{R}^{16\\times d^{ \\prime}}\\)\n3. Project back to the standard basis: \\(\\bar{h}^{(l+1)}=B\\,z^{(l+1)}\\).\n4. Perform pointwise nonlinearity: \\(h^{(l+1)}=\\sigma(\\bar{h}^{(l+1)})\\). return\\(h^{(l+1)}\\)\n```\n\n**Algorithm 2** Equivariant layer \\(f_{\\mathcal{A}_{G}}:\\mathbb{R}^{16\\times d}\\rightarrow\\mathbb{R}^{16\\times d^ {\\prime}}\\) for \\(\\mathcal{A}_{G}\\cong(\\mathcal{S}_{2})^{2}\\)\n\n**Algorithm 3** The basis \\(B\\in\\mathbb{R}^{16\\times 16}\\) in (66) for isotypic decomposition of \\(\\mathcal{A}_{G}=(\\mathcal{S}_{2})^{2}\\), input \\(h^{(l)}\\in\\mathbb{R}^{16\\times d}\\).\n\n**Algorithm 4** The basis \\(B\\in\\mathbb{R}^{16\\times 16}\\) in (66) for isotypic decomposition of \\(\\mathcal{A}_{G}=(\\mathcal{S}_{2})^{2}\\), input \\(h^{(l)}\\in\\mathbb{R}^{16\\times d}\\).\n\n**Algorithm 5** The basis \\(B\\in\\mathbb{R}^{16\\times 16}\\) in (66) for isotypic decomposition of \\(\\mathcal{A}_{G}=(\\mathcal{S}_{2})^{2}\\), input \\(h^{(l)}\\in\\mathbb{R}^{16\\times d}\\).\n\n#### d.4.2 Experiment Details\n\n**Data.** We use the standard benchmark dataset, Human3.6M [77], with the same protocol as in [78]: We train the models on \\(1.56\\)M poses (from human subjects \\(S1,S5,S6,S7,S8\\)) and evaluate them on \\(0.54\\)M poses (from human subjects \\(S9,S11\\)). We use the method described in [88] to normalize the inputs (2D joint poses) to \\([-1,1]\\) and align the targets (3d joint poses) with the root joint.\n\n**Model.** We give a detailed description of \\(\\mathcal{G}\\)-Net and its variants used in the experiments. Figure inset illustrates the architecture of \\(\\mathcal{G}\\)-Net.\n\nFor the human skeleton graph with \\(N=16\\), we have \\(f_{\\mathcal{G}}:\\mathbb{R}^{16\\times d}\\rightarrow\\mathbb{R}^{16\\times k}\\), where \\(d,k\\) represent the input dimension and output dimension (for each layer). Let \\(f_{\\mathcal{G}}[i,j]:\\mathbb{R}^{16}\\rightarrow\\mathbb{R}^{16}\\) denote its \\((i,j)\\)-th slice.\n\n1. \\(\\mathcal{G}\\)-Net with strict equivariance using equivariant linear map \\(f_{\\mathcal{G}}\\) (see Table 3): * \\(\\mathcal{S}_{16}\\): \\(f_{S_{16}}[i,j]\\in\\mathbb{R}^{16\\times 16}\\) is a diagonal matrix, with one learnable scalar \\(a\\) on diagonal and another learnable scalar \\(b\\) off diagonal. * Relax-\\(\\mathcal{S}_{16}\\): We relax \\(f_{S_{16}}[i,j]\\) by having \\(16\\) different pairs of scalars \\((a_{i},b_{i}),i\\in[16]\\), such that each node \\(i\\) can map to itself and communicate to its neighbors in a different way (controlled by \\((a_{i},b_{i})\\)), while still treat all neighbors equally (by using the same \\(b_{i}\\) for nodes \\(j\\neq i\\)). * \\(\\mathcal{A}_{G}=\\mathcal{S}_{2}{}^{2}\\): We use Algorithm 2. * Trivial: We allow \\(f[i,j]\\in\\mathbb{R}^{16\\times 16}\\) to be arbitrary, i.e., it has \\(16\\times 16\\) learnable scalars.\n\nWe remark that for \\(\\mathcal{S}_{16}\\) and Relax-\\(\\mathcal{S}_{16}\\), we implement them by tying weights; for \\(\\mathcal{A}_{G}\\), we implement them by projecting to isotypic component as shown in Algorithm 2.\n\n2. \\(\\mathcal{G}\\)-Net augmented with graph convolution \\(Af_{\\mathcal{G}}(x)\\), denoted as \\(\\mathcal{G}\\)-Net(gc) (see Table 3): We apply the equivariant linear map \\(f_{\\mathcal{G}}\\) in 1. and obtain the output \\(f_{\\mathcal{G}}(x)\\in\\mathbb{R}^{16\\times k}\\); We then apply graph convolution by multiplication from the left, i.e., \\(Af_{\\mathcal{G}}(x)\\in\\mathbb{R}^{16\\times k}\\).\n\n3. \\(\\mathcal{G}\\)-Net augmented with graph convolution and learnable edge weights, denoted as \\(\\mathcal{G}\\)-Net(gc+ew) (see Table 4): We further learn the edge weights for the adjacency matrix \\(A\\), by \\(\\texttt{softmax}(M\\odot A)\\) where \\(M\\in\\mathbb{R}^{16}\\) represents the learnable edge weights, and \\(M_{i,j}\\) is nonzero when \\(A_{i,j}\\neq 0\\) and \\(0\\) elsewhere. This is inspired from SemGCN [78]. Besides the groups discussed above, we also implemented Relax-\\((\\mathcal{S}_{6})^{2}\\) which corresponds to tying weights among the coarsened graph orbits, consists of \\(4\\) spline nodes (singleton orbits) and \\(2\\) orbits for the left/right arm and leg nodes.\n\n4. \\(\\mathcal{G}\\)-Net augmented with graph locality constraints \\((A\\odot f_{\\mathcal{G}})(x)\\) and learnable edge weights, denoted as \\(\\mathcal{G}\\)-Net(pt+ew) (see Table 3): We perform pointwise multiplication \\(A\\odot f_{\\mathcal{G}}[i,j]\\) at each \\((i,j)\\)-th slice of \\(f_{\\mathcal{G}}\\). In practice, we also allow learnable edge weights as done in 3.\n\n**Experimental Set-up.** We design \\(\\mathcal{G}\\)-Net to have \\(4\\) layers (with batch normalization and residual connections in between the hidden layers), \\(128\\) hidden units, and use ReLU nonlinearity. This allows \\(\\mathcal{G}\\)-Net(gc+ew) to recover SemGCN [78] when choosing \\(\\mathcal{G}=\\mathcal{S}_{16}\\). We train our models for at most \\(30\\)\n\nFigure 9: Human skeleton graph \\(G\\), its coarsened graph \\(G^{\\prime}\\) (clustering leg joints), and blow-up of \\(G^{\\prime}\\)epochs with early stopping. For comparison purpose, we use the same optimization routines as in SemGCN [78] and perform the hyper-parameter search of learning rates \\(\\{0.001,0.002\\}\\).\n\n**Evaluation.** Table 3 shows results of \\(\\mathcal{G}\\)-Net and its variants when varying the choice of \\(\\mathcal{G}\\). We observe that using the automorphism group \\(\\mathcal{A}_{G}\\) does not give the best performance, while imposing no symmetries (Trivial) or a relaxed version of \\(\\mathcal{S}_{16}\\) yields better results. Here, enforcing no symmetry achieves better performance since the human skeleton graph is very small with \\(16\\) nodes only. As shown in other experiments with larger graphs (e.g. image inpainting), enforcing symmetries indeed yields better performance.\n\n**Additional Evaluation.** Table 4 shows the experiments when we keep the number of parameters roughly the same across different choices of \\(\\mathcal{G}\\).\n\n\\begin{table}\n\\begin{tabular}{c c c c c} \\hline \\hline \\(\\mathcal{G}\\)-Net & Number of Parameters & Number of Epochs & MPJPE & P-MPJPE \\\\ \\hline \\(\\mathcal{S}_{16}\\) & 0.27M (\\(d=128\\)) & \\(50\\) & \\(43.48\\) & \\(34.96\\) \\\\ Relax-\\(\\mathcal{S}_{16}\\) & 0.27M (\\(d=32\\)) & \\(20\\) & \\(\\mathbf{40.08}\\) & \\(\\mathbf{32.08}\\) \\\\ \\(\\mathcal{A}_{G}=(\\mathcal{S}_{2})^{2}\\) & 0.22M (\\(d=16\\)) & \\(30\\) & \\(44.10\\) & \\(34.12\\) \\\\ Trivial & 0.22M (\\(d=10\\)) & \\(30\\) & \\(45.05\\) & \\(34.79\\) \\\\ \\hline \\hline \\end{tabular}\n\\end{table}\nTable 4: 3D human pose prediction using \\(\\mathcal{G}\\)-Net(gc+ew), where the models induced from each choice of \\(\\mathcal{G}\\) are set to have roughly the same number of parameters. \\(d\\) denotes the number of hidden units.\n\n\\begin{table}\n\\begin{tabular}{l c c c c} \\hline \\hline\n**MPJPE**\\(\\downarrow\\) & \\(\\mathcal{S}_{16}\\) & Relax-\\(\\mathcal{S}_{16}\\) & \\(\\mathcal{A}_{G}=(\\mathcal{S}_{2})^{2}\\) & Trivial \\\\ \\hline \\(\\mathcal{G}\\)-Net & NA & \\(\\mathbf{47.97\\pm 0.47}\\) & \\(48.30\\pm 0.69\\) & \\(\\mathbf{42.86\\pm 0.64}\\) \\\\ \\(\\mathcal{G}\\)-Net(gc) & NA & \\(54.50\\pm 4.33\\) & \\(\\mathbf{49.40\\pm 1.37}\\) & \\(\\mathbf{43.24\\pm 0.82}\\) \\\\ \\(\\mathcal{G}\\)-Net(pt+ew) & \\(41.54\\pm 0.47\\) & \\(\\mathbf{40.44\\pm 0.61}\\) & \\(40.63\\pm 0.26\\) & \\(\\mathbf{38.41\\pm 0.31}\\) \\\\ \\hline \\hline\n**P-MPJPE**\\(\\downarrow\\) & \\(\\mathcal{S}_{16}\\) & Relax-\\(\\mathcal{S}_{16}\\) & \\(\\mathcal{A}_{G}=(\\mathcal{S}_{2})^{2}\\) & Trivial \\\\ \\hline \\(\\mathcal{G}\\)-Net & NA & \\(\\mathbf{36.45\\pm 0.56}\\) & \\(37.17\\pm 0.59\\) & \\(\\mathbf{32.59\\pm 0.62}\\) \\\\ \\(\\mathcal{G}\\)-Net(gc) & NA & \\(40.61\\pm 0.99\\) & \\(\\mathbf{37.62\\pm 1.32}\\) & \\(\\mathbf{33.05\\pm 0.81}\\) \\\\ \\(\\mathcal{G}\\)-Net(pt+ew) & \\(32.31\\pm 0.03\\) & \\(\\mathbf{31.11\\pm 0.68}\\) & \\(31.35\\pm 0.14\\) & \\(\\mathbf{29.68\\pm 0.22}\\) \\\\ \\hline \\hline \\end{tabular}\n\\end{table}\nTable 3: 3D human pose prediction using \\(\\mathcal{G}\\)-Net and its variants. Error (\\(\\pm\\) std) measured by Mean Per-Joint Position Error (MPJPE) and MPJPE after rigid alignment (P-MPJPE) across \\(3\\) runs. All methods use the same hidden dimension \\(d=128\\). Bold type indicates the top-\\(2\\) performance among each variant. “NA” indicates the loss fails to converge.", "response": "## Review\n\n### Summary\n\nThis paper explores the complexities of equivariant graph learning, specifically addressing the bias-variance tradeoff in models that leverage symmetries of fixed graphs. The authors introduce novel concepts such as utilizing supergroups of natural symmetry groups to optimize model performance and employing graphon theory to define approximate symmetry groups through graph coarsening. Their theoretical analysis is complemented by numerical experiments, validating the proposed methodologies in contexts like human pose estimation and traffic flow prediction, although further elucidation of the connection between theoretical foundations and experimental validation is suggested.\n\n### Strengths\n\n- Contains several new ideas, including the use of supergroups to optimize bias-variance tradeoff, and explicitly considering fixed graph symmetries instead of global permutation symmetry.\n- Introduces novel theoretical results on generalization with approximate symmetries, particularly through graphon distances.\n- The experiments are clean tests of the presented ideas, demonstrating the existence of the proposed bias-variance trade-off across various settings.\n- The use of Graphons for defining metrics over graph spaces is innovative and potentially impactful.\n- The theoretical claims are articulated clearly and detailed well throughout the paper.\n\n### Weaknesses\n\n- Some sections, particularly regarding the risk gap and clustering methods, lack clarity and adequate motivation.\n- There are significant mismatches between theoretical claims and empirical results, particularly concerning the combination of symmetries used in experiments.\n- The paper requires a thorough proofread for typos and grammatical errors.\n- The experimental section is difficult to follow, with critical details relegated to the appendix, and the architecture of the neural networks used is insufficiently described.\n- The results on small, specific tasks do not adequately demonstrate the generalizability of the proposed methods.\n\n### Questions\n\n- How do the authors reconcile the bias-variance tradeoff with the existing literature on equivariant neural networks?\n- What specific methods or criteria are used for coarsening the graphs, particularly in the context of traffic flow?\n- Could the authors clarify the practical implications of the approximated symmetry in broader graph tasks beyond those tested?\n- How do the authors apply results from complex representation theory to real-number applications, especially concerning intertwiners?\n- Could the authors elaborate on the significance of the inner product properties mentioned in the analysis?\n\n### Soundness\n\n**Score:** 3\n\n**Description:** Good: The theoretical foundation is largely sound, but some claims lack empirical support and clarity in their presentation.\n\n### Presentation\n\n**Score:** 2\n\n**Description:** Fair: The paper suffers from structural issues and requires clearer exposition, especially in the methodology and experimental sections.\n\n### Contribution\n\n**Score:** 4\n\n**Description:** Excellent: The work presents significant advancements in the understanding of equivariant graph networks with potential broad applications.\n\n### Rating\n\n**Score:** 7\n\n**Description:** Accept: The paper is technically solid with notable contributions, but requires improvements in clarity and empirical validation to fully support its claims.\n\n### Paper Decision\n\n**Decision:** Accept (poster)\n\n**Reasons:** The paper introduces significant novel concepts in equivariant graph learning and presents a strong theoretical foundation backed by clean experiments. While there are areas for improvement, particularly in clarity and the robustness of empirical validation, the overall contributions are compelling and merit acceptance.\n"}
{"query": "You are a highly experienced, conscientious, and fair academic reviewer, please help me review this paper. The review should be organized into nine sections: \n1. Summary: A summary of the paper in 100-150 words.\n2. Strengths/Weaknesses/Questions: The Strengths/Weaknesses/Questions of paper, which should be listed in bullet points, with each point supported by specific examples from the article where possible.\n3. Soundness/Contribution/Presentation: Rate the paper's Soundness/Contribution/Presentation, and match this score to the corresponding description from the list below and provide the result. The possible scores and their descriptions are: \n    1 poor\n    2 fair\n    3 good\n    4 excellent\n4. Rating: Give this paper an appropriate rating, match this rating to the corresponding description from the list below and provide the result. The possible Ratings and their descriptions are: \n    1 strong reject\n    2 reject, significant issues present\n    3 reject, not good enough\n    4 possibly reject, but has redeeming facets\n    5 marginally below the acceptance threshold\n    6 marginally above the acceptance threshold\n    7 accept, but needs minor improvements \n    8 accept, good paper\n    9 strong accept, excellent work\n    10 strong accept, should be highlighted at the conference   \n5. Paper Decision: It must include the Decision itself(Accept or Reject) and the reasons for this decision, based on the criteria of originality, methodological soundness, significance of results, and clarity and logic of presentation.\n\nHere is the template for a review format, you must follow this format to output your review result:\n**Summary:**\nSummary content\n\n**Strengths:**\n- Strength 1\n- Strength 2\n- ...\n\n**Weaknesses:**\n- Weakness 1\n- Weakness 2\n- ...\n\n**Questions:**\n- Question 1\n- Question 2\n- ...\n\n**Soundness:**\nSoundness result\n\n**Presentation:**\nPresentation result\n\n**Contribution:**\nContribution result\n\n**Rating:**\nRating result\n\n**Paper Decision:**\n- Decision: Accept/Reject\n- Reasons: reasons content\n\n\nPlease ensure your feedback is objective and constructive. The paper is as follows:\n\n# Evolving Standardization for Continual Domain Generalization over Temporal Drift\n\n Mixue Xie\\({}^{1}\\)  Shuang Li\\({}^{1,}\\)1  Longhui Yuan\\({}^{1}\\)  Chi Harold Liu\\({}^{1}\\)  Zehui Dai\\({}^{2}\\)\n\n\\({}^{1}\\)Beijing Institute of Technology, China \\({}^{2}\\)Lazada Search & Monetisation Tech, China\n\n{mxxie,shuangli,longhuiyuan}@bit.edu.cn, liuchi02@gmail.com\n\nzehui.dzh@alibaba-inc.com\n\nCorresponding author.\n\nFootnote 1: footnotemark:\n\n###### Abstract\n\nThe capability of generalizing to out-of-distribution data is crucial for the deployment of machine learning models in the real world. Existing domain generalization (DG) mainly embarks on offline and discrete scenarios, where multiple source domains are simultaneously accessible and the distribution shift among domains is abrupt and violent. Nevertheless, such setting may not be universally applicable to all real-world applications, as there are cases where the data distribution gradually changes over time due to various factors, e.g., the process of aging. Additionally, as the domain constantly evolves, new domains will continually emerge. Re-training and updating models with both new and previous domains using existing DG methods can be resource-intensive and inefficient. Therefore, in this paper, we present a problem formulation for _Continual Domain Generalization over Temporal Drift_ (CDGTD). CDGTD addresses the challenge of gradually shifting data distributions over time, where domains arrive sequentially and models can only access the data of the current domain. The goal is to generalize to unseen domains that are not too far into the future. To this end, we propose an _Evolving Standardization_ (EvoS) method, which characterizes the evolving pattern of feature distribution and mitigates the distribution shift by standardizing features with generated statistics of corresponding domain. Specifically, inspired by the powerful ability of transformers to model sequence relations, we design a multi-scale attention module (MSAM) to learn the evolving pattern under sliding time windows of different lengths. MSAM can generate statistics of current domain based on the statistics of previous domains and the learned evolving pattern. Experiments on multiple real-world datasets including images and texts validate the efficacy of our EvoS.\n\n## 1 Introduction\n\nIn real-world applications, the assumption that training and testing data conform to the same distribution, a prerequisite for the success of contemporary deep learning methods, is seldom valid. A prime example of this can be found in autonomous driving, where a vehicle may traverse environments that switch from daylight to nightfall or from urban to rural. As environmental conditions change, the issue of distribution shift [5, 4, 51, 62] arises. Moreover, directly utilizing a model trained on in-distribution (ID) data in an out-of-distribution (OOD) context often results in catastrophic performance deterioration. Consequently, ensuring that models perform well on OOD data has emerged as a crucial challenge for the widespread deployment of machine learning models in the real world.\n\nTo cope with the distribution shift, two dominant paradigms have been systematically explored depending on the availability of target (test) domain. One is domain adaptation (DA), which aims to assist the model learning on an unlabeled or label-scarce target domain by transferring the knowledgefrom a related and label-rich source domain [15; 34; 29; 35]. Yet, target data are not always known or available in advance. The other is domain generalization (DG), the goal of which is to learn a model capable of generalizing to any unseen domain by using date from multiple related but distinct source domains [62; 36; 27; 63; 26]. Though the second paradigm DG has attained some encouraging outcomes, its configuration is limited to offline and discrete scenarios where multiple source domains can be accessed simultaneously, and the distribution shift among domains is sudden and severe. For instance, the prevalent benchmark PACS [25] in current DG methods comprises four distinct domain styles - \"Art\", \"Cartoon\", \"Photo\" and \"Sketch\". Nevertheless, this type of configuration may not be suitable for all real-world applications. There are also some cases that the data distribution gradually evolves over time, the distribution shift arising from which is referred to as _temporal drift_[56].\n\nOn one hand, the real-world scenarios often exhibit underlying evolutionary patterns [59; 44]. For example, annual or monthly weather data can be utilized for weather forecasting [12]. The evolutionary patterns can be exploited to enhance generalization capabilities towards future unseen domains that are not too distant. However, current DG methods often fail to consider these patterns, leading to suboptimal performance. On the other hand, since data distribution is constantly evolving, new domains will continue to emerge. Consequently, it is imperative to efficiently utilize these new domains to further enhance the model's performance for practical applications. For instance, in the context of advertisement recommendation, user browsing data for various products continually surfaces. How can we leverage these newly collected data to enable more accurate advertisement recommendations tailored to each user's preferences in the days to come? One simple approach is to store data from previous domains and retrain the model using both the new and previous domains with the existing DG techniques. However, such way may consume huge training resources due to the accumulation of data and is of low efficiency, especially for scenarios with rapid data accumulation.\n\nIn this paper, we formulate the aforementioned problems as Continual Domain Generalization over Temporal Drifts (CDGTD), where the data distribution gradually drifts with the passage of time and the domain arrives sequentially. And the goal of CDGTD is to generalize well on future unseen domains that are not too far under the circumstance that only current domain is available at current times point while data from previous domains are inaccessible. Although some temporal / evolving DG methods [44; 59; 3; 41] have been proposed to handle the temporal drift, most of them [59; 44; 41] works in non-incremental setting, i.e., multiple domains can be accessed simultaneously. Instead, we design an Evolving Standardization (EvoS) approach specialized for CDGTD.\n\nFor CDGTD, there are two main challenges: how to characterize the evolving pattern in the incremental setting and how to achieve generalization using the learned evolving pattern. For the former, we draw inspiration from the sequence modeling capability of transformers [31] and design a multi-scale attention module (MSAM) to learn the evolving pattern underlying the feature distribution of domains. Specifically, we store the statistics (i.e., mean and variance of features) of previous domains and use sliding time windows of different lengths over the statistic tokens to obtain multi-scale information. Then multi-scale statistic tokens are fed into MSAM to generate statistics of current domain. MSAM is trained over the whole sequence of domains to learn the evolving pattern. Here, we integrate multi-scale information, with the consideration that some evolving patterns may be better characterized at different time intervals, e.g., seasonal climate and daily weather. For the latter challenge, in order to mitigate the temporal drift, each domain is transformed into a common normal distribution by conducting feature standardization with the generated statistics of corresponding domain. Besides, considering that the feature encoder may suffer from catastrophic forgetting and overfitting to current domain, we constrain it to learn a shared feature space among domains via the adversarial learning.\n\n**Contributions: 1)** We formulate a promising but challenging problem of continual domain generalization over temporal drift (CDGTD), which has seldom been explored, compared with traditional DG. **2)** An evolving standardization (EvoS) approach is specially proposed for CDGTD, which can characterize the evolving pattern and further achieve generalization by conducting the feature standardization. **3)** Experiments on multiple real-world datasets with different models verify the effectiveness of EvoS and the flexibility to be applied on different models.\n\n## 2 Related Work\n\n**Domain Generalization (DG)** aims to learn a model that can generalize well to any unseen target domains by leveraging data from multiple source domains. In recent years, a wide range of DGmethods have been proposed [6; 39; 27; 26; 63; 19; 37]. According to the strategies of improving generalization, existing DG methods roughly fall into the three categories. Representation learning based methods [7; 39; 50; 27; 2; 2; 17; 42] aim to learn domain-invariant or domain-shared representations to enable models to generalize to unseen target domains. Data augmentation/generation based methods [53; 61; 63; 57; 32] focus on manipulating inputs to facilitate learning general representations. Differently, other DG methods instead employ general learning strategies like meta-learning [26] and ensemble learning [21] to improve generalization ability. With full access to all source domains, these DG methods have achieved promising performance on unseen domains with abrupt and violent distribution shifts. Unfortunately, without the consideration of the evolutionary pattern of domains over time, existing DG methods are usually less efficient under the scenario of temporal drift.\n\n**Continual Learning (CL)** is a machine learning paradigm, where models learn continuously from a stream of tasks over time and meanwhile try to retain performances on all seen tasks [22; 1; 24; 9; 46; 47]. Existing CL methods can be roughly categorized into replay-based [46; 10; 52], regularization-based [22; 60; 30] and structure-based [45; 14] methods. In addition, the combination of CL and DA has attracted lots of attention [23; 17; 33; 43; 55]. Yet, combining CL and DG remains underdeveloped. In this work, we propose CDGTD which continually trains the model on sequential domains, but the goal is to generalize well on novel domains in the near future. This distinct objective from CL yields different challenges: the modeling of underlying evolutionary patterns of temporal domains and how to utilize these patterns to mitigate the distribution shifts in forthcoming domains.\n\n**Test Time Adaptation (TTA)** focuses on adapting the source-pretrained model during testing phase with test data [40; 49; 20; 54; 58]. For example, [49] corrects the activation statistics of batch normalization using test data. T3A [20] uses a back-propagation-free manner to adjust only the weights of the linear classifier during test time. By contrast, CDGTD optimizes the model during training phase with sequential domains and emphasizes on generalization without using test data.\n\n**Evolving / Temporal Domain Generalization** has emerged in recent years, aiming to tackle the problem of generalizing on temporally drifted domains, where the environment evolves dynamically over time [41; 3; 44; 59]. GI [41] proposes a time-sensitive model architecture to capture the time-varying data distribution and the model is supervised with the first-order Taylor expansion of the learned function to advance the generalization in the near feature. DRAIN [3] launches a Bayesian framework to model the concept drift and utilizes a recurrent neural network to dynamically generate network parameters to adapt the evolving pattern of domains. LSSAE [44] incorporates variational inference to explore the evolving patterns of covariate shift and label shift in the latent space. The goal of learning evolutionary patterns from source domains and generalizing to domains in the near future is similar to our EvoS. The main difference is that the model in our EvoS is incrementally trained on sequentially arriving domains, considering the low efficiency of \"offline\" training with accumulated domains. By contrast, the aforementioned methods [41; 44] require multiple source domains to be simultaneously available. Besides, the Taylor expansion in [41], variational inference in [44] and network parameters generation in [3] make them hard to expand on large models.\n\n## 3 Evolving Standardization for Continual Domain Generalization over Temporal Drift\n\n### Problem Formulation of CDGTD\n\nHere, we take the \\(C\\)-class classification problem as an example, where \\(X\\) and \\(Y\\) denote the data and label space, respectively. Suppose that \\(T\\) source (training) domains \\(\\{\\mathcal{D}^{1},\\mathcal{D}^{2},\\cdots,\\mathcal{D}^{T}\\}\\) sequentially arrive, which are sampled from distributions at \\(T\\) different times points \\(\\mathfrak{t}_{1}<\\mathfrak{t}_{2}<\\cdots<\\mathfrak{t}_{T}\\). At time point \\(\\mathfrak{t}_{i}\\), \\(t\\in\\{1,2,\\cdots,T\\}\\), only the domain \\(\\mathcal{D}^{t}=\\{\\mathbf{x}_{i}^{t},y_{i}^{t}\\}_{i=1}^{N^{t}}\\) is accessible, where \\(\\mathbf{x}_{i}^{t}\\in X\\), \\(y_{i}^{t}\\in Y\\) and \\(N^{t}\\) is the number of training samples in \\(\\mathcal{D}^{t}\\). Previous and future domains are unavailable. In addition, as previous temporal/evolving DG methods [44; 41; 3], we further assume that the data distribution \\(P(X,Y)\\) of domains evolves temporally, i.e., the distribution of domains changes along the time following certain patterns. The goal of CDGTD is to enable the model, composed of a feature encoder \\(\\mathcal{E}:\\mathbf{x}\\rightarrow\\mathbf{f}\\in\\mathbb{R}^{d_{f}}\\) (\\(d_{f}\\) is the dimension of features) and a classifier \\(\\mathcal{C}:\\mathbf{f}\\rightarrow{y}\\in\\{0,1,\\cdots,C-1\\}\\), to generalize well on \\(K\\) unseen target (test) domains in the near future: \\(\\{\\mathcal{D}^{T+k}\\}_{k=1}^{K}\\). To this end, two main challenges need to be addressed. One is to characterize the evolving pattern of domains, which we address by designing a multi-scale attention module (MSAM)inspired from the sequence modeling ability of transformers. The other is the generalization, which is realized by transforming domains into a common normal distribution via the feature standardization.\n\n### EvoS: Evolving Standardization\n\n**Single-scale Attention.** Inspired by the capability of the transformer to model the relationships among sequences [31], we introduce its attention mechanism to model the evolving pattern among sequential domains. Before presenting our multi-scale attention module (MSAM), we first introduce how single-scale attention works. Without loss of much generality, we assume that the feature distribution of domain \\(\\mathcal{D}^{t}\\) follows the normal distribution, which is characterized by the mean vector \\(\\mathbf{\\mu}^{t}\\in\\mathbb{R}^{d_{f}}\\) and the standard deviation vector \\(\\mathbf{\\sigma}^{t}\\in\\mathbb{R}^{d_{f}}\\). Now, our goal turns out to be learning the evolving pattern of the feature statistics (i.e., \\(\\mathbf{\\mu}^{t}\\) and \\(\\mathbf{\\sigma}^{t}\\)). In practice, to ensure that \\(\\mathbf{\\sigma}^{t}\\) is non-negative, we choose to learn \\(\\mathbf{a}^{t}\\) (the logarithm of \\(\\mathbf{\\sigma}^{t}\\)). In this way, \\(\\mathbf{\\sigma}^{t}=\\exp(\\mathbf{a}^{t})\\) is always non-negative. Besides, considering the unavailability of previous domains, we store the learned statistics at time point \\(\\mathsf{t}_{t}\\) into a memory pool \\(\\mathcal{M}\\) for future uses once the training procedure finishes on domain \\(\\mathcal{D}^{t}\\). Having the feature statistics of previous \\(t-1\\) domains ready, we then utilize the multi-head self-attention module \\(\\mathcal{A}\\) to generate the statistics of domain \\(\\mathcal{D}^{t}\\) at time point \\(\\mathsf{t}_{t}\\).\n\nConcretely, given statistic tokens \\(\\{\\mathbf{s}^{i}\\}_{i=1}^{t-1}\\) of previous \\(t-1\\) domains, where \\(\\mathbf{s}^{i}=[\\mathbf{\\mu}^{i},\\mathbf{a}^{i}]\\in\\mathbb{R}^{2d_{f}}\\) is the token of concatenated statistics from domain \\(\\mathcal{D}^{t}\\), the output of \\(\\mathcal{A}\\) is expressed as\n\n\\[\\mathbf{\\hat{S}}^{t} =\\mathcal{A}(\\mathbf{S}^{t-1})=[\\mathrm{SA}_{1}(\\mathbf{S}^{t-1}),\\mathrm{ SA}_{2}(\\mathbf{S}^{t-1}),\\cdots,\\mathrm{SA}_{n_{h}}(\\mathbf{S}^{t-1})]\\mathbf{W}_{fc}, \\quad\\mathbf{W}_{fc}\\in\\mathbb{R}^{(n_{h}\\cdot d_{h})\\times 2d_{f}},\\] \\[=[\\mathbf{\\hat{s}}^{1};\\mathbf{\\hat{s}}^{2};\\cdots;\\mathbf{\\hat{s}}^{t-1}],\\] \\[\\mathbf{S}^{t-1} =[\\mathbf{s}^{1};\\mathbf{\\hat{s}}^{2};\\cdots;\\mathbf{s}^{t-1}],\\quad\\mathbf{S}^{t -1}\\in\\mathbb{R}^{(t-1)\\times 2d_{f}} \\tag{1}\\]\n\nwhere \\(\\mathbf{\\hat{s}}^{i}\\) is the \\(i\\)-th output statistic token of \\(\\mathcal{A}\\), \\(n_{h}\\) and \\(d_{h}\\) are the number and feature dimension of heads in the multi-head self-attention module and \\(\\mathbf{W}_{fc}\\) is the learnable parameters of \\(\\mathcal{A}\\) to convert feature dimensions. \\(\\mathrm{SA}_{i}(\\cdot)\\) is the self-attention of the \\(i\\)-th head, which operates as follows:\n\n\\[[\\mathbf{S}^{t-1}_{i,q},\\mathbf{S}^{t-1}_{i,k},\\mathbf{S}^{t-1}_{i,v}]=\\mathbf{S}^{t-1}\\mathbf{ W}^{i}_{qkv},\\quad\\mathbf{W}^{i}_{qkv}\\in\\mathbb{R}^{2d_{f}\\times 3d_{h}}\\] \\[\\mathrm{SA}_{i}(\\mathbf{S}^{t-1})=\\mathrm{softmax}(\\mathbf{S}^{t-1}_{i,q} {\\mathbf{S}^{t-1}_{i,k}}^{\\top}/\\sqrt{d}_{h})\\mathbf{S}^{t-1}_{i,v}, \\tag{2}\\]\n\nFigure 1: Overview of EvoS. Memory pool \\(\\mathcal{M}\\) stores previously generated domain statistics (i.e., \\(\\mathbf{\\mu}^{i}\\) (mean) and \\(\\mathbf{a}^{i}\\) (logarithm of standard deviation) of features, \\(1\\leq i\\leq t-1\\)). With the \\(t\\)-th domain \\(\\mathcal{D}^{t}\\) available, loss \\(\\mathcal{L}^{t}_{stan}\\) is minimized to train the multi-scale attention module (MSAM) to generate statistics \\(\\mathbf{\\hat{\\mu}}^{t}\\) and \\(\\mathbf{\\hat{a}}^{t}\\) that approach the real ones via leveraging historical statistics in \\(\\mathcal{M}\\). Besides, we sample features from historical feature distributions as the proxy of previous domains to conduct adversarial training with current domain, encouraging to learn a shared feature space.\n\nwhere \\(\\mathbf{W}^{i}_{qkv}\\) is the learnable parameters of the \\(i\\)-th head and \\(\\mathbf{S}^{t-1}_{i,q},\\mathbf{S}^{t-1}_{i,k},\\mathbf{S}^{t-1}_{i,v}\\) are the query, key and value embeddings of \\(\\mathbf{S}^{t-1}\\). Finally, we use the average of all output statistics tokens of the attention module \\(\\mathcal{A}\\) as the generated statistic token \\(\\mathbf{\\hat{s}}^{t}\\) for time points \\(\\mathsf{t}_{t}\\):\n\n\\[[\\mathbf{\\hat{\\mu}}^{t},\\mathbf{\\hat{a}}^{t}]=\\mathbf{\\hat{s}}^{t}=avg(\\mathbf{\\hat{S}}^{t})= \\frac{1}{t-1}\\sum_{i=1}^{t-1}\\mathbf{\\hat{s}}^{i}. \\tag{3}\\]\n\nBy continuously generating new statistics from historical ones throughout the whole sequence of domains, the attention module \\(\\mathcal{A}\\) is expected to learn the evolving pattern of feature distributions. Note that we directly introduce learnable statistic vectors \\(\\mathbf{\\hat{\\mu}}^{1},\\mathbf{\\hat{a}}^{1}\\) and \\(\\mathbf{\\hat{\\mu}}^{2},\\mathbf{\\hat{a}}^{2}\\) for time points \\(\\mathsf{t}_{1}\\) and \\(\\mathsf{t}_{2}\\), respectively, since the tokens of historical statistics are not enough for the attention module \\(\\mathcal{A}\\) to work. That is, the attention module \\(\\mathcal{A}\\) is only used at time points \\(\\mathsf{t}_{t},t\\geq 3\\).\n\n**Multi-scale Attention Module (MSAM).** The above details how to model the evolving pattern using single-scale attention. However, it is limited to learn the pattern using a sliding time window of length 1 over the statistic tokens. Sometimes, the evolutionary pattern may be better captured by considering longer time intervals. For instance, the data of a season in each time window would be more suitable for capturing the evolving pattern of seasonal climate, rather than data of a single day in each time window. Considering this, we introduce multi-scale attention in Fig. 1, which leverages information from observation windows of different lengths to better model the evolving pattern.\n\nSpecifically, for the multi-head self-attention module \\(\\mathcal{A}_{w}\\) responsible for the time window of length \\(w\\), we slide the time window with stride 1 over historical \\(t-1\\) statistics tokens and concatenate the statistic tokens in each window to obtain the input \\(\\mathbf{\\hat{S}}^{t-1}_{w}\\) for the attention module \\(\\mathcal{A}_{w}\\) at time point \\(\\mathsf{t}_{t}\\):\n\n\\[\\mathbf{\\hat{s}}^{i}_{w} =[\\mathbf{s}^{i},\\mathbf{s}^{i+1}\\cdots,\\mathbf{s}^{i+w-1}],\\quad\\mathbf{\\hat{s}}^ {i}_{w}\\in\\mathbb{R}^{w\\cdot 2d_{f}},\\quad(1\\leq i\\leq t-w)\\wedge(t\\geq w+2)\\] \\[\\mathbf{\\hat{S}}^{t-1}_{w} =[\\mathbf{\\hat{s}}^{1}_{w};\\mathbf{\\hat{s}}^{2}_{w};\\cdots;\\mathbf{\\hat{s}}^ {t-w}_{w}],\\quad\\mathbf{\\hat{S}}^{t-1}_{w}\\in\\mathbb{R}^{(t-w)\\times(w\\cdot 2d_{f})}. \\tag{4}\\]\n\nAnd similarly, we use the average output \\(\\mathbf{\\hat{s}}^{t-w+1}_{w}\\) of the attention module \\(\\mathcal{A}_{w}\\) as the generated statistics for the sliding time window at the next time point, the formulation of which is expressed as\n\n\\[\\mathbf{\\hat{S}}^{t}_{w} =\\mathcal{A}_{w}(\\mathbf{\\hat{S}}^{t-1}_{w})=[\\mathbf{s}^{1}_{w};\\mathbf{ \\hat{s}}^{2}_{w};\\cdots;\\mathbf{\\hat{s}}^{t-w}_{w}],\\quad\\mathbf{\\hat{S}}^{t}_{w}\\in \\mathbb{R}^{(t-w)\\times(w\\cdot 2d_{f})}\\] \\[\\mathbf{\\hat{s}}^{t-w+1}_{w} =avg(\\mathbf{\\hat{S}}^{t}_{w})=\\frac{1}{t-w}\\sum_{i=1}^{t-w}\\mathbf{\\hat {s}}^{i}_{w},\\quad\\mathbf{\\hat{s}}^{t-w+1}_{w}\\in\\mathbb{R}^{w\\cdot 2d_{f}}. \\tag{5}\\]\n\nNote that in MSAM, the attention module \\(\\mathcal{A}_{w}\\) is to predict the statistics in the next sliding time window of length \\(w\\), instead of one statistic token as in the single-scale attention. This may encourage the attention module to also capture the domain relationships within the window.\n\nThen, we split \\(\\mathbf{\\hat{s}}^{t-w+1}_{w}\\) into \\(w\\) parts: \\([\\mathbf{\\hat{s}}^{t-w+1}_{w},\\cdots,\\mathbf{\\hat{s}}^{t-1}_{w},\\mathbf{\\hat{s}}^{t}_{w}]= \\mathbf{\\hat{s}}^{t-w+1}_{w}\\), where \\(\\mathbf{\\hat{s}}^{j}_{w}\\in\\mathbb{R}^{2d_{f}}\\) can be regarded as the predicted statistic token for time point \\(\\mathsf{t}_{j}\\) using the attention module \\(\\mathcal{A}_{w},j=t-w+1,\\cdots,t-1,t\\). Finally, the generated statistic token \\(\\mathbf{\\hat{s}}^{t}\\) for time point \\(\\mathsf{t}_{t}\\) using MSAM is denoted as the average of predicted statistic tokens for time point \\(\\mathsf{t}_{t}\\) at different time window lengths:\n\n\\[[\\mathbf{\\hat{\\mu}}^{t},\\mathbf{\\hat{a}}^{t}]=\\mathbf{\\hat{s}}^{t}=\\frac{1}{W}\\sum_{w=1}^{W }\\mathbf{\\hat{s}}^{t}_{w}, \\tag{6}\\]\n\nwhere \\(W\\) is the maximum length of the sliding time window. Generally, MSAM integrates evolving patterns learned at different scales, contributing better estimation of future feature distributions.\n\n**Feature Standardization.** For CDGTD, the second main challenge is generalization. To this end, we leverage the generated feature statistics by MSAM to transform the distribution of corresponding domain into a standard normal distribution, by which the temporal drift is mitigated. Concretely, for feature \\(\\mathbf{f}^{t}_{i}=\\mathcal{E}(\\mathbf{x}^{t}_{i})\\), its standardized feature \\(\\mathbf{z}^{t}_{i}\\) via the feature standardization is formulated as\n\n\\[\\mathbf{z}^{t}_{i}=\\frac{\\mathbf{f}^{t}_{i}-\\mathbf{\\hat{\\mu}}^{t}}{\\mathbf{\\hat{\\sigma}}^{t}} =\\frac{\\mathbf{f}^{t}_{i}-\\mathbf{\\hat{\\mu}}^{t}}{\\exp(\\mathbf{\\hat{a}}^{t})}. \\tag{7}\\]\n\nAnother benefit of feature standardization is that it allows classifier to be trained on a common feature distribution, thus avoiding the problem of overfitting to current domain and catastrophic forgetting.\n\n### Model Training\n\nThis section outlines the four losses \\(\\mathcal{L}^{t}_{ce}\\), \\(\\mathcal{L}^{t}_{stan}\\), \\(\\mathcal{L}^{t}_{con}\\) and \\(\\mathcal{L}^{t}_{adv}\\) involved in the training stage at time point \\(\\mathbf{t}_{t}\\). The first one is the essential loss to supervise the model learning for specific tasks, e.g., the following cross-entropy loss \\(\\mathcal{L}^{t}_{ce}\\) for classification tasks in this paper.\n\n\\[\\min_{\\mathcal{E},\\mathcal{E}}\\mathcal{L}^{t}_{ce}=\\frac{1}{N^{t}}\\sum_{i=1}^{ N^{t}}\\mathcal{CE}(\\mathbf{p}^{t}_{i},y^{t}_{i}),\\quad\\mathbf{p}^{t}_{i}=\\mathrm{softmax} \\left(\\mathcal{C}\\Big{(}\\frac{\\mathbf{f}^{t}_{i}-\\mathrm{sg}(\\mathbf{\\hat{\\mu}}^{t})}{ \\mathrm{exp}(\\mathrm{sg}(\\mathbf{\\hat{a}}^{t}))}\\Big{)}\\right), \\tag{8}\\]\n\nwhere \\(\\mathcal{CE}(\\cdot,\\cdot)\\) is the cross-entropy. Here, stopping gradient \\(\\mathrm{sg}(\\cdot)\\) is adopted to stabilize training.\n\nLosses \\(\\mathcal{L}^{t}_{stan}\\) and \\(\\mathcal{L}^{t,w}_{con}\\) are responsible for the training of the attention module \\(\\mathcal{A}_{w}\\) at time point \\(\\mathbf{t}_{t}\\), \\(w=1,2,\\cdots,W\\). The former loss \\(\\mathcal{L}^{t}_{stan}\\) in Eq. 10 is minimized to ensure that the standardized feature \\(\\mathbf{z}^{t}_{i}\\) follows a standard normal distribution. Specifically, we first calculate the mean vector and variance vector of the standardized features as\n\n\\[mean:\\mathbf{\\hat{\\mu}}^{t}=\\frac{1}{N^{t}}\\sum_{i=1}^{N_{t}}\\frac{\\mathrm{sg}(\\bm {f}^{t}_{i})-\\mathbf{\\hat{\\mu}}^{t}}{\\mathrm{exp}(\\mathbf{\\hat{a}}^{t})},\\quad variance :\\mathbf{\\hat{\\epsilon}}^{t}=\\frac{1}{N^{t}-1}\\sum_{i=1}^{N_{t}}\\left(\\frac{ \\mathrm{sg}(\\mathbf{f}^{t}_{i})-\\mathbf{\\hat{\\mu}}^{t}}{\\mathrm{exp}(\\mathbf{\\hat{a}}^{t}) }-\\mathbf{\\hat{\\mu}}^{t}\\right)^{2}. \\tag{9}\\]\n\nThen the loss \\(\\mathcal{L}^{t}_{stan}\\) at time point \\(\\mathbf{t}_{t}\\) is expressed as\n\n\\[\\left\\{\\begin{aligned} &\\min_{\\mathbf{\\hat{\\mu}}^{t},\\mathbf{\\hat{a}}^{t}} \\mathcal{L}^{t}_{stan}=\\|\\mathbf{\\hat{\\mu}}^{t}-\\mathbf{0}\\|_{2}+\\|\\mathbf{\\hat{\\nu}}^{t}- \\mathbf{1}\\|_{2},\\quad 1\\leq t\\leq 2\\\\ &\\min_{\\mathcal{A}_{w}}\\mathcal{L}^{t}_{stan}=\\|\\mathbf{\\hat{\\mu}}^{t }-\\mathbf{0}\\|_{2}+\\|\\mathbf{\\hat{\\nu}}^{t}-\\mathbf{1}\\|_{2},\\quad t\\geq w+2\\end{aligned} \\right., \\tag{10}\\]\n\nwhere \\(d_{f}\\) is the dimension of features and \\(w\\) is the length of the sliding time window. Here, we just want the learnable/generated \\(\\mathbf{\\hat{\\mu}}^{t},\\mathbf{\\hat{a}}^{t}\\) to approach the statistics of true feature distribution, so stopping gradient operation \\(\\mathrm{sg}(\\cdot)\\) is conducted on \\(\\mathbf{f}^{t}_{i}\\) to prevent the gradient of \\(\\mathcal{L}^{t}_{stan}\\) flowing to the feature encoder \\(\\mathcal{E}\\). Otherwise, the gradient of \\(\\mathcal{L}^{t}_{stan}\\) may distort the feature distribution. Ideally, if \\(\\mathcal{L}^{t}_{stan}\\) is minimized, the generated statistics \\(\\mathbf{\\hat{\\mu}}^{t}\\) and \\(\\mathrm{exp}(\\mathbf{\\hat{a}}^{t})\\) by our MSAM would be equal to the mean vector and standard deviation vector of the true feature distribution of \\(\\mathcal{D}^{t}\\). This means that our MSAM can predict future feature distributions based on historical feature statistics.\n\nAnd the loss \\(\\mathcal{L}^{t}_{con}\\) is to ensure the consistency between generated statistic tokens in the next sliding time window and real statistic tokens. Specifically, we calculate \\(\\mathcal{L}^{t,w}_{con}\\) for attention module \\(\\mathcal{A}_{w}\\) as\n\n\\[\\min_{\\mathcal{A}_{w}}\\mathcal{L}^{t,w}_{con}=\\frac{1}{w-1}\\sum_{k=1}^{w-1}\\| \\mathbf{\\hat{s}}^{t-w+k}_{w}-\\mathbf{s}^{t-w+k}\\|_{2},\\quad(t\\geq w+2)\\wedge w\\geq 2, \\tag{11}\\]\n\nwhere \\(\\mathbf{\\hat{s}}^{j}_{w}\\) and \\(\\mathbf{s}^{j}_{w}\\), \\(j=t-w+1,\\cdots,t-1\\) are respectively the predicted statistic token for time point \\(\\mathbf{t}_{j}\\) using the attention module \\(\\mathcal{A}_{w}\\) and the real statistic token from memory pool \\(\\mathcal{M}\\). This loss is expected to also capture the evolutionary pattern within the time window.\n\nThe fourth loss \\(\\mathcal{L}^{t}_{adv}\\) is introduced with this consideration that the feature extractor \\(\\mathcal{E}\\) may overfit to current domain and lacks generalizability. To avoid this, we additionally conduct adversarial training between the feature encoder \\(\\mathcal{E}\\) and a discriminator \\(\\mathfrak{D}\\). Nevertheless, historical data are unavailable. So we instead resort to the learned/generated statistics at previous time points. Concretely, we randomly sample a batch of \\(B\\) samples \\(\\{\\mathbf{f^{\\prime}}_{1}^{m},\\cdots,\\mathbf{f^{\\prime}}_{B}^{m}\\}\\) from each distribution \\(\\mathcal{N}(\\mathbf{\\mu}^{m},(\\mathrm{exp}(\\mathbf{a}^{m}))^{2})\\), \\(m=1,2,\\cdots,t-1\\), via Eq. 12 at each iteration, and use them as the proxy of historical domains.\n\n\\[\\mathbf{f^{\\prime}}_{i}^{m}=\\mathbf{\\mu}^{m}+\\epsilon\\cdot\\mathrm{exp}(\\mathbf{a}^{m}), \\quad\\epsilon\\sim\\mathcal{N}(0,1)\\wedge-\\alpha\\leq\\epsilon\\leq\\alpha, \\tag{12}\\]\n\nwhere \\(\\alpha\\) is a truncation hyper-parameter to control the sampling area. Then discriminator \\(\\mathfrak{D}\\) is trained to distinguish \\(\\{\\mathbf{f^{\\prime}}_{i}\\}_{i=1}^{N^{t}}\\) from \\(\\{\\mathbf{f^{\\prime}}_{1}^{m},\\cdots,\\mathbf{f^{\\prime}}_{B}^{m}\\}_{m=1}^{t-1}\\) and the feature extractor tries to confuse \\(\\mathfrak{D}\\). Such adversarial process is achieved by the following loss \\(\\mathcal{L}^{t}_{adv}\\):\n\n\\[\\max_{\\mathcal{E}}\\min_{\\mathfrak{D}}\\mathcal{L}^{t}_{adv}=\\frac{1}{2}\\Big{(} \\frac{1}{B\\times(t-1)}\\sum_{m=1}^{t-1}\\sum_{j=1}^{B}-\\mathrm{log}(\\mathfrak{D}( \\mathbf{f^{\\prime}}_{j}^{m}))+\\frac{1}{N^{t}}\\sum_{j=1}^{N^{t}}-\\mathrm{log}(1- \\mathfrak{D}(\\mathbf{f^{\\prime}}_{j}^{t}))\\Big{)},\\quad t\\geq 2. \\tag{13}\\]\n\nIn practice, the gradient reverse layer (GRL) [15] is used to achieve the adversarial training. To sum up, at time point \\(\\mathbf{t}_{t}\\), the model is trained to minimize the following total loss \\(\\mathcal{L}^{t}_{total}\\):\n\n\\[\\mathcal{L}^{t}_{total}=\\mathcal{L}^{t}_{ce}+\\mathcal{L}^{t}_{stan}+\\sum_{w=1}^ {W}\\mathcal{L}^{t,w}_{con}+\\lambda\\mathcal{L}^{t}_{adv}, \\tag{14}\\]where \\(\\lambda\\) is a hyper-parameter to balance the loss tradeoff. Once the training procedure at time point \\(\\mathfrak{t}_{t}\\) is finished, we store the learned/generated statistics of current domain \\(\\mathcal{D}^{t}\\) into memory pool \\(\\mathcal{M}\\) by\n\n\\[\\boldsymbol{\\mu}^{t},\\boldsymbol{a}^{t}\\leftarrow\\boldsymbol{\\hat{\\mu}}^{t}, \\boldsymbol{\\hat{a}}^{t}. \\tag{15}\\]\n\nIn the inference stage, we use MSAM to generate future statistics \\(\\boldsymbol{\\hat{s}}^{T+k}\\) based on the statistics \\(\\{s^{t}\\}_{t=1}^{T+k-1}\\) in memory pool \\(\\mathcal{M}\\) and store it into \\(\\mathcal{M}\\) as Eq. 15 for the generation at next time point. Due to space limitation, the training and testing procedures are provided in the appendix.\n\n## 4 Experiments\n\n### Experimental Setup\n\nThanks to the work in [56], several real-world datasets with distribution shifts over time have been available. And we evaluate EvoS on three image classification datasets (**Yearbook** and **fMoW** from [56] and **RMNIST**) and two text classification datasets (**Huffpost** and **Arxiv** from [56]). Yearbook collects data from 1930 to 2013, where we treat every four years as a domain and use the first 16 domains for training (\\(T=16\\)), the last 5 domains for testing (\\(K=5\\)). fMoW includes data of 16 years and we set \\(T=13,K=3\\). RMNIST contains 9 domains with \\(T=6,K=3\\). Huffpost includes data of 7 years with \\(T=4,K=3\\). Arxiv collects data of 172 categories for 16 years, with \\(T=9,K=7\\). For each training domain of all datasets, we randomly select 90% data as training split and 10% data as validation split. Following the backbones in [56], we use a 4-layer convolutional network [56] for Yearbook, DenseNet-121 [18] pretrained on ImageNet for fMoW, pretrained DistilBERT base model [48] for Huffpost and Arxiv, and the ConvNet in [44] for RMNIST. For each attention module \\(\\mathcal{A}_{w}\\) in MSAM, its dimension of head \\(d_{h}\\) is set to 8 and its number of heads is set to \\(w\\cdot n_{h}\\). Specifically, \\(n_{h}\\) is set to 16 for Yearbook, 32 for RMNIST, 64 for fMoW and 128 for Huffpost and Arxiv. For optimization, we use the Adam optimizer with \\(lr=1e-3\\) for Yearbook and RMNIST, \\(lr=2e-4\\) for fMoW and \\(lr=2e-5\\) for Huffpost and Arxiv. The batch size is set to 64 for all datasets. As for hyper-parameters, we select them via grid search using the validation splits of training domains and finally use \\(\\alpha=2.0\\) for RMNIST, \\(\\alpha=1.0\\) for others, \\(\\lambda=1.0,W=3\\) for all datasets. For all tasks, we report the mean of 3 random trials. Due to space limitations, please refer to appendix for more details. Code is available at [https://github.com/BIT-DA/EvoS](https://github.com/BIT-DA/EvoS).\n\n### Main Results\n\nIn addition to the incremental training scenario, we also provide results in non-incremental scenario with all source domains simultaneously available, which serves as upper bounds. Among compared baselines, \"Offline\" denotes merging all source domains into one domain and training the model with the merged domain, while \"IncFinetune\" represents incrementally training the model in a domain-by-domain fashion. For each dataset, we report the accuracy on the nearest target domain (\\(\\mathcal{D}^{T+1}\\)),\n\n\\begin{table}\n\\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|} \\hline \\multirow{3}{*}{Method} & \\multirow{3}{*}{Conference} & \\multirow{3}{*}{Incremental training} & \\multirow{3}{*}{Access training} & \\multicolumn{3}{c|}{Yearbook} & \\multicolumn{3}{c|}{MRNIST} & \\multicolumn{3}{c|}{2fMoW} \\\\ \\cline{4-11}  & & & & & & & & & & & & & \\\\ \\cline{4-11}  & & & & & & & & & & & & \\\\ \\hline Offline & & ✓ & & 89.30 & 88.46 & 86.81 & 98.15 & 92.14 & 83.89 & 72.43 & 99.76 & 49.85 \\\\ IRM [22] & arXiv’19 & ✗ & ✓ & 97.09 & 94.52 & 92.58 & 95.10 & 85.05 & 82.52 & 64.71 & 54.92 & 46.51 \\\\ CORAL [56] & ECCV Workshops’16 & ✗ & ✓ & 95.94 & 91.79 & 88.84 & 90.74 & 90.10 & 62.96 & 62.14 & 51.42 & 42.19 \\\\ Mixup [61] & ICLR’18 & ✗ & ✓ & 94.98 & 91.12 & 88.35 & 91.71 & 89.66 & 79.20 & 72.77 & 57.33 & 48.04 \\\\ LISA [57] & ICML’22 & ✗ & ✓ & 95.51 & 92.97 & 91.29 & 92.1 & 58.04 & 75.15 & 70.08 & 55.52 & 44.61 \\\\ CDOT [43] & arXiv’19 & ✗ & ✓ & 95.17 & 92.90 & 91.46 & 97.96 & 90.19 & 76.7 & - & - \\\\ CIDA [55] & ICML’20 & ✗ & ✓ & 92.36 & 90.67 & 88.45 & 97.43 & 89.19 & 78.32 & - & - \\\\ GI [41] & NeurIPS’21 & ✗ & ✓ & 97.42 & 96.37 & 97.53 & 97.78 & 91.00 & 82.46 & 61.62 & 50.83 & 42.78 \\\\ LSSAE [44] & ICML’22 & ✗ & ✓ & 93.93 & 92.12 & 88.75 & 96.73 & 90.36 & 82.13 & 59.15 & 48.66 & 41.38 \\\\ \\hline IncFinetune & & ✓ & ✗ & 96.61 & 94.72 & 94.88 & 98.62 & 92.80 & 84.61 & 65.52 & 53.99 & 45.23 \\\\ Mixup [61] & ICLR’18 & ✓ & ✗ & 90.21 & 98.83 & 88.43 & 98.43 & 92.38 & 84.55 & 64.84 & 52.00 & 42.54 \\\\ SimCLR [11] & ICML’20 & ✓ & ✗ & 95.94 & 93.07 & 89.65 & 98.23 & 90.88 & 81.05 & 64.97 & 53.20 & 44.71 \\\\ SiaWV [8] & NeurIPS’20 & ✓ & ✗ & **97.37** & 94.27 & 94.14 & 94.88 & 90.88 & 90.85 & 66.47 & 54.51 & 45.29 \\\\ EWC [22] & arXiv’16 & ✓ & ✗ & 97.18 & 95.12 & 96.34 & 98.56 & 92.20 & 82.80 & 66.23 & 54.55 & 45.80 \\\\ S1 [60] & ICML’17 & ✓ & ✗ & 97.09 & 94.67 & 93.48 & 98.61 & 92.27 & 86.55 & 66.61 & 54.29 & **46.46** \\\\ A-GEM [10] & ICLR’19 & ✓ & ✗ & 94.36 & 99.66 & 88.88 & 95.99 & 89.56 & 75.45 & 54.47 & 47.61 & 41.13 \\\\ SGP [47] & AAAI’23 & ✓ & ✗ & 95.65 & 92.92 & 91.39 & 97.12 & 88.97 & 78.05 & - & - \\\\ DRAN [3] & ICLR’23 & ✓ & ✗ & 96.23 & 94.71 & 97.33 & 98.52 & 93.09 & 87.55 & **67.22** & **55.05** & 46.24 \\\\ \\hline\n**EvoS** & - & ✓ & ✗ & **97.37** & **95.53** & **94.78** & **98.64** & **93.84** & **97.04** & 61.78 & 54.64 & 45.86 \\\\ \\hline \\end{tabular}\n\\end{table}\nTable 1: Accuracy (%) on Yearbook, RMNIST and fMoW. The best and second-best results on CDGTD setup are bolded and underlined. (Yearbook: \\(K=5\\), RMNIST: \\(K=3\\), fMoW: \\(K=3\\))the average and worst accuracy of future \\(K\\) domains (\"OOD avg.\": \\(\\frac{1}{K}\\sum_{k=1}^{K}Acc(\\mathcal{D}^{T+k})\\), \"OOD worst\": \\(\\min_{k\\in\\{1,2,\\cdots,K\\}}Acc(\\mathcal{D}^{T+k})\\)).\n\n**Results on image classification tasks** are provided in Table 5, where we can observe that EvoS consistently achieves superior performance across three reported metrics on Yearbook and RMNIST datasets, compared with the methods in incremental training scenario. Especially, EvoS outperforms DRAIN by over \\(1\\%\\) according the metric \"OOD worst\" on Yearbook and RMNIST, which means that our EvoS learns more robust evolving patterns. In addition, we notice that conventional DG methods IRM [2] and CORAL [50] perform worse than temporal DG method GI [41] on Yearbook and RMNIST, showing the importance of leveraging evolutionary patterns for generalization over temporal drifts. Though being effective for the small networks used in Yearbook and RMNIST, the advantage of GI disappears on fMoW dataset with backbone DenseNet-121, where its first-order Taylor expansion requires extremely huge computing resources (over 80G). And our available resources cannot afford it. When ablating the Taylor expansion, the performance of GI on fMoW is not the best. So GI is hard to expand on larger models and the similar issue also exists in DRAIN and LSSAE. Finally, we find that for dataset fMoW, not only our method performs unsatisfactorily but also other methods, except for the \"offline\" method (i.e., merging all source domain into one domain to train the model). We infer this may be due to the temporal drift is not well presented in this dataset.\n\n**Results on text classification tasks** are shown in Table 6, where EvoS consistently achieves the best performances according to the average and worst accuracy of future \\(K\\) domains under the CDGTD setting. This can be owed to the more robust evolving patterns captured by our multi-scale attention module (MSAM). Similarly, IRM and CORAL show obvious performance drop compared with DRAIN, once again demonstrating the necessity to learn evolving patterns for the problem of CDGTD. And the large performance gap between continual learning methods (i.e., SI and A-GEM) and our method EvoS on dataset Arxiv verifies that our method EvoS fully utilizes the historical knowledge to learn evolutionary patterns, while SI and A-GEM do not consider this.\n\n### Analytical Experiments\n\n**Ablation Study.** Firstly, we study the influence of the stopping gradient \\(\\mathrm{sg}(\\cdot)\\) in the losses \\(\\mathcal{L}_{ce}^{t}\\) and \\(\\mathcal{L}_{stan}^{t}\\). The results of variant A and B in Table 3 almost degenerate to random predictions, compared with variant C. This suggests that using \\(\\mathrm{sg}(\\cdot)\\) in loss \\(\\mathcal{L}_{stan}^{t}\\) is essential. Otherwise, the gradient of \\(\\mathcal{L}_{stan}^{t}\\) would largely distort the learning of feature encoder, causing training collapse. Meanwhile, the inferior performance of variant C to EvoS also indicates that the gradient of \\(\\mathcal{L}_{ce}^{t}\\) should not interfere with the learning of MSAM. Secondly, we ablate the losses \\(\\mathcal{L}_{adv}^{t}\\) and \\(\\mathcal{L}_{con}^{t,w}\\) to testify their necessity. We can see that the results of\n\n\\begin{table}\n\\begin{tabular}{|c|c|c|c|c|c|c|c c c|} \\hline \\multirow{2}{*}{Method} & \\multirow{2}{*}{Conference} & \\multirow{2}{*}{\\begin{tabular}{c} Incremental \\\\ training \\\\ \\end{tabular} } & \\multirow{2}{*}{\\begin{tabular}{c} Access \\\\ multiple \\\\ \\end{tabular} } & \\multicolumn{3}{c|}{\\begin{tabular}{c} \\\\ Latency (\\%) \\\\ \\end{tabular} } & \\multicolumn{3}{c|}{\\begin{tabular}{c} Arxiv \\\\ Accuracy (\\%) \\\\ \\end{tabular} } \\\\ \\cline{4-10}  & & & & & \\multicolumn{2}{c|}{\\begin{tabular}{c} \\\\ \\end{tabular} } & \\multicolumn{2}{c|}{\\begin{tabular}{c} \\\\ \\end{tabular} } & \\multicolumn{2}{c|}{\\begin{tabular}{c} \\\\ \\end{tabular} } & \\multicolumn{2}{c|}{\\begin{tabular}{c} \\\\ \\end{tabular} } & \\multicolumn{2}{c|}{\\begin{tabular}{c} \\\\ \\end{tabular} } & \\multicolumn{2}{c|}{\\begin{tabular}{c} \\\\ \\end{tabular} } & \\multicolumn{2}{c|}{\\begin{tabular}{c} \\\\ \\end{tabular} } & \\multicolumn{2}{c|}{\\begin{tabular}{c} \\\\ \\end{tabular} } & \\multicolumn{2}{c|}{\\begin{tabular}{c} \\\\ \\end{tabular} } & \\multicolumn{2}{c|}{\n\\begin{tabular}{c} \\\\ \\end{tabular} } \\\\ \\hline Offline & - & ✗ & ✓ & 72.74 & 71.50 & 69.63 & 57.49 & 52.38 & 49.28 \\\\ IRM [2] & arXiv’19 & ✗ & ✓ & 71.04 & 70.31 & 68.97 & 51.11 & 45.89 & 42.86 \\\\ CORAL [50] & ECCV Workshops’16 & ✗ & ✓ & ✓ & 71.34 & 70.08 & 68.68 & 50.98 & 45.77 & 42.71 \\\\ Mixup [61] & ICLR’18 & ✗ & ✓ & ✓ & 73.34 & 71.16 & 69.29 & 57.58 & 52.77 & 49.62 \\\\ LISA [57] & ICML’22 & ✗ & ✓ & 72.19 & 70.24 & 68.60 & 56.53 & 52.41 & 49.67 \\\\ GI [41] & NeurIPS’21 & ✗ & ✓ & 68.06 & 66.32 & 64.54 & 53.43 & 49.19 & 46.13 \\\\ \\hline InCintenue & - & ✓ & ✗ & 23.57 & 71.28 & 69.80 & 56.22 & 52.43 & 49.37 \\\\ Mixup [61] & ICLR’18 & ✓ & ✗ & 73.07 & 71.52 & 69.44 & **56.64** & 52.95 & 49.97 \\\\ EWC [22] & arXiv’16 & ✓ & ✗ & **73.64** & 71.53 & 68.99 & 56.60 & 52.78 & 49.73 \\\\ SI [60] & ICML’17 & ✓ & ✗ & 72.58 & 71.50 & 69.61 & 49.98 & 47.27 & 44.77 \\\\ A-GEM [10] & ICLR’19 & ✓ & ✗ & 72.23 & 71.16 & 69.10 & 52.02 & 48.91 & 46.03 \\\\ DRAIN [3] & ICLR’23 & ✓ & ✗ & 73.42 & 71.75 & 69.69 & 56.04 & 52.07 & 48.97 \\\\ \\hline\n**EvoS** & - & ✓ & ✗ & 73.42 & **72.36** & **70.19** & 56.60 & **53.15** & **50.19** \\\\ \\hline \\end{tabular} \n\\begin{tabular}{c} For Huffpost and Arxiv, backbone DistilBERT-base is too big to apply the full GI and DRAIN. So we apply DRAIN only to the classifier and apply GI without the fine-tuning stage. \\\\ \\end{tabular}\n\\end{table}\nTable 2: Accuracy (%) on Huffpost and Arxiv. The best and second-best results in CDGTD setup are bolded and underlined. (Huffpost: \\(K=3\\), Arxiv: \\(K=7\\))\n\n\\begin{table}\n\\begin{tabular}{|c|c|c|c|c|c|c|c|c|} \\hline \\multirow{2}{*}{Method} & \\multicolumn{2}{c|}{Scale} & \\multicolumn{2}{c|}{Loss} & \\multicolumn{2}{c|}{\\begin{tabular}{c} Using sg(\\cdot) \\\\ \\end{tabular} } & \\multicolumn{2}{c|}{\\begin{tabular}{c} Using sg(\\cdot) \\\\ \\end{tabular} } & \\multicolumn{2}{c|}{\\begin{tabular}{c} \\\\ \\end{tabular} } & \\multicolumn{2}{c|}{\\begin{tabular}{c} \\\\ \\end{tabular} } & \\multicolumn{2}{c|}{\\begin{tabular}{c} \\\\ \\end{tabular} } & \\multicolumn{2}{c|}{\n\\begin{tabular}{c} \\\\ \\end{tabular} } \\\\ \\hline Variant A & - & ✓ & ✓ & ✓ & No & Yes & 51.28 \\\\ Variant B & - & ✓ & ✓ & ✓ & Yes & No & Yes & 55.37 \\\\ Variant C & - & ✓ & ✓ & ✓ & No & Yes & Yes & 93.97 \\\\ \\hline Variant D & - & ✓ & - & - & Yes & Yes & Yes & 93.25 \\\\ Variant E & - & ✓ & ✓ & ✓ & Yes & Yes & Yes & 94.75 \\\\ Variant F & - & - & ✓ & - & ✓ & Yes & Yes & 94.27 \\\\ \\hline Variant G & - & ✓ & ✓ & ✓ & Yes & Yes & No & 94.46 \\\\ EvoS & - & ✓ & ✓ & ✓ & Yes & Yes & Yes & **95.53** \\\\ Variant H & ✓ & - & ✓ & ✓ & Yes & Yes & Yes & 94.09 \\\\ \\hline \\end{tabular}\n\\end{table}\nTable 3: Ablation study of EvoS on dataset Yearbook.\n\nvariant D, E and F are all inferior to EvoS. Such results verify two points. One is that the adversarial training conduces to better generalization and mitigating the problem of overfitting. And the other is that the consistency between generated statistic tokens in the window and real statistic tokens in the memory pool imposes stronger constraint to attention module \\(\\mathcal{A}_{w}\\), helping to model the evolving pattern more accurately. Thirdly, we testify the influence of truncation when sampling features from normal distributions. The worse result of variant G than EvoS may be due to the sampled outliers hinder the learning of a shared feature space. Finally, the results of variant H and EvoS compare single-scale and multi-scale attention. The better result of EvoS shows the superiority of MSAM.\n\nLast but not least, in Table 4, we investigate the effect of randomly selecting a historical domain distribution _vs_ using all historical domain distributions in the adversarial training. From the results, we see that randomly selecting one historical domain distribution performs worse. This may be because the feature space to be aligned frequently changes if using this manner, making the optimization challenging. By contrast, it is more stable to simultaneously leverage all the preserved domain distributions in the memory pool \\(\\mathcal{M}\\) in each iteration for the adversarial training.\n\n**Choices of \\(\\mathcal{L}_{stan}^{t}\\).** In addition to training \\(\\mathcal{A}_{w}\\) via loss \\(\\mathcal{L}_{stan}^{t}\\) in Eq. 10, we also try another way. Concretely, we directly minimize the \\(l\\)-2 norm of the difference between statistics calculated in each batch and that generated by MSAM, i.e., \\(\\mathcal{L}_{stan}^{t}=\\|\\frac{1}{B}\\operatorname{sg}(\\boldsymbol{f}_{i}^{t} )-\\boldsymbol{\\hat{\\mu}}^{t}\\|_{2}+\\|\\frac{1}{B-1}\\sum_{i=1}^{B}(sg( \\boldsymbol{f}_{i}^{t})-\\frac{1}{B}\\sum_{j=1}^{B}\\operatorname{sg}( \\boldsymbol{f}_{j}^{t}))^{2}-\\left(\\boldsymbol{\\hat{\\sigma}}^{t}\\right)^{2}\\| _{2}\\). Fig. 2(a) plots the discrepancy between generated and real statistics on each test domain of Yearbook when using the two ways to train the attention module. We see that the way used in Sec. 3.3 yields a smaller discrepancy between generated and real statistics, implying that the learned evolving pattern is more accurate. So we choose to use it throughout the experiments.\n\n**Effect of Number of Heads and Scale.** We further investigate the effect of number of heads and scale of MSAM in Fig. 2(b). Varying the number of heads, the performance exhibits an inverted V-shaped trend, because too few heads would hamper the learning capability of the attention module while too many heads could lead to overfitting. As for difference scales (i.e., the maximum length of sliding time window), we find that multi-scales (\\(W\\geq 1\\)) yields better performance than single-scale (\\(W=1\\)), showing the superiority of MSAM. Yet, more scales do not always produce better results. One possible explanation is that an excessively large scale relative to the domain sequence length leads to an inadequate number of sliding windows, hampering the learning of attention module.\n\n**Visualization of Decision Boundary.** In this qualitative experiment, we visualize the change of decison boundary on the inter-twinning moons 2D problem in a gradual manner. Concretely, the 2-Moons dataset in [41] is used, where 10 domains (0 to 9) are obtained by rotating data points counter-clockwise in units of \\(18^{\\circ}\\). In Figure 3, the model is sequentially trained using EvoS until the \\(t\\)-th domain is finished, and then we visualize the decision boundary on current domain \\(\\mathcal{D}^{t}\\) and the next future domain \\(\\mathcal{D}^{t+1}\\). From the results, we can observe that the decision boundary successfully adapts to future domains, showing that EvoS can truly capture the underlying temporal drift of data.\n\n\\begin{table}\n\\begin{tabular}{|c|c|c c c c c c c c c c c c c c|} \\hline \\multirow{2}{*}{Method} & \\multicolumn{4}{c|}{Yearbook} & \\multicolumn{4}{c|}{RMSIST} & \\multicolumn{4}{c|}{RMoW} & \\multicolumn{4}{c|}{Ruffpool} & \\multicolumn{4}{c|}{Arav} \\\\ \\cline{3-14}  & Used historical domain & Accuracy (\\%) & \\(\\uparrow\\) & Accuracy (\\%) & \\(\\uparrow\\) & Accuracy (\\%) & \\(\\uparrow\\) & Accuracy (\\%) & \\(\\uparrow\\) & Accuracy (\\%) & \\(\\uparrow\\) & Accuracy (\\%) & \\(\\uparrow\\) \\\\  & distribution in \\(\\mathcal{L}_{ads}^{t}\\) & \\(D^{t+1}\\) & \\(D\\) & OOD & OPD\\({}^{t+1}\\) & OOD & OPD\\({}^{t+1}\\) & OOD & OOD & OPD\\({}^{t+1}\\) & OOD & OOD & OPD\\({}^{t+1}\\) & OOD & OOD \\\\ \\hline EvoS\\({}^{t}\\) & random one & 96.50 & 95.40 & 94.46 & 98.03 & 93.84 & 94.66 & 60.62 & 53.81 & 44.76 & 72.99 & 71.37 & 90.14 & 55.39 & 32.45 & 49.52 \\\\ EvoS & all historical & 97.37 & **95.55** & **94.78** & **98.64** & **93.84** & **87.04** & **67.18** & **84.64** & **45.86** & **73.42** & **72.36** & **70.19** & **56.60** & **53.15** & **50.19** \\\\ \\hline \\multicolumn{14}{l}{EvoS\\({}^{t}\\) denotes that we randomly select a domain distribution from the memory pool \\(\\mathcal{M}\\) to sample a batch of \\(B\\) features for participating in the adversarial training.} \\\\ \\end{tabular}\n\\end{table}\nTable 4: Results of using historical domain distributions in different ways during adversarial training.\n\nFigure 2: (a): The \\(l\\)-2 norm of the difference \\(\\Delta\\) between the statistics calculated on each test domain and the statistics generated by MSAM on Yearbook. “Ours” refers to minimizing the \\(l\\)-2 norm of \\(\\Delta\\) between statistics calculated in each batch and that generated by MSAM. (b): Effects of using different numbers of heads and scales on Yearbook.\n\n**Effect of Memory Pool Length.** In the main experiments, we do not restrict the size of the memory pool \\(\\mathcal{M}\\), since the datasets used in our paper has a moderate number of domains and the memory cost is small. Nevertheless, a fixed memory pool size is more practical when considering a lifelong process, i.e., \\(T\\rightarrow\\infty\\). Thus, we additionally conduct the experiment where the memory pool \\(\\mathcal{M}\\) is implemented as a FIFO queue with different fixed length \\(L\\). That is, only the statistics for up to the \\(L\\) most recent historical domains can be stored. The results on Yearbook in Fig. 4(b) demonstrate that our method generally performs well and a relatively large memory pool length is better.\n\n**t-SNE Visualization of Standardized Features.** To verify our method appropriate for generalization, we visualize the standardized features of all target domains via t-SNE [38] on RMNIST and Yearbook in Fig. 4(a). It can be observed that the standardized features of target domains are well aligned, suggesting that MASM can capture evolving patterns effectively and feature standardization helps address temporal drift appropriately. These enable EvoS to achieve generalization on future domains.\n\n**Hyper-parameter Sensitivity.**\\(\\alpha\\) and \\(\\lambda\\) control the truncation range and the tradeoff the adversarial loss \\(\\mathcal{L}_{adv}\\), respectively. Fig. 4(c) shows the sensitivity of EvoS to them on Yearbook, where \\(\\alpha\\in\\{0.1,0.5,1.0,1.5,2.0,2.5,3.0\\}\\) and \\(\\lambda\\in\\{0.1,0.5,1.0,1.5,2.0,5.0,10.0,15.0,20.0\\}\\). We see that EvoS is more sensitive to \\(\\lambda\\) than \\(\\alpha\\), and large \\(\\lambda\\) values significantly worsen performance. In practice, we suggest selecting \\(\\lambda\\) from a range \\(\\leq\\)1. For \\(\\alpha\\), its effect is modest and \\(\\alpha=1\\) generally works well.\n\n## 5 Conclusion\n\nThis paper considers a more challenging problem of continual domain generalization over temporal drift (CDGTD) than conventional DG, where the model is incrementally trained with sequential domains and is required to generalize to unseen domains that are not too far into the future. To this end, we propose an Evolving Standardization (EvoS) method to learn the evolving pattern of sequential domains over the temporal drift and hope to achieve the generalization by transforming the domain distribution into a common normal distribution via feature standardization. Experiments on real-world datasets including images and texts verify the efficacy of EvoS. Since existing DG works focus on conventional setting, we hope this work can encourage more research works on CDGTD.\n\nFigure 4: (a): Visualization of standardized features from each test domain on Yearbook and RMNIST. Different colors denote different domains. (b): Effect of the memory pool length \\(L\\) on dataset Yearbook. (c):Hyper-parameter sensitivity of \\(\\alpha\\) and \\(\\lambda\\) on Yearbook\n\nFigure 3: Visualization of decision boundaries for the model at current time point \\(\\mathbf{t}_{t}\\) on inter-twinning moons 2D problem. Blue line and points are the decision boundary and samples in current domain \\(\\mathcal{D}^{t}\\), and red line and points are the decision boundary and samples in the next future domain \\(\\mathcal{D}^{t+1}\\).\n\n## Acknowledgements\n\nThis paper was supported by National Key R&D Program of China (No. 2021YFB3301503), the National Natural Science Foundation of China (No. 62376026), and also sponsored by Beijing Nova Program (No. 20230484296).\n\n## References\n\n* [1] R. Aljundi, M. Lin, B. Goujaud, and Y. Bengio. Gradient based sample selection for online continual learning. In _NeurIPS_, pages 11816-11825, 2019.\n* [2] M. Arjovsky, L. Bottou, I. Gulrajani, and D. Lopez-Paz. Invariant risk minimization. _arXiv:1907.02893_, 2019.\n* [3] G. Bai, C. Ling, and L. Zhao. Temporal domain generalization with drift-aware dynamic neural networks. In _ICLR_, 2023.\n* [4] S. Ben-David, J. Blitzer, K. Crammer, A. Kulesza, F. Pereira, and J. W. Vaughan. A theory of learning from different domains. _Mach. Learn._, 79(1-2):151-175, 2010.\n* [5] S. Ben-David, J. Blitzer, K. Crammer, and F. Pereira. Analysis of representations for domain adaptation. In _NeurIPS_, pages 137-144, 2006.\n* [6] G. Blanchard, G. Lee, and C. Scott. Generalizing from several related classification tasks to a new unlabeled sample. In _NeurIPS_, pages 2178-2186, 2011.\n* [7] G. Blanchard, G. Lee, and C. Scott. Generalizing from several related classification tasks to a new unlabeled sample, 2011.\n* [8] M. Caron, I. Misra, J. Mairal, P. Goyal, P. Bojanowski, and A. Joulin. Unsupervised learning of visual features by contrasting cluster assignments. In _NeurIPS_, 2020.\n* [9] F. M. Castro, M. J. Marin-Jimenez, N. Guil, C. Schmid, and K. Alahari. End-to-end incremental learning. In _ECCV_, pages 233-248, 2018.\n* [10] A. Chaudhry, M. Ranzato, M. Rohrbach, and M. Elhoseiny. Efficient lifelong learning with A-GEM. In _ICLR_, 2019.\n* [11] T. Chen, S. Kornblith, M. Norouzi, and G. E. Hinton. A simple framework for contrastive learning of visual representations. In _ICML_, pages 1597-1607, 2020.\n* [12] X. Chen, W. Chen, V. Dinavahi, Y. Liu, and J. Feng. Short-term load forecasting and associated weather variables prediction using resnet-lstm based deep learning. _IEEE Access_, 11:5393-5405, 2023.\n* [13] L. Deng. The MNIST database of handwritten digit images for machine learning research [best of the web]. _SPM_, 29(6):141-142, 2012.\n* [14] S. Ebrahimi, F. Meier, R. Calandra, T. Darrell, and M. Rohrbach. Adversarial continual learning. In _ECCV_, volume 12356, pages 386-402, 2020.\n* [15] Y. Ganin and V. Lempitsky. Unsupervised domain adaptation by backpropagation. In _ICML_, pages 1180-1189, 2015.\n* [16] S. Ginosar, K. Rakelly, S. Sachs, B. Yin, and A. A. Efros. A century of portraits: A visual historical record of american high school yearbooks. In _ICCV Workshops_, pages 652-658, 2015.\n* [17] R. Gong, W. Li, Y. Chen, and L. Van Gool. Dlow: Domain flow for adaptation and generalization. In _CVPR_, 2020.\n* [18] G. Huang, Z. Liu, L. van der Maaten, and K. Q. Weinberger. Densely connected convolutional networks. In _CVPR_, pages 2261-2269, 2017.\n* [19] Y. Iwasawa and Y. Matsuo. Test-time classifier adjustment module for model-agnostic domain generalization. In _NeurIPS_, pages 2427-2440, 2021.\n\n* [20] Y. Iwasawa and Y. Matsuo. Test-time classifier adjustment module for model-agnostic domain generalization. In _NeurIPS_, pages 2427-2440, 2021.\n* [21] P. Izmailov, D. Podoprikhin, T. Garipov, D. P. Vetrov, and A. G. Wilson. Averaging weights leads to wider optima and better generalization. In _UAI_, pages 876-885, 2018.\n* [22] J. Kirkpatrick, R. Pascanu, N. C. Rabinowitz, J. Veness, G. Desjardins, A. A. Rusu, K. Milan, J. Quan, T. Ramalho, A. Grabska-Barwinska, D. Hassabis, C. Clopath, D. Kumaran, and R. Hadsell. Overcoming catastrophic forgetting in neural networks. _arXiv:1612.00796_, 2016.\n* [23] A. Kumar, T. Ma, and P. Liang. Understanding self-training for gradual domain adaptation. In _ICML_, pages 5468-5479, 2020.\n* [24] M. D. Lange, R. Aljundi, M. Masana, S. Parisot, X. Jia, A. Leonardis, G. G. Slabaugh, and T. Tuytelaars. A continual learning survey: Defying forgetting in classification tasks. _TPAMI_, 44(7):3366-3385, 2022.\n* [25] D. Li, Y. Yang, Y. Song, and T. M. Hospedales. Deeper, broader and artier domain generalization. In _ICCV_, pages 5543-5551, 2017.\n* [26] D. Li, Y. Yang, Y. Song, and T. M. Hospedales. Learning to generalize: Meta-learning for domain generalization. In _AAAI_, pages 3490-3497, 2018.\n* [27] H. Li, S. J. Pan, S. Wang, and A. C. Kot. Domain generalization with adversarial feature learning. In _CVPR_, pages 5400-5409, 2018.\n* [28] S. Li, C. H. Liu, Q. Lin, Q. Wen, L. Su, G. Huang, and Z. Ding. Deep residual correction network for partial domain adaptation. _TPAMI_, 43(7):2329-2344, 2021.\n* [29] S. Li, C. H. Liu, B. Xie, L. Su, Z. Ding, and G. Huang. Joint adversarial domain adaptation. In _ACM MM_, pages 729-737, 2019.\n* [30] Z. Li and D. Hoiem. Learning without forgetting. _TPAMI_, 40(12):2935-2947, 2018.\n* [31] T. Lin, Y. Wang, X. Liu, and X. Qiu. A survey of transformers. _AI Open_, 3:111-132, 2022.\n* [32] A. Liu, Y.-C. Liu, Y.-Y. Yeh, and Y.-C. Wang. A unified feature disentangler for multi-domain image translation and manipulation, 2018.\n* [33] H. Liu, M. Long, J. Wang, and Y. Wang. Learning to adapt to evolving domains, 2020.\n* [34] M. Long, Z. Cao, J. Wang, and M. I. Jordan. Conditional adversarial domain adaptation. In _NeurIPS_, pages 1647-1657, 2018.\n* [35] F. Lv, J. Liang, K. Gong, S. Li, C. H. Liu, H. Li, D. Liu, and G. Wang. Pareto domain adaptation. In _NeurIPS_, pages 12917-12929, 2021.\n* [36] F. Lv, J. Liang, S. Li, B. Zang, C. H. Liu, Z. Wang, and D. Liu. Causality inspired representation learning for domain generalization. In _CVPR_, pages 8036-8046, 2022.\n* [37] F. Lv, J. Liang, S. Li, J. Zhang, and D. Liu. Improving generalization with domain convex game. In _CVPR_, pages 24315-24324, 2023.\n* [38] L. v. d. Maaten and G. Hinton. Visualizing data using t-sne. _JMLR_, 9:2579-2605, 2008.\n* [39] K. Muandet, D. Balduzzi, and B. Scholkopf. Domain generalization via invariant feature representation. In _ICML_, pages 10-18, 2013.\n* [40] Z. Nado, S. Padhy, D. Sculley, A. D'Amour, B. Lakshminarayanan, and J. Snoek. Evaluating prediction-time batch normalization for robustness under covariate shift. _arXiv:2006.10963_, 2020.\n* [41] A. Nasery, S. Thakur, V. Piratla, A. De, and S. Sarawagi. Training for the future: A simple gradient interpolation loss to generalize along time. In _NeurIPS_, pages 19198-19209, 2021.\n\n* [42] A. Nguyen, T. Tran, Y. Gal, and A. Baydin. Domain invariant representation learning with domain density transformations, 2021.\n* [43] G. Ortiz-Jimenez, M. E. Gheche, E. Simou, H. P. Maretic, and P. Frossard. CDOT: continuous domain adaptation using optimal transport. _arXiv:1909.11448_, 2019.\n* [44] T. Qin, S. Wang, and H. Li. Generalizing to evolving domains with latent structure-aware sequential autoencoder. In _ICML_, pages 18062-18082, 2022.\n* [45] J. Rajasegaran, M. Hayat, S. H. Khan, F. S. Khan, and L. Shao. Random path selection for incremental learning. _arXiv:906.01120_, 2019.\n* [46] S. Rebuffi, A. Kolesnikov, G. Sperl, and C. H. Lampert. icarl: Incremental classifier and representation learning. In _CVPR_, pages 5533-5542, 2017.\n* [47] G. Saha and K. Roy. Continual learning with scaled gradient projection. In _AAAI_, pages 9677-9685, 2023.\n* [48] V. Sanh, L. Debut, J. Chaumond, and T. Wolf. Distilbert, a distilled version of BERT: smaller, faster, cheaper and lighter. _arXiv:1910.01108_, 2019.\n* [49] S. Schneider, E. Rusak, L. Eck, O. Bringmann, W. Brendel, and M. Bethge. Improving robustness against common corruptions by covariate shift adaptation. In _NeurIPS_, 2020.\n* [50] B. Sun and K. Saenko. Deep coral: Correlation alignment for deep domain adaptation. In _ECCV Workshops_, pages 443-450, 2016.\n* [51] R. Taori, A. Dave, V. Shankar, N. Carlini, B. Recht, and L. Schmidt. Measuring robustness to natural distribution shifts in image classification. In _NeurIPS_, 2020.\n* [52] R. Tiwari, K. Killamsetty, R. K. Iyer, and P. Shenoy. GCR: gradient coreset based replay buffer selection for continual learning. In _CVPR_, pages 99-108, 2022.\n* [53] J. Tobin, R. Fong, A. Ray, J. Schneider, W. Zaremba, and P. Abbeel. Domain randomization for transferring deep neural networks from simulation to the real world. In _IROS_, 2017.\n* [54] D. Wang, E. Shelhamer, S. Liu, B. A. Olshausen, and T. Darrell. Tent: Fully test-time adaptation by entropy minimization. In _ICLR_, 2021.\n* [55] H. Wang, H. He, and D. Katabi. Continuously indexed domain adaptation. In _ICML_, volume 119, pages 9898-9907, 2020.\n* [56] H. Yao, C. Choi, B. Cao, Y. Lee, P. W. Koh, and C. Finn. Wild-time: A benchmark of in-the-wild distribution shift over time. In _NeurIPS_, 2022.\n* [57] H. Yao, Y. Wang, S. Li, L. Zhang, W. Liang, J. Zou, and C. Finn. Improving out-of-distribution robustness via selective augmentation. In _ICML_, pages 25407-25437, 2022.\n* [58] L. Yuan, B. Xie, and S. Li. Robust test-time adaptation in dynamic scenarios. In _CVPR_, pages 15922-15932, 2023.\n* [59] Q. Zeng, W. Wang, F. Zhou, C. Ling, and B. Wang. Foresee what you will learn: Data augmentation for domain generalization in non-stationary environments. _arXiv:2301.07845_, 2023.\n* [60] F. Zenke, B. Poole, and S. Ganguli. Continual learning through synaptic intelligence. In _ICML_, pages 3987-3995, 2017.\n* [61] H. Zhang, M. Cisse, Y. N. Dauphin, and D. Lopez-Paz. mixup: Beyond empirical risk minimization. In _ICLR_, 2018.\n* [62] K. Zhou, Z. Liu, Y. Qiao, T. Xiang, and C. C. Loy. Domain generalization: A survey. _TPAMI_, 45(4):4396-4415, 2023.\n* [63] K. Zhou, Y. Yang, Y. Qiao, and T. Xiang. Domain generalization with mixstyle. In _ICLR_, 2021.\n\n## Appendix A Broader Impacts & Limitations\n\n1. [label=.]\n2. Ethics Statement & Licenses\n3. Algorithm of EvoS\n4. Experimental Setup Details\n5. Experimental Results with Error Bars\n6. Complexity Analysis\n7. Additional Results\n\n## Appendix B Ethics Statement & Licenses\n\nAll the datasets used in the paper are publicly available and are only intended to compare the performances of different algorithms on classification tasks, adhering to the following licenses of these datasets:\n\n* Yearbook: MIT licensed\n* MNIST: CC BY-SA 3.0\n* fMoW: The Functional Map of the World Challenge Public License\n* Huffpost: CC0: Public Domain\n* arXiv: CC0: Public Domain\n\n## Appendix C Algorithm of EvoS\n\nThe training and inference procedure for EvoS is provided in Algorithm 1 and 2, respectively.\n\n## Appendix D Experimental Setup Details\n\n### Dataset Description\n\n**Yearbook** used in our paper is from [56]. It is based on the Portraits [16] dataset (MIT license), containing the \\(32\\times 32\\) grayscale images of yearbook portraits from 128 American high schools in 27 states. The data spanning from 1930 to 2013 reflects the evolving fashion styles and shifting social norms throughout the decades. The task is the binary classification of genders. Note that it is only used for comparing the generalization performances of different algorithms on classification tasks. For this dataset, we treat every four years as a domain, resulting in 16 domains. Table 11 provides the number of samples in each domain. The first 16 domains are used for training (\\(T=16\\)), the last 5 domains for testing (\\(K=5\\)).\n\n**RMNIST** is a variant of the MNIST [13] dataset, which comprises 9 domains generated by applying the rotations with degree of \\(0^{\\circ},10^{\\circ},\\cdots,80^{\\circ}\\) in order to MNIST to simulate the temporal drift. The task is to classify a \\(28\\times 28\\) grayscale digit image from 0 to 9. We use the first 6 domains for training (\\(T=6\\)) and the last 3 for testing (\\(K=3\\)).\n\n**fMoW** used in our paper is from [56]. It collects the \\(224\\times 224\\) RGB satellite images from 2002 to 2017 over 200 countries, where the visual features present in satellite data undergo changes over time due to both human and environmental activities. The task to classify the functional purpose of the buildings or the land in the images into one of 62 categories. For this dataset, we treat every year as a domain and Table 12 provides the number of samples in each domain. The first 13 domains are used for training (\\(T=13\\)) and the last 3 domains are used for testing (\\(K=3\\)).\n\n**Huffpost** in [56] comprises news headlines from the Huffington Post from 2012 to 2018, the task of which is to classify the news headline into one of 11 news categories (\"Black Voices\", \"Business\", \"Comedy\", \"Crime\", \"Entertainment\", \"Impact\", \"Queer Voices\", \"Science\", \"Sports\", \"Tech\", \"Travel\"). The data spanning for 2012 to 2018 presents changes in the content or style of news along the time dimension. For this dataset, we use the first 4 years for training (\\(T=4\\)) and the last 3 years for testing (\\(K=3\\)). The number of samples in each domain is provided in Table 13.\n\n**Arxiv** in [56] contains paper titles and their corresponding primary categories spanning from 2007 to 2022. The content of Arxiv preprints evolves over time, reflecting the dynamic nature of research fields. And the task is to classify a research paper into one of 172 categories based on its title. We use the data from the first 9 domains for training (\\(T=9\\)) and the data from the last 7 years for testing (\\(K=7\\)). The number of samples in each domain is presented in Table 14.\n\nFor each training domain of all datasets, we randomly select 90% of data as training split and 10% of data as validation split. The evaluation of generalization performance is based on the whole data of each test domain.\n\n### Implementation Details\n\nAll experiments are implemented via PyTorch and the backbones we use mainly adhere to [56].\n\n**Yearbook** uses a 4-layer convolutional network from [56] as the backbone. For multi-scale attention module (MSAM), we set the dimension of head as \\(d_{h}=8\\) for each attention module \\(\\mathcal{A}_{w}\\) and the number of heads for \\(\\mathcal{A}_{w}\\) as \\(w\\cdot n_{h}\\), with \\(n_{h}=16,w=1,\\cdots,W\\). And the Adam optimizer with \\(lr=1e-3\\) is used to optimize the model, where the batch size \\(B\\) is set to 64 and the training epochs of each domain are set to 50. As for the hyper-parameters, we use \\(\\alpha=1.0,\\lambda=1.0,W=3\\).\n\n**RMNIST** adopts the ConvNet in [44] as the backbone, and we set the dimension of head as \\(d_{h}=8\\) for each attention module \\(\\mathcal{A}_{w}\\) and the number of heads for \\(\\mathcal{A}_{w}\\) as \\(w\\cdot n_{h}\\), with \\(n_{h}=32,w=1,\\cdots,W\\). The Adam optimizer with \\(lr=1e-3\\) is adopted for model optimization. The batch size and training epochs of each domain are set to 64 and 50, respectively. And \\(\\alpha=2.0,\\lambda=1.0,W=3\\) is used.\n\n**fMoW** employ the DenseNet-121 [18] pretrained on ImageNet as the backbone. Besides, we use a bottleneck layer [28] to reduce the feature dimensions into 256, and set the dimension of head as \\(d_{h}=8\\) for each attention module \\(\\mathcal{A}_{w}\\) and the number of heads for \\(\\mathcal{A}_{w}\\) as \\(w\\cdot n_{h}\\), with \\(n_{h}=64,w=1,\\cdots,W\\). Similarly, the Adam optimizer is used, where the learning rate is set to \\(lr=2e-4\\). The batch size is set to \\(B=64\\) and each training domain is trained for 25 epochs. As for the hyper-parameters, we set \\(\\alpha=1.0,\\lambda=1.0,W=3\\) for fMoW dataset.\n\n**Huffpost** and **Arxiv** use the pretrained DistilBERT base model [48] as the backbone. For MSAM, we set the dimension of head as \\(d_{h}=8\\) for each attention module \\(\\mathcal{A}_{w}\\) and the number of heads for \\(\\mathcal{A}_{w}\\) as \\(w\\cdot n_{h}\\), with \\(n_{h}=128,w=1,\\cdots,W\\). Also the Adam optimizer is used with the learning rate \\(lr=2e-5\\) and batch size \\(B=64\\). And \\(\\alpha=1.0,\\lambda=1.0,W=3\\) is used for the two dataset. For the Huffpost dataset, each training is trained for 50 epochs, while the training epochs of each domain are set to 5 for the Arxiv dataset.\n\nWe run each task on a single NVIDIA GeForce RTX 3090 GPU for three random trials. For baselines, we also select their hyper-parameters via the grid search using the validation splits of training domains.\n\n## Appendix E Experimental Results with Error Bars\n\nIn this section, we report the mean and standard derivation (denoted as mean (std) ) for each task, when running with 3 random trials. The results with error bars on Yearbook, RMNIST and fMoW are provided in Table 5, and the error bars on Huffpost and Arxiv are given in Table 6.\n\n## Appendix F Complexity Analysis\n\n**Time Complexity of MSAM.** Taking one of the multi-head self-attention module \\(\\mathcal{A}_{w}\\) in MSAM as an example, we denote \\(d_{f}\\) as the feature dimension of its input tokens, and \\(d_{h}\\), \\(n_{h}\\) and \\(n_{i}\\) denote the feature dimension of its heads, the number of its heads and the number of its input tokens, respectively. Then the time complexity of \\(\\mathcal{A}_{w}\\) is \\(\\mathcal{O}((n_{i}^{2}+n_{i}\\cdot d_{f})\\cdot(d_{h}\\cdot n_{h}))\\). Since \\(n_{i}\\) will be no larger than the number of training domains \\(T\\) and \\(d_{h}\\cdot n_{h}\\) is usually set to \\(d_{f}\\) in transformers, the time complexity of MSAM can be roughly approximated as \\(\\mathcal{O}(W\\cdot(T^{2}d_{f}+T\\cdot d_{f}^{2}))\\), where \\(W\\) is the number of multi-head self-attention modules in MSAM. It is roughly equivalent to the time complexity of a single multi-head self-attention layer in conventional transformers multiplied by \\(W\\).\n\n**Memory Complexity of \\(\\mathcal{M}\\).** Assuming that there are \\(T\\) historical domains and the dimension of statistic vectors is \\(d_{f}\\), then the memory complexity of the memory pool \\(\\mathcal{M}\\) is \\(\\mathcal{O}(T\\cdot d_{f})\\). In practice, after being processed by deep neural networks, the dimension of pooled features is usually much smaller than that of original inputs. Moreover, only two vectors need to be stored per domain. Hence, the memory cost of \\(\\mathcal{M}\\) is relatively small, compared with sample replay-based CL methods. Concretely, Table 7 provides the memory cost of \\(\\mathcal{M}\\) and the increment of GPU memory cost for EvoS on Yearbook, RMNIST and fMoW datasets with batch size 64.\n\n\\begin{table}\n\\begin{tabular}{|c|c|c|c|c c c|c c c|} \\hline \\multirow{2}{*}{Method} & \\multirow{2}{*}{\\begin{tabular}{c} Incremental \\\\ training \\\\ \\end{tabular} } & \\multirow{2}{*}{\\begin{tabular}{c} Access \\\\ multiple \\\\ \\end{tabular} } & \\multicolumn{3}{c|}{\\begin{tabular}{c} MultiPots \\\\ \\(\\mathcal{O}\\)OD \\\\ \\end{tabular} } & \\multicolumn{3}{c|}{\\begin{tabular}{c} Arxiv \\\\ Accuracy (\\%) \\(\\uparrow\\) \\\\ \\end{tabular} } \\\\  & & & \\multicolumn{3}{c|}{\\begin{tabular}{c} Accuracy (\\%) \\(\\uparrow\\) \\\\ \\end{tabular} } & \\multicolumn{3}{c|}{\\begin{tabular}{c} Accuracy (\\%) \\(\\uparrow\\) \\\\ \\end{tabular} } \\\\  & & & \\multicolumn{3}{c|}{\\begin{tabular}{c} \\(\\mathcal{O}\\)OD \\\\ \\end{tabular} } & \\multicolumn{3}{c|}{\\begin{tabular}{c} OOD \\\\ \\(\\mathcal{D}^{T+1}\\) \\\\ \\end{tabular} } & \\multicolumn{3}{c|}{\n\\begin{tabular}{c} OOD \\\\ avg. \\\\ \\end{tabular} } \\\\ \\hline Offline & ✗ & ✓ & 72.74 (0.14) & 71.50 (0.56) & 69.63 (0.81) & 57.49 (0.15) & 52.38 (0.43) & 49.28 (0.67) \\\\ IRM [2] & ✗ & ✓ & 71.04 (0.45) & 70.31 (0.67) & 69.87 (0.87) & 51.11 (0.14) & 45.89 (2.77) & 42.86 (3.98) \\\\ CORAL [50] & ✗ & ✓ & 71.34 (0.51) & 70.08 (0.68) & 69.68 (0.94) & 50.88 (1.34) & 45.77 (2.92) & 47.14 (1.15) \\\\ Mixup [61] & ✗ & ✓ & 73.34 (0.02) & 71.16 (0.07) & 69.29 (0.12) & 57.58 (0.03) & 52.77 (0.19) & 49.62 (0.24) \\\\ LISA [37] & ✗ & ✓ & 72.19 (0.06) & 70.24 (0.56) & 68.60 (0.91) & 56.53 (0.02) & 52.41 (0.11) & 49.67 (0.23) \\\\ GI [41] & ✗ & ✓ & 68.06 (1.51) & 66.32 (7.8) & 64.64 (3.67) & 53.33 (1.65) & 49.19 (2.49) & 46.13 (3.06) \\\\ \\hline InFinetune & ✓ & ✗ & 73.57 (0.02) & 71.98 (0.06) & 69.80 (0.11) & 56.22 (0.02) & 52.43 (0.14) & 49.37 (0.21) \\\\ Mixup [61] & ✓ & ✗ & 73.07 (0.08) & 71.52 (0.21) & 69.44 (0.29) & **56.44 (0.02)** & 52.95 (0.06) & 49.97 (0.11) \\\\ EWC [22] & ✓ & ✗ & **73.64** (0.02) & 71.53 (0.29) & 68.99 (0.46) & 56.60 (0.01) & 52.78 (0.07) & 49.33 (0.12) \\\\ SI [60] & ✓ & ✗ & 72.58 (0.07) & 71.50 (0.15) & 69.61 (0.34) & 49.98 (0.19) & 47.27 (2.46) & 44.77 (3.11) \\\\ A-GEM [10] & ✓ & ✗ & 72.23 (0.06) & 71.16 (0.14) & 69.10 (0.28) & 52.02 (0.09) & 48.91 (2.67) & 46.03 (0.29) \\\\ DRAIN [3] & ✓ & ✗ & 73.42 (0.02) & 71.75 (0.16) & 69.69 (0.20) & 56.04 (0.06) & 52.07 (0.18) & 48.97 (0.34) \\\\ \\hline\n**EvoS** & & & & & & & & & & \\\\ \\hline \\end{tabular}\n\\end{table}\nTable 6: Accuracy (%) on Huffpost and Arxiv. The best and second-best results in CDGTD setup are bolded and underlined. (Huffpost: \\(K=3\\), Arxiv: \\(K=7\\))\n\n**Model Complexity.** Here, we measure the model complexity by the number of parameters. Specifically, Table 8 presents the model complexity of our method EvoS and other two temporal DG methods LSSAE [44] and DRAIN [3] on Yearbook, RMNIST and fMoW datasets. LSSAE [44] introduces sequential autoencoder to explore the underlying continuous structure in the latent space of deep neural networks, where the complicated VAE and LSTM networks require lots of parameters. And DRAIN [3] needs to encode and decode the entire network parameters, which also requires a great number of parameters for large backbone networks. By contrast, our EvoS is overall less complex than these temporal DG methods, and is more friendly to the relatively large backbone network.\n\n## Appendix G Additional Results\n\n**Results under Eval-Stream Manner.** In this part, we additionally provide the results when adopting the evaluation manner of Eval-Stream in [56]. Specifically, Eval-Stream denotes the evaluation with domain stream, i.e., the model is evaluated at each timestamp using the average and worst performance on the next \\(K\\) timestamps. Formally, given a sequence of domains with total length \\(\\mathcal{T}\\), the average performance \\(Avg_{stream}\\) and worst performance \\(Worst_{stream}\\) under the strategy of Eval-Stream are defined as \\(Avg_{stream}=\\frac{1}{\\mathcal{T}-K}\\sum_{i=1}^{\\mathcal{T}-K}\\frac{1}{K}\\sum _{j=i+1}^{i+K}Acc_{i}(\\mathcal{D}^{j})\\), \\(Worst_{stream}=\\frac{1}{\\mathcal{T}-K}\\sum_{i=1}^{\\mathcal{T}-K}\\min_{j\\in\\{i+ 1,\\cdots,i+K\\}}Acc_{i}(\\mathcal{D}^{j}),\\) where \\(Acc_{i}(\\mathcal{D}^{j})\\) is the accuracy on domain \\(\\mathcal{D}^{j}\\) when using the model at the \\(i\\)-th timestamp. And Table 9 shows the results on Yearbook and Huffpost datasets when using the Eval-Stream evaluation manner. According to the results, we see that EvoS still outperforms other baselines, showing that our method is better at handling the problem of continual domain generalization over temporal drift.\n\n\\begin{table}\n\\begin{tabular}{|c|c|c|c|} \\hline \\multicolumn{4}{|c|}{GPU memory cost (GB) of different methods} \\\\ \\hline \\multirow{3}{*}{Method} & \\multicolumn{3}{c|}{Dataset} \\\\ \\cline{2-4}  & Yearbook & RMNIST & fMoW \\\\ \\cline{2-4}  & Backbone & & \\\\ \\cline{2-4}  & 4-layer CNN [56] & ConvNet [44] & DenseNet-121 [18] \\\\ \\hline IncFinetune & 1.72 & 1.84 & 10.69 \\\\ EvoS & 1.94 & 2.09 & 11.04 \\\\ Increment \\(\\Delta\\) (GB) & 0.22 & 0.25 & 0.35 \\\\ \\hline \\multicolumn{4}{|c|}{Memory cost (MB) of the memory pool \\(\\mathcal{M}\\)} \\\\ \\hline EvoS & 5.25 & 9.00 & 32.00 \\\\ \\hline \\end{tabular}\n\\end{table}\nTable 7: Memory cost on Yearbook, RMNIST and fMoW datasets.\n\n\\begin{table}\n\\begin{tabular}{|c|c|c|c|} \\hline \\multirow{3}{*}{Method} & \\multicolumn{3}{c|}{Dataset} \\\\ \\cline{2-4}  & Yearbook & RMNIST & fMoW \\\\ \\cline{2-4}  & \\multicolumn{3}{c|}{Backbone} \\\\ \\cline{2-4}  & 4-layer CNN [56] & ConvNet [44] & DenseNet-121 [18] \\\\ \\hline LSSAE [44] & 4.70 & 23.25 & 90.92 \\\\ DRAIN [3] & 7.51 & 184.29 & 1113.85 \\\\ EvoS & 1.94 & 15.81 & 56.11 \\\\ \\hline \\end{tabular}\n\\end{table}\nTable 8: Model parameters (MB) of different methods on Yearbook, RMNIST and fMoW.\n\n\\begin{table}\n\\begin{tabular}{|c|c|c|c|} \\hline \\multirow{3}{*}{Method} & \\multicolumn{3}{c|}{Dataset} \\\\ \\cline{2-4}  & Yearbook & RMNIST & fMoW \\\\ \\cline{2-4}  & \\multicolumn{3}{c|}{Backbone} \\\\ \\cline{2-4}  & 4-layer CNN [56] & ConvNet [44] & DenseNet-121 [18] \\\\ \\hline LSSAE [44] & 4.70 & 23.25 & 90.92 \\\\ DRAIN [3] & 7.51 & 184.29 & 1113.85 \\\\ EvoS & 1.94 & 15.81 & 56.11 \\\\ \\hline \\end{tabular}\n\\end{table}\nTable 9: Accuracy (%) on Yearbook and Huffpost under the evaluation manner of **Eval-Stream**. (Yearbook: \\(\\mathcal{T}=21,K=5\\), Huffpost: \\(\\mathcal{T}=7,K=3\\))\n\n**Results on More Datasets.** In addition to the datasets provided in [56], we also run our method on the _2-Moons_, _ONP_ and _Elec2_ datasets used in [3]. Specifically, _2-Moons_ is a variant of the 2-entangled moons dataset by rotating data counter-clockwise in units of \\(18^{\\circ}\\) to construct 10 domains, where the rotation angle is used to simulate the temporal shift. For 2-Moons, we use Adam optimizer and a MLP with two hidden layers of hidden size 64 and 128, and \\(B=64,lr=1e-3,\\alpha=1.0,\\lambda=1.0,W=3,d_{h}=8,n_{h}=32,T=9,K=1\\). _Online News Popularity (ONP)_ summarizes a heterogeneous set of features about articles published by Mashable in a period of two years. The dataset is split into 6 domains by time and the goal is to predict the number of shares in social networks (popularity). For ONP, we use Adam optimizer and a MLP with one hidden layer of hidden size 128, and \\(B=64,lr=1e-4,\\alpha=1.0,\\lambda=1.0,W=3,d_{h}=8,n_{h}=32,T=5,K=1\\). _Electrical Demand (Elec2)_ contains information about the demand of electricity in a particular province. Following [3, 41], the first 30 domains in Elec2 are used (two weeks as one time domain) and the task is to predict whether the demand of electricity in each period (of 30 mins) is higher or lower than the average demand over the last day. For Elec2, we use Adam optimizer and a MLP with two hidden layers of hidden size 128 and 128, and \\(B=64,lr=1e-4,\\alpha=1.0,\\lambda=1.0,W=3,d_{h}=8,n_{h}=32,T=29,K=1\\). For these datasets, as in [3, 41], we use the last domain for testing and the rest for training. Please refer to [3] for more dataset details. The experimental results are given in Table 10, where the misclassification errors of compared baselines are all reported from DRAIN [3]. From Table 10, we can observe that our method EvoS still surpasses the most recent method DRAIN, affirming its effectiveness in temporal domain generalization.\n\n\\begin{table}\n\\begin{tabular}{|c|c|c|c|c|} \\hline \\multirow{2}{*}{Method} & \\multirow{2}{*}{Conference} & \\multicolumn{3}{c|}{Misclassification error (in \\%) \\(\\downarrow\\)} \\\\ \\cline{3-5}  & & 2-Moons & \\multicolumn{1}{c|}{ONP} & \\multicolumn{1}{c|}{Elec2} \\\\ \\hline Offline & - & 22.4\\(\\pm\\)4.6 & 33.8\\(\\pm\\)0.6 & 23.0\\(\\pm\\)3.1 \\\\ LastDomain & - & 14.9\\(\\pm\\)0.9 & 36.0\\(\\pm\\)0.2 & 25.8\\(\\pm\\)0.6 \\\\ IncFinetune & - & 16.7\\(\\pm\\)3.4 & 34.0\\(\\pm\\)0.3 & 27.3\\(\\pm\\)4.2 \\\\ CDOT [43] & arXiv’19 & 9.3\\(\\pm\\)1.0 & 34.1\\(\\pm\\)0.0 & 17.8\\(\\pm\\)0.6 \\\\ CIDA [55] & ICML’20 & 10.8\\(\\pm\\)1.6 & 34.7\\(\\pm\\)0.6 & 14.1\\(\\pm\\)0.2 \\\\ GI [41] & NeurIPS’21 & 3.5\\(\\pm\\)1.4 & 36.4\\(\\pm\\)0.8 & 16.9\\(\\pm\\)0.7 \\\\ DRAIN [3] & ICLR’23 & 3.2\\(\\pm\\)1.2 & 38.3\\(\\pm\\)1.2 & 12.7\\(\\pm\\)0.8 \\\\\n**EvoS** & - & **2.5\\(\\pm\\)1.0** & **33.1\\(\\pm\\)0.6** & **11.6\\(\\pm\\)0.7** \\\\ \\hline \\end{tabular}\n\\end{table}\nTable 10: Misclassification error (%) on 2-Moons, ONP and Elec2. (\\(K=1\\))\n\n\\begin{table}\n\\begin{tabular}{|c|c|c c c|} \\hline Domain & Interval & Training Split & Validation Split & All \\\\ \\hline\n\n[MISSING_PAGE_POST]\n\n \\hline total & 1930 - 2013 & 33431 & 3758 & 37189 \\\\ \\hline \\end{tabular}\n\\end{table}\nTable 11: Data Subset Size for the Yearbook Dataset.\n\n\\begin{table}\n\\begin{tabular}{|c|c|c c c|} \\hline Domain & Year & Training Split & Validation Split & All \\\\ \\hline\n1 & 2012 & 6701 & 744 & 7446 \\\\\n2 & 2013 & 7492 & 832 & 8325 \\\\\n3 & 2014 & 9539 & 1059 & 10599 \\\\\n4 & 2015 & 11826 & 1313 & 13140 \\\\ \\hline\n5 & 2016 & 10548 & 1172 & 11721 \\\\\n6 & 2017 & 7907 & 878 & 8786 \\\\\n7 & 2018 & 3501 & 388 & 3890 \\\\ \\hline total & 2012-2018 & 57514 & 6386 & 63907 \\\\ \\hline \\end{tabular}\n\\end{table}\nTable 13: Data Subset Size for the Huffpost Dataset.\n\n\\begin{table}\n\\begin{tabular}{|c|c|c c c|} \\hline Domain & Year & Training Split & Validation Split & All \\\\ \\hline\n1 & 2002 & 1676 & 227 & 1903 \\\\\n2 & 2003 & 2279 & 276 & 2555 \\\\\n3 & 2004 & 1755 & 240 & 1995 \\\\\n4 & 2005 & 2512 & 324 & 2836 \\\\\n5 & 2006 & 3155 & 406 & 3561 \\\\\n6 & 2007 & 1497 & 190 & 1687 \\\\\n7 & 2008 & 2261 & 298 & 2559 \\\\\n8 & 2009 & 7439 & 935 & 8374 \\\\\n9 & 2010 & 18957 & 2456 & 21413 \\\\\n10 & 2011 & 22111 & 2837 & 24948 \\\\\n11 & 2012 & 24704 & 3138 & 27842 \\\\\n12 & 2013 & 3465 & 385 & 3850 \\\\\n13 & 2014 & 5572 & 620 & 6192 \\\\ \\hline\n14 & 2015 & 8885 & 988 & 9873 \\\\\n15 & 2016 & 14363 & 1596 & 15959 \\\\\n16 & 2017 & 5534 & 615 & 6149 \\\\ \\hline total & 2002-2017 & 126165 & 15531 & 141696 \\\\ \\hline \\end{tabular}\n\\end{table}\nTable 12: Data Subset Size for the fMoW Dataset.\n\n\\begin{table}\n\\begin{tabular}{|c|c|c c c|} \\hline Domain & Year & Training Split & Validation Split & All \\\\ \\hline\n1 & 2007 & 131550 & 14616 & 146167 \\\\\n2 & 2008 & 62460 & 6939 & 69400 \\\\\n3 & 2009 & 206244 & 22916 & 229161 \\\\\n4 & 2010 & 50665 & 5629 & 56295 \\\\\n5 & 2011 & 55741 & 6193 & 61935 \\\\\n6 & 2012 & 51678 & 5741 & 57420 \\\\\n7 & 2013 & 64951 & 7216 & 72168 \\\\\n8 & 2014 & 79498 & 8833 & 8832 \\\\\n9 & 2015 & 193979 & 21553 & 215533 \\\\ \\hline\n10 & 2016 & 120682 & 13409 & 134092 \\\\\n11 & 2017 & 111024 & 12336 & 123361 \\\\\n12 & 2018 & 123891 & 13765 & 137657 \\\\\n13 & 2019 & 142767 & 15862 & 158630 \\\\\n14 & 2020 & 166014 & 18445 & 184460 \\\\\n15 & 2021 & 201241 & 22360 & 223602 \\\\\n16 & 2022 & 89765 & 9973 & 99739 \\\\ \\hline total & 2007-2022 & 1852150 & 205786 & 2057952 \\\\ \\hline \\end{tabular}\n\\end{table}\nTable 14: Data Subset Size for the Arxiv Dataset.", "response": "## Review\n\n### Summary\n\nThis paper addresses the challenge of Continual Domain Generalization over Temporal Drift (CDGTD), where data distributions evolve over time, requiring models to generalize to unseen future domains. The authors propose a novel method known as Evolving Standardization (EvoS), which utilizes a multi-scale attention module to predict and adapt to these changing distributions. The method standardizes features based on generated statistics from the current and historical domains. The effectiveness of EvoS is demonstrated through extensive experiments on various real-world datasets, achieving state-of-the-art performance. Overall, the paper tackles a significant problem in machine learning with innovative solutions and thorough evaluations.\n\n### Strengths\n\n- 1. The introduction of the CDGTD problem is innovative and highly relevant, as it mimics real-world scenarios where domains arrive sequentially.\n- 2. The proposed method, EvoS, is novel and utilizes a multi-scale attention module effectively to capture evolving feature statistics.\n- 3. The paper is well-structured, clearly written, and provides thorough evaluations with ablation studies.\n- 4. The experimental results demonstrate strong performance across diverse datasets, showcasing the method's effectiveness.\n\n### Weaknesses\n\n- 1. The technical contribution appears somewhat incremental, as some concepts like domain-invariant adversarial loss are common in the field.\n- 2. The complexity of the EvoS method is not thoroughly analyzed, particularly regarding computational requirements and memory complexity.\n- 3. The choice of baselines for comparison lacks recent state-of-the-art methods, potentially undermining the evaluation's robustness.\n- 4. The paper could benefit from additional visualizations to qualitatively demonstrate the method's performance over time.\n\n### Questions\n\n- 1. Clarification needed on the number of heads and feature dimensions in the multi-scale attention module.\n- 2. Concerns about the sampling process for proxy samples and its impact on diversity.\n- 3. Inquiry regarding the potential imbalance in adversarial training as the number of historical domains increases.\n- 4. Suggestions for visualizing trained models to demonstrate the ability to capture temporal drift.\n- 5. Clarification requested on the rationale behind hyperparameter choices and the differences in experimental settings compared to existing benchmarks.\n\n### Soundness\n\n**Score:** 3\n\n**Description:** 3 = good; the paper presents a sound methodology, though some technical aspects could be further strengthened.\n\n### Presentation\n\n**Score:** 3\n\n**Description:** 3 = good; the paper is clearly written and mostly well-organized, but some areas could benefit from additional detail.\n\n### Contribution\n\n**Score:** 3\n\n**Description:** 3 = good; while the paper presents novel ideas and addresses an important problem, some contributions appear incremental.\n\n### Rating\n\n**Score:** 6\n\n**Description:** 6 = Weak Accept; the paper is technically solid with moderate-to-high impact, but there are several areas for improvement.\n\n### Paper Decision\n\n**Decision:** Accept (poster)\n\n**Reasons:** The paper is innovative, addressing a significant challenge in machine learning with a novel method. It demonstrates good performance across multiple datasets and is well-presented. While there are some weaknesses, such as potential incremental contributions and the need for more recent baseline comparisons, the strengths outweigh these concerns, making it suitable for acceptance.\n"}
{"query": "You are a highly experienced, conscientious, and fair academic reviewer, please help me review this paper. The review should be organized into nine sections: \n1. Summary: A summary of the paper in 100-150 words.\n2. Strengths/Weaknesses/Questions: The Strengths/Weaknesses/Questions of paper, which should be listed in bullet points, with each point supported by specific examples from the article where possible.\n3. Soundness/Contribution/Presentation: Rate the paper's Soundness/Contribution/Presentation, and match this score to the corresponding description from the list below and provide the result. The possible scores and their descriptions are: \n    1 poor\n    2 fair\n    3 good\n    4 excellent\n4. Rating: Give this paper an appropriate rating, match this rating to the corresponding description from the list below and provide the result. The possible Ratings and their descriptions are: \n    1 strong reject\n    2 reject, significant issues present\n    3 reject, not good enough\n    4 possibly reject, but has redeeming facets\n    5 marginally below the acceptance threshold\n    6 marginally above the acceptance threshold\n    7 accept, but needs minor improvements \n    8 accept, good paper\n    9 strong accept, excellent work\n    10 strong accept, should be highlighted at the conference   \n5. Paper Decision: It must include the Decision itself(Accept or Reject) and the reasons for this decision, based on the criteria of originality, methodological soundness, significance of results, and clarity and logic of presentation.\n\nHere is the template for a review format, you must follow this format to output your review result:\n**Summary:**\nSummary content\n\n**Strengths:**\n- Strength 1\n- Strength 2\n- ...\n\n**Weaknesses:**\n- Weakness 1\n- Weakness 2\n- ...\n\n**Questions:**\n- Question 1\n- Question 2\n- ...\n\n**Soundness:**\nSoundness result\n\n**Presentation:**\nPresentation result\n\n**Contribution:**\nContribution result\n\n**Rating:**\nRating result\n\n**Paper Decision:**\n- Decision: Accept/Reject\n- Reasons: reasons content\n\n\nPlease ensure your feedback is objective and constructive. The paper is as follows:\n\n# LinGCN: Structural Linearized Graph Convolutional Network for Homomorphically Encrypted Inference\n\nHongwu Peng\\({}^{1}\\), \\({}^{*}\\)Ran Ran\\({}^{2}\\), Yukui Luo\\({}^{3}\\), Jiahui Zhao\\({}^{1}\\), Shaoyi Huang\\({}^{1}\\), Kiran Thorat\\({}^{1}\\), Tong Geng\\({}^{4}\\), Chenghong Wang\\({}^{5}\\), Xiaolin Xu\\({}^{6}\\), Wujie Wen\\({}^{2}\\), Caiwen Ding\\({}^{1}\\)\n\n\\({}^{1}\\)University of Connecticut, \\({}^{2}\\)North Carolina State University,\n\n\\({}^{3}\\) University of Massachusetts Dartmouth, \\({}^{4}\\)University of Rochester,\n\n\\({}^{5}\\)Indiana University Bloomington, \\({}^{6}\\)Northeastern University\n\n{hongwu.peng, jiahui.zhao, shaoyi.huang, kiran_gautam.thorat}@uconn.edu, {rran, wwen2}@ncsu.edu,yluo2@umassd.edu, tgeng@ur.rochester.edu cw166@iu.edu,x.xu@northeastern.edu, caiwen.ding@uconn.edu\n\n###### Abstract\n\nThe growth of Graph Convolution Network (GCN) model sizes has revolutionized numerous applications, surpassing human performance in areas such as personal healthcare and financial systems. The deployment of GCNs in the cloud raises privacy concerns due to potential adversarial attacks on client data. To address security concerns, Privacy-Preserving Machine Learning (PPML) using Homomorphic Encryption (HE) secures sensitive client data. However, it introduces substantial computational overhead in practical applications. To tackle those challenges, we present LinGCN, a framework designed to reduce multiplication depth and optimize the performance of HE based GCN inference. LinGCN is structured around three key elements: (1) A differentiable structural linearization algorithm, complemented by a parameterized discrete indicator function, co-trained with model weights to meet the optimization goal. This strategy promotes fine-grained node-level non-linear location selection, resulting in a model with minimized multiplication depth. (2) A compact node-wise polynomial replacement policy with a second-order trainable activation function, steered towards superior convergence by a two-level distillation approach from an all-ReLU based teacher model. (3) an enhanced HE solution that enables finer-grained operator fusion for node-wise activation functions, further reducing multiplication level consumption in HE-based inference. Our experiments on the NTU-XVIEW skeleton joint dataset reveal that LinGCN excels in latency, accuracy, and scalability for homomorphically encrypted inference, outperforming solutions such as CryptoGCN. Remarkably, LinGCN achieves a 14.2\\(\\times\\) latency speedup relative to CryptoGCN, while preserving an inference accuracy of ~75% and notably reducing multiplication depth. Additionally, LinGCN proves scalable for larger models, delivering a substantial 85.78% accuracy with 6371s latency, a 10.47% accuracy improvement over CryptoGCN. The codes are shared on Github1.\n\nFootnote 1: [https://github.com/harveyp123/LinGCN-Neurips23](https://github.com/harveyp123/LinGCN-Neurips23)\n\n## 1 Introduction\n\nGraph learning, a deep learning subset, aids real-time decision-making and impacts diverse applications like computer vision [1], traffic forecasting [2], action recognition [3, 4], recommendation systems [5], and drug discovery [6]. However, the growth of Graph Convolution Network (GCN)model sizes [7, 8] introduces challenges in integrating encryption into graph-based machine learning services. Platforms such as Pinterest's PinSAGE [9] and Alibaba's AliGraph [10] operate on extensive user/item data graphs, increasing processing time. Furthermore, deploying GCN-based services in the cloud raises privacy concerns [11, 12] due to potential adversarial attacks on client data, such as gradient inversion attacks [13, 14]. Given the sensitive information in graph embeddings, it's crucial to implement lightweight, privacy-focused strategies for cloud-based GCN services.\n\nPrivacy-Preserving Machine Learning (PPML) with Homomorphic Encryption (HE) helps protect sensitive graph embeddings, allowing client data encryption before transmission and enabling direct data processing by the cloud server. However, this security comes with substantial computational overhead from HE operations such as rotations, multiplications, and additions. For instance, using the Cheon-Kim-Kim-Song (KKS) [15], a Leveled HE (LHE) [16] scheme, the computational overhead escalates significantly (e.g. \\(\\sim 3\\times\\)) with the total number of successive multiplications, which is directly related to the layers and non-linear implementations in deep GCNs.\n\nThree research trends have emerged to lessen computational overhead and inference latency for HE inference. First, _leveraging the graph sparsity_, as in CryptoGCN [12], reduces multiplication level consumption. Second, _model quantization_, seen in TAPAS [17] and DiNN [18], binarizes activation and weight to 1 bit, leading to a 3%-6.2% accuracy loss on smaller datasets and limited scalability. Third, _nonliear operation level reduction_, as in CryptoNet [19] and Lola [20], substitutes low-degree polynomial operators like square for ReLU in CNN-based models. However, these methods struggle with performance due to absent encryption context of graph features, and the latter two are not suitable for GCN-based models.\n\nTo bridge the research gap, in this paper, we propose LinGCN, an end-to-end policy-guided framework for non-linear reduction and polynomial replacement. This framework is designed to optimize deep Spatial-Temporal Graph Convolution Network (STGCN)-based models for HE-based inference. We conduct extensive experiments on the NTU-XVIEW [21] skeleton joint [22, 23, 24] dataset and compare our approach with CryptoGCN [12]. The pareto frontier comparison of accuracy-latency between LinGCN and CryptoGCN [12] demonstrates that LinGCN surpasses CryptoGCN [12] by over 5% accuracy under the same latency budget, and achieves a 14.2-fold latency reduction under the same accuracy constraint (~75% accuracy).\n\nOur proposed framework, LinGCN, is principally influenced by two observations: the need to conserve levels in the CKKS scheme and the crucial role of synchronized linearization. In the CKKS scheme, deeper networks demand a larger polynomial degree, thereby increasing the latency of homomorphically encrypted operators. Non-linear layer pruning can mitigate this by reducing level consumption. However, unstructured non-linear pruning is insufficient; structural linearization becomes essential for effective level reduction, underscoring synchronized linearization's significance in optimizing CKKS performance. We summarize our contributions as follows:\n\n1. We present a **differentiable structural linearization** algorithm, paired with a parameterized discrete indicator function, which guides the structural polarization process. This methodology is co-trained with model weights until the optimization objective is attained. Our approach facilitates fine-grained node-level selection of non-linear locations, ultimately yielding a model with reduced multiplication levels.\n2. We introduce a node-wise polynomial replacement policy utilizing a second-order trainable polynomial activation function. This process is further steered by an all-ReLU based teacher model employing a two-level distillation approach, thus promoting superior convergence.\n3. We have engineered a corresponding HE solution that facilitates finer-grained operator fusion for node-wise activation functions. This development further mitigates multiplication level consumption during our HE-based inference process.\n\nFigure 1: Frontier of LinGCN vs. CryptoGCN [12]\n\nBackground and Related Work\n\n**CKKS Homomorphic Encryption Scheme.** The CKKS scheme [15], based on the ring learning with errors (RLWE) problem, allows arithmetic operations on encrypted fixed-point numbers. It provides configurable precision via encryption noise as natural error \\(e\\)[25]. We denote the cyclotomic polynomial degree as \\(N\\) and the polynomial coefficient modulus as \\(Q\\), both cryptographic parameters. A ciphertext \\(ct\\) encrypts a message \\(m\\) with noise \\(e\\) as \\(ct=m+e\\) (mod \\(Q\\)). The CKKS scheme's security level [26], measured in bits, suggests \\(2^{128}\\) operations to break a \\(\\lambda=128\\) encryption. CKKS supports operations including ciphertext addition _Add_(\\(ct_{1},ct_{2}\\)), ciphertext multiplication _CMult_(\\(ct_{1},ct_{2}\\)), scalar multiplication _PMult_(\\(ct\\), _pt_), rotation _Rot_(\\(ct\\), _k_) and rescaling _Rescale_(\\(ct\\)). Scalar multiplication multiplies a ciphertext with plaintext, and rotation cyclically shifts the slot vector. For example, \\(Rot(ct,k)\\) transforms \\((v_{0},...,v_{N/2-1})\\) into \\((v_{k},...,v_{N/2-1},v_{0},...,v_{k-1})\\). The level of a ciphertext (\\(L\\)), the number of successive multiplications it can undergo without bootstrapping, is reduced by the \\(Rescale\\) operation after each multiplication. With the level-reduced target network, we do not consider using bootstrapping [27] in this work.\n\n**STGCN for Graph Dataset with Timing Sequence.** The focus of this study revolves around the STGCN, a highly esteemed category of GCNs proposed by [4]. The STGCN model primarily employs two core operators, Spatial graph convolution (GCNConv) and Temporal convolution, designed to extract spatial and temporal information from input graph data, respectively. The Spatial convolution operation, as defined in Equation 1, precisely illustrates the process of extracting spatial information. In Equation 1, the variables \\(\\mathbf{A}\\), \\(\\mathbf{D}\\), \\(\\mathbf{X}_{i}\\), \\(\\mathbf{W}_{i}\\), and \\(\\mathbf{X}_{i,\\text{out}}\\) correspond to the adjacent matrix, degree matrix, input feature, weight parameter, and output feature, respectively [4].\n\n\\[\\mathbf{X}_{i,out}=\\mathbf{D}^{-\\frac{1}{2}}(\\mathbf{A}+\\mathbf{I})\\mathbf{D}^{-\\frac{1}{2}}\\mathbf{X} _{i}\\mathbf{W}_{i} \\tag{1}\\]\n\n**Related Work.** CryptoNets [19] pioneered Privacy-Preserving Machine Learning (PPML) via Homomorphic Encryption (HE) but suffered from extended inference latency for large models. Subsequent studies, like SHE [28] which realizes realize the TFHE-based PPML, and those employing Multi-Party Computation (MPC) solutions[29; 30; 31], reduced latency but faced high communication overhead and prediction latency. Research like TAPAS [17] and XONN [32] compressed models into binary neural network formats, while studies like CryptoNAS [30], Sphynx [33], and SafeNet [34] used Neural Architecture Search (NAS) to find optimal architectures. Delphi [29], SNL [35], and DeepReduce [36] replaced ReLU functions with low order polynomial or linear functions to accelerate MPC-based Private Inference (PI). Nevertheless, these strategies proved suboptimal for homomorphic encrypted inference. Solutions like LoLa [20], CHET [37], and HEAR [38] sought to use ciphertext packing techniques but were not optimized for GCN-based models.\n\n**Threat Model.** In line with other studies [12; 37; 39], we presume a semi-honest cloud-based machine learning service provider. The well-trained STGCN model's weight, including adjacency matrices, is encoded as plaintexts in the HE-inference process. Clients encrypt data via HE and upload it to the cloud for privacy-preserving inference service. The server completes encrypted inference without data decryption or accessing the client's private key. Finally, clients decrypt the returned encrypted inference results from the cloud using their private keys.\n\n## 3 LinGCN Framework\n\n### Motivation\n\nWe present two pivotal observations that serve as the foundation for our LinGCN.\n\n**CKKS Based Encrypted Multiplication.** As described in section 2, the message \\(m\\) is scaled by a factor \\(\\Delta=2^{p}\\) before encoding, resulting in \\(\\Delta m\\). Any multiplicative operations square the ciphertext's scale, including inherent noise \\(e\\). The \\(Rescale\\) operation reduces this scale back to \\(\\Delta\\) by modswitch [15], reducing \\(Q\\) by \\(p\\) bits and lowering the level, as depicted in Figure 2. When \\(Q\\to q_{o}\\) after \\(L\\) multiplications and rescaling, the ciphertext's multiplicative level reaches 0. At this stage, bootstrapping [12] is necessary for further multiplication, unless a higher \\(Q\\) is used to increase the multiplicative level \\(L\\).\n\n**Observation 1: Saving Level is Important in CKKS Scheme.** Within a STGCN model, the graph embedding is encrypted with an initial polynomial degree, denoted as \\(N\\), and a coefficient modulus,represented by \\(Q\\). These factors are determined by the network's multiplicative depth. As the multiplication depth increases, the coefficient modulus must also increase, subsequently leading to a higher polynomial degree \\(N\\). To maintain a security level above a certain threshold (for example, 128-bit) in the face of deeper multiplicative depths, a larger \\(N\\) is necessitated [37; 38; 12]. A larger \\(N\\) inherently results in increased complexity in each homomorphic encryption operation such as Rot and CMult. These operations are integral to the computation of convolution (Conv) and graph convolution (GCNConv). We present a comparative analysis of operator latency with differing polynomial degrees \\(N\\) in Figure 2. Thus, it can be inferred that pruning certain operators to conserve levels not only diminishes the latency of the pruned operators but also reduces other operators' latency.\n\n**Observation 2: Synchronized Linearization Matters.** Our study mainly targets reducing multiplication depth in standard STGCN layers, comprising an initial spatial GCNConv operation, two non-linear operators, and a temporal convolution. Under the CKKS base HE scheme, each operator's input needs synchronized multiplication depths, with any misalignment adjusted to the minimal depth. This is particularly vital for the GCNConv operator, while later node-wise separable operators enable structured non-linear reduction to boost computational efficiency.\n\nUnstructured non-linear reduction strategies, common in MPC setups like AutoReP [40], PASNet [41] SNL [35], RRNet [42], DELPHI [29], and SAFENet [43], prove suboptimal for homomorphic encrypted inference. Our example in Figure 3(b) illustrates node-wise unstructured non-linear reduction in an STGCN layer, leading to unsynchronized remaining multiplication depth budgets for nodes after the layer. This approach is ineffective in reducing levels consumption for homomorphic encrypted inference, emphasizing the necessity for a more structured non-linear reduction approach. To address the limitations of unstructured non-linear reduction, we propose a structured approach, illustrated in Figure 3(c), where we apply an equal level of non-linear reduction to each target node, reducing the overall multiplication depth. In contrast to CryptoGCN's layer-wise pruning method [12], our scheme provides nodes with the flexibility to perform non-linear operations at preferred positions. This fine-grained, structural non-linear reduction may improve the non-linear reduced structure, potentially augmenting the efficiency and effectiveness of HE inference.\n\n### Learning for Structural Linearization\n\n**Problem Formulation.** Our methodology is aimed at the node-wise structural dropping of non-linear operators within a given \\(L\\)-layered STGCN model \\(f_{W}\\), parameterized by \\(W=\\{W_{i}\\}_{i=0}^{2L}\\). This model maps the input \\(X_{0}\\in R^{V\\times C\\times T}\\) to the target \\(Y\\in R^{d}\\). Here, \\(V\\), \\(C\\), and \\(T\\) represent the number of nodes, channels, and frames, respectively. The primary objective of our approach is to eliminate the non-linear operator (designated as \\(\\sigma_{n}\\)) in a structured, node-wise manner. The ultimate aim is to achieve a reduced multiplication level while minimizing any resultant drop in model accuracy. This goal serves to optimize the efficiency of homomorphic encrypted inference for STGCN model.\n\nWe introduce an indicator parameter, denoted as \\(h\\), to mark the node-wise non-linear operator dropping. The following conditions apply: (1) If \\(h_{i,k}=0\\), the non-linear operator \\(\\sigma_{n}\\) is substituted by the linear function \\(f(x)=x\\); (2) If \\(h_{i,k}=1\\), the operator \\(\\sigma_{n}\\) is utilized. Here, \\(h_{i,k}\\) signifies\n\nFigure 3: Unstructured vs. structural linearization. Unstructured one doesn’t lead to effective level reduction.\n\nFigure 2: Top: Rescale decreases the ciphertext level. Bottom: Higher polynomial degree leads to longer HE operator’s latency.\n\nthe \\(k_{th}\\) node of the \\(i_{th}\\) non-linear layer. The proposed node-wise non-linear indicator parameter \\(h_{i,k}\\) facilitates the expression of the \\(i_{th}\\) non-linear layer with partial linearization, as given by \\(X_{i,k}=h_{i,k}\\odot\\sigma_{n}(Z_{(i-1),k})+(1-h_{i,k})\\odot Z_{(i-1),k}\\). Here, \\(Z_{i-1}\\) denotes the input of the \\(i_{th}\\) non-linear layer. Consequently, the issue of structural linearization can be formulated as follows:\n\n\\[\\underset{W,h}{\\text{argmin}}\\mathcal{L}=\\underset{W,h}{\\text{ argmin}}\\ \\ \\ \\mathcal{L}_{acc}(f_{W}(X_{0}),Y)+\\mu\\cdot\\sum_{i=1}^{2L}||h_{i}||_{0}\\] (2) subject to \\[\\forall j,k\\in[1,V],\\ (h_{2i,j}+h_{2i+1,j})=(h_{2i,k}+h_{2i+1,k})\\]\n\nIn our formulation, \\(||h_{r}||0\\) represents the \\(L_{0}\\) norm or the count of remaining non-linear operators. \\(\\mathcal{L}acc\\) signifies cross-entropy loss, while \\(\\mu\\) is the \\(L_{0}\\) norm penalty term in the linearization process. The second term of Eq. 2 poses a challenge due to its non-differentiability, stemming from zero norm regularization and additional constraints on the discrete indicator parameter \\(h_{i}\\). Hence, this issue becomes difficult to handle using traditional gradient-based methods.\n\n**Differentiable Structural Linearization.** To handle the non-differentiable discrete indicator parameter, we introduce a trainable auxiliary parameter, \\(h_{w}\\), to denote its relative importance. However, the transformation from \\(h_{w}\\) to the final indicator \\(h\\) must comply with the structural constraint in Eq.2, ruling out simple threshold checks used in SNL [35]. Structural pruning methods [44, 45], useful for weight or neuron pruning, are ill-suited for our problem due to its exponential complexity arising from 2 non-linear selections of 25 nodes, and its distinct nature from weight or neuron pruning.\n\nTo tackle this challenge, we propose a structural polarization forward process, detailed in Algorithm 1. The algorithm first ranks the relative importance of two auxiliary parameters (\\(h_{w(2i,j)}\\) and \\(h_{w(2i+1,j)}\\)) for every \\(j_{th}\\) node within the \\(i_{th}\\) STGCN layer, obtaining the higher and lower values and their respective indices. We then sum each \\(i_{th}\\) layer's higher auxiliary parameter value into \\(s_{h}\\), and lower auxiliary parameter value into \\(s_{l}\\). Next, we conduct a threshold check of \\(s_{h}\\) and assign the final polarization output of the indicator value into the corresponding locations of higher value indices, applying the same polarization for lower values. The proposed structural polarization enforces the exact constraint as given in Eq. 2, where each STGCN layer maintains a synchronized non-linear count across all nodes. The structural polarization is also capable of capturing the relative importance of the auxiliary parameter within each node and applying the proper polarization to every position, all with a complexity of only \\(O(V)\\). Each STGCN layer retains the flexibility to choose all positions to conduct the non-linear operation, or just select one position for each node to conduct the non-linear operation, or opt not to perform the non-linear operation at all. This selection is based on the importance of their auxiliary parameters.\n\n```\n0:\\(h_{w}\\): auxiliary parameter\n0:\\(h\\): final indicator\n1:for\\(i=0\\) to \\(L\\)do\n2:\\(s_{h},s_{l}=0\\) and \\(ind_{h},ind_{l}\\leftarrow\\emptyset\\)\n3:for\\(j=1\\) to \\(V\\)do\n4:if\\(h_{w(2i,j)}>h_{w(2i+1,j)}\\)then\n5:\\(s_{h}+=\\ h_{w(2i,j)}\\), \\(s_{l}+=\\ h_{w(2i+1,j)}\\)\n6:\\(ind_{h}\\gets(2i,j)\\), \\(ind_{l}\\gets(2i+1,j)\\)\n7:else\n8:\\(s_{h}+=\\ h_{w(2i+1,j)},s_{l}+=\\ h_{w(2i,j)}\\)\n9:\\(ind_{h}\\gets(2i+1,j)\\), \\(ind_{l}\\gets(2i,j)\\)\n10:endif\n11:endfor\n12:\\(h_{ind_{h}}=s_{h}>0\\) and \\(h_{ind_{l}}=s_{l}>0\\)\n13:endfor\n```\n\n**Algorithm 1** Structural Polarization.\n\nDespite its benefits, the structural binarization in Algorithm 1 is discontinuous and non-differentiable. To approximate its gradient, we use coarse gradients [46] via straight through estimation (STE), a good fit for updating \\(h_{w}\\) and reflecting the optimization objective in Eq.2. Among various methods such as Linear STE[47], ReLU and clip ReLU STE [48], we adopt Softplus STE [49] due to its smoothness and recoverability discovered in [49], to update \\(h_{w}\\) as follows:\n\n\\[\\frac{\\partial\\mathcal{L}}{\\partial h_{w(i,k)}}=\\frac{\\partial\\mathcal{L}_{acc }}{\\partial X_{i,k}}(\\sigma_{n}(Z_{i-1})-Z_{i-1})\\frac{\\partial h_{i,k}}{ \\partial h_{w(i,k)}}+\\mu\\frac{\\partial h_{i,k}}{\\partial h_{w,(i,k)}},\\ \\ \\frac{\\partial h_{i,k}}{\\partial h_{w(i,k)}}=Softplus(h_{w(i,k)}) \\tag{3}\\]\n\nThe gradient of the auxiliary parameter, as displayed in Eq. 3, consists of two components: the gradient from the accuracy constraint and the gradient from the linearization ratio constraint. The gradient stemming from the accuracy constraint will be negative if it attempts to counteract the linearization process, thereby achieving a balance with the \\(L_{0}\\) penalty term \\(\\mu\\cdot\\sum_{i=1}^{2L}||h_{i}||_{0}\\). The entirety of the linearization process is recoverable, which allows for the exploration of a near-optimal balance between accuracy and linearization ratio via the back-propagation process.\n\n### Learnable Polynomial Replacement with Teacher Guidance\n\nCryptoNet [19] and DELPHI [29] suggest using quadratic functions \\(y=x^{2}\\) as ReLU replacements for private inference. A second-order polynomial replacement [50]\\(y=x^{2}+x\\) for ReLU was proposed but resulted in decreased performance. CryptoGCN [12] uses a layer-wise trainable polynomial function \\((ax^{2}+bx+c)\\) for the STGCN model, but it suffers from significant accuracy loss and scalability problems. To tackle these issues, we offer a two-tiered strategy: node-wise trainable polynomial replacement and a distillation technique to facilitate the convergence of replacement.\n\n**Node-wise Trainable Polynomial Replacement.** In our HE setup (see Figure 4), each node can independently perform non-linear, convolution, and further non-linear operations without increasing the multiplication depth. We suggest using a node-wise trainable polynomial function as the non-linear function (see Eq. 4). To prevent gradient explosion from quadratic terms, we include a small constant \\(c\\) to adjust the gradient scale of the \\(w_{2}\\) parameter [41].\n\n\\[\\sigma_{n}(x)=c\\cdot w_{2}x^{2}+w_{1}x+b \\tag{4}\\]\n\n**Improved Convergence Through Distillation.** Training a polynomial neural network from scratch often overfits, leading to accuracy loss [50; 12]. To counteract this, we recommend initially training a baseline ReLU model as the teacher, then transferring its knowledge to the polynomial student model to lessen accuracy degradation. The student model's parameters are initialized from the teacher's and polynomial function parameters start at \\((w_{2}=0,w_{1}=1,b=0)\\). To aid knowledge transfer, we use KL-divergence loss [51] and peer-wise normalized feature map difference penalty. The loss function for polynomial replacement is as follows:\n\n\\[\\mathcal{L}_{p}=(1-\\eta)\\mathcal{L}_{CE}(f_{W,s}(X_{0}),Y)+\\eta\\mathcal{L}_{KL }(f_{W,s}(X_{0}),f_{W,t}(X_{0}))+\\frac{\\mathscr{L}}{2}\\sum_{i=1}^{L}MSE(\\frac {X_{i,s}}{||X_{i,s}||_{2}},\\frac{X_{i,t}}{||X_{i,t}||_{2}}) \\tag{5}\\]\n\nIn the above equation, \\(\\eta\\) is employed to balance the significance between the CE loss and the KL divergence loss components, while \\(\\varphi\\) acts as the penalty weight for the peer-wise feature map normalized \\(L_{2}\\) distance term, analogous to the approach in [52].\n\n### Put it Together\n\nThe comprehensive workflow of our proposed LinGCN framework is outlined in Algorithm 2. Initially, we construct the student model, \\(M_{S}\\), utilizing the parameters of the teacher model, \\(M_{T}\\), and subsequently initialize the indicator auxiliary parameter \\(h_{w}\\) for \\(M_{S}\\). Following this, \\(M_{S}\\) undergoes training iterations involving updates for structural linearization. Upon\n\nFigure 4: Overview of single layer STGCN with non-linear reduction and computation.\n\nachieving convergence, the indicator function \\(h\\) is fixed, and the ReLU function in \\(M_{S}\\) is replaced with a polynomial function. Subsequently, we initialize \\(w_{poly}\\) and train the final model using a combination of CE loss and the two-level distillation loss as defined in Eq. 5. The resulting model output will feature structurally pruned node-wise polynomial functions, rendering it primed for the final stage of homomorphic encryption-based inference.\n\n**Further Operator Fusion to Save Level.** Figure 4 shows an example of the single-layer STGCN structure generated by our LinGCN framework. The STGCN utilizes both input nodes information and aggregates them to generate output node features. We adopt the same AMA format [12] for GCNConv computation, as such, we are able to fuse the plaintext weights \\(c\\cdot w_{2}\\) of node-wise polynomial function into GCNConv layer and save one multiplication depth budget. Layer operators involve another temporal convolution layer and non-linear/linear layer, we can still fuse the plaintext weights \\(c\\cdot w_{2}\\) from the second polynomial function into the convolution operator and save the multiplication depth budget. As such, the ReLU-reduced example shown in Figure 4 only consumes 3 multiplication level of ciphertext as opposed to the original 4 multiplication depth budget. For the convolution operator, we adopt the same computation flow from [12; 38].\n\n## 4 Experiment\n\n### Experiment Setting\n\n**HE Parameter Setting.** We performed experiments with two sets of encryption parameters: one without and one with non-linear reduction. We used a scaling factor \\(\\Delta=2^{33}\\) in both cases to ensure comparable accuracy. This scaling consumes 33 bits from the ciphertext modulus \\(Q\\) per level. Without non-linear reduction, a 3-layer STGCN network needs 14 levels, and a 6-layer network needs 27. For a security level of 128-bit, we set \\(Q\\) to 509 (932) and the polynomial degree \\(N\\) to \\(2^{15}(2^{16})\\) for 3 (6) layers STGCN network. In the non-linear reduction case, the total level count ranges from 14 to 9 for 3-layer, and 27 to 16 for 6-layer networks. With the reduced models, we can use smaller Q and \\(N\\) while maintaining 128-bit security. Specifically, we set \\(Q\\) to 410 to 344 and \\(N\\) to \\(2^{14}\\) for the 3-layer network (from 3 non-linear reduced), and \\(Q\\) to 767 to 569 and \\(N\\) to \\(2^{15}\\) for the 6-layer network (starting from 5 non-linear reduced).\n\n**Experiment Dataset.** The **NTU-RGB+D** dataset [21], the largest available with 3D joint annotations for human action recognition, is utilized in our study. With 56,880 action clips in 60 classes, each is annotated with \\((X,Y,Z)\\) coordinates of 25 joints (nodes) per subject. We chose the NTU-cross-View (NTU-XView) benchmark given its representativeness as a human skeleton joint dataset, containing 37,920 training and 18,960 evaluation clips. For a thorough evaluation, 256 frames were used from each clip, resulting in a \\(2\\times 3\\times 256\\times 25\\) input tensor for two individuals each with 3 channels.\n\n**Baseline Model.** Our experiment runs on 8*A100 GPU server, and uses the state-of-the-art STGCN architecture for human action recognition, which combines GCN and CNN. We tested three configurations: STGCN-3-128, STGCN-3-256, and STGCN-6-256, where the first number represents the layer count, and the second the last STGCN layer's channel count. These networks contain a stack of three or six STGCN layers, followed by a global average pooling layer and a fully-connected layer. They were trained for 80 epochs using SGD, with a mini-batch size of 64, a momentum of 0.9, weight decay of \\(10^{-4}\\), and dropout 0.5. The initial learning rate (LR) was 0.1, with a decay factor of 0.1 at the 10th and 50th epochs. The baseline accuracy of models can also be found in Table 1.\n\n**LinGCN Algorithm Setting.** In our LinGCN algorithm, we used the baseline model from Table 1 as the teacher model in Algorithm 2. For structural linearization, we used SGD optimizer with learning rate (LR) 0.01 for both weight and auxiliary parameters. The \\(L_{0}\\) norm penalty \\(\\mu\\) was varied from 0.1to 10 for desired linearized layers, over 25 epochs. Then, ReLU was replaced with a second-order polynomial in the polynomial replacement iterations. We set scaling factor \\(c\\) at 0.01, parameters \\(\\eta\\) and \\(\\varphi\\) at 0.2 and 200 respectively. This process took 90 epochs, with SGD optimizer and initial LR of 0.01, decaying by a factor of 0.1 at the \\(40th\\) and \\(80th\\) epochs.\n\n**Private Inference Setup.** Our experiments used an AMD Ryzen Threadripper PRO 3975WX machine, single threaded. The RNS-variant of the CKKS scheme [53] was employed using Microsoft SEAL version 3.7.2 [54]. We adopted the Adjacency Matrix-Aware (AMA) data formatting from CryptoGCN [12] to pack input graph data and performed GCNConv layer inference via multiplying with sparse matrices \\(A_{i}\\) split from the adjacency matrix \\(A\\). The temporal convolution operation was optimized using the effective convolution algorithm from [38; 12; 55]. The computation for the global average pooling layer and fully-connected layer also follows the SOTA [38; 12].\n\n### Experiment Result and Comparison\n\n**LinGCN Outperforms CryptoGCN [12].** The proposed **LinGCN** algorithm demonstrates superior performance compared to CryptoGCN [12]. The STGCN-3-128 and STGCN-3-256 models, which are the basis of this comparison, share the same backbone as those evaluated in CryptoGCN [12]. Hence, we have undertaken a comprehensive cross-work comparison between the LinGCN framework and CryptoGCN [12]. The latter employs a heuristic algorithm for layer-wise activation layer pruning, which necessitates a sensitivity ranking across all layers. This approach, however, proves to be ineffective, leading to significant accuracy drops as the number of pruned activation layers increases. Comparative results between LinGCN and CryptoGCN [12] are provided in Table 2 and Table 3. In the context of LinGCN, the number of non-linear layers presented in the table represents the effective non-linear layers post-structural linearization. For the STGCN-3-128 and STGCN-3-256 models, LinGCN yielded an accuracy of 77.55% and 80.28% respectively for the baseline 6 non-linear layers model, exceeding CryptoGCN [12] by 3.3% and 4.98%. This outcome signifies the superiority of our proposed teacher-guided node-wise polynomial replacement policy over the naive layer-wise polynomial replacement provided in CryptoGCN [12].\n\nImportantly, the LinGCN model for the 6 non-linear layers employs a more fine-grained node-wise polynomial function, which is fused into the preceding Conv or GCNConv layers. Thus, the encryption level required is lower than that of CryptoGCN [12], resulting in reduced latency. When considering non-linear layer reduction performance, LinGCN experiences only a 1.2 to 1.7 % accuracy decline with a 2 effective non-linear layer reduction, whereas CryptoGCN [12] displays approximately 4% accuracy degradation when pruned for 2 non-linear layers. Remarkably, LinGCN achieves an accuracy improvement of more than 6% over CryptoGCN [12] for models with 4 non-linear layers. Even in models with only 2 non-linear layers, LinGCN exhibits exceptional accuracy\n\n\\begin{table}\n\\begin{tabular}{c|c|c} \\hline Model & Layer-wise number of channel configuration & Accuracy (\\%) \\\\ \\hline STGCN-3-128 & 3-64-128-128 & 80.64 \\\\ \\hline STGCN-3-256 & 3-128-256-256 & 82.80 \\\\ \\hline STGCN-6-256 & 3-64-64-128-128-256-256 & 84.52 \\\\ \\hline \\end{tabular}\n\\end{table}\nTable 1: All ReLU based model architecture and accuracy\n\n\\begin{table}\n\\begin{tabular}{c|c|c|c} \\hline Model & \\multicolumn{3}{c}{STGCN-3-256} \\\\ \\hline \\multirow{2}{*}{Methods} & Non-linear & Test acc & Latency \\\\  & layers & (\\%) & (s) \\\\ \\hline LinGCN & 6 & 77.55 & 1856.95 \\\\ \\hline LinGCN & 5 & 75.48 & 1663.13 \\\\ \\hline LinGCN & 4 & 76.33 & 1458.95 \\\\ \\hline LinGCN & 3 & 74.27 & 850.22 \\\\ \\hline\n**LinGCN** & **2** & **75.16** & **741.55** \\\\ \\hline LinGCN & 1 & 69.61 & 642.06 \\\\ \\hline CryptoGCN & 6 & 74.25 & 4273.89 \\\\ \\hline CryptoGCN & 5 & 73.12 & 1863.95 \\\\ \\hline CryptoGCN & 4 & 70.21 & 1856.36 \\\\ \\hline \\end{tabular}\n\\end{table}\nTable 2: STGCN-3-128 comparison\n\n\\begin{table}\n\\begin{tabular}{c|c|c|c} \\hline Model & \\multicolumn{3}{c}{STGCN-3-256} \\\\ \\hline \\multirow{2}{*}{Methods} & Non-linear & Test acc & Latency \\\\  & layers & (\\%) & (s) \\\\ \\hline LinGCN & 6 & 80.29 & 4632.05 \\\\ \\hline LinGCN & 5 & 79.07 & 4166.12 \\\\ \\hline\n**LinGCN** & **4** & **78.59** & **3699.49** \\\\ \\hline LinGCN & 3 & 76.41 & 2428.88 \\\\ \\hline LinGCN & 2 & 74.74 & 2143.46 \\\\ \\hline LinGCN & 1 & 71.98 & 1873.40 \\\\ \\hline CryptoGCN & 6 & 75.31 & 10580.41 \\\\ \\hline CryptoGCN & 5 & 73.78 & 4850.93 \\\\ \\hline CryptoGCN & 4 & 71.36 & 4831.93 \\\\ \\hline \\end{tabular}\n\\end{table}\nTable 3: STGCN-3-256 comparison (75.16% and 74.74%). While this accuracy aligns with the performance of 6 non-linear layer models in CryptoGCN [12], LinGCN significantly outperforms it in terms of private inference latency, providing more than a five-fold reduction compared to CryptoGCN [12].\n\n**Non-linear Layer Sensitivity.** We conduct a sensitivity analysis on the STGCN-3-256 model to investigate the effect of non-linear layers. The results are displayed in Figure 5, wherein the \\(i\\) layers in the figure represent the remaining effective non-linear layers in the model, and every \\(2i-1\\) and \\(2i\\) exhibit a total number of non-linear nodes summing up to an integer multiple of 25. During the automatic structural linearization process, the gradient propagation is orchestrated to strike a balance between two goals: (i) preserving the feature structural information at the deeper layers, and (ii) mitigating the over-smoothing effect [56] resulting from the deep linearized GCN layers. Consequently, the middle 4th layer's nonlinearity is most important within the STGCN model.\n\n**LinGCN is Scalable.** Beyond the evaluation of the 3-layer model, we also apply our LinGCN to a 6-layer STGCN-6-256 model, which no prior work evaluated, maintaining the same settings to assess the trade-off between accuracy and latency. An intriguing finding is that the teacher-guided full polynomial replacement not only maintains model performance but also surpasses the accuracy of the teacher baseline by 0.95%, achieving an accuracy of 85.47%. This suggests that the STGCN model equipped with a smooth polynomial activation function may enhance model expressivity. Experiments reveal a significant redundancy in the non-linear layers of the STGCN-6-256 model, as it exhibits no accuracy degradation up to the reduction of 8 non-linear layers. When the pruning extends to 10 non-linear layers, the accuracy experiences a slight dip of 3.2% compared to the non-reduction one.\n\n**Pareto Frontier of LinGCN.** Figure 1 compares the Pareto frontier between LinGCN and CryptoGCN [12], showing LinGCN consistently surpassing CryptoGCN by at least 4% in accuracy across all latency ranges. As latency constraints ease (around 6000s), LinGCN's accuracy notably increases, improving by roughly 10% over CryptoGCN. Impressively, LinGCN maintains an accuracy of 75.16% while achieving a private inference latency of 742s, making it 14.2 times faster than CryptoGCN.\n\n### Ablation Studies\n\n**Non-linear Reduction & Replacement Sequence matters.** Our LinGCN algorithm employs a sequence of non-linear reduction and polynomial replacement, proving more efficacious in terms of search space convergence compared to the inverse sequence of polynomial replacement followed by non-linear reduction. The reasoning behind this lies in the possibility that the polynomial replacement process may produce a model optimized for a given architecture, hence subsequent changes to the architecture, such as non-linear reduction, could lead to significant accuracy degradation. Keeping all other parameters constant and merely altering the replacement sequence, we present the accuracy evaluation results of STGCN-3-256 model in Figure 5(a). As depicted in the figure, the sequence of polynomial replacement followed by non-linear reduction incurs an accuracy degradation exceeding 2% compared to the baseline sequence across all effective non-linear layer ranges.\n\n**LinGCN Outperforms Layer-wise Non-linear Reduction.** A significant innovation in our LinGCN framework is the structural linearization process. This process allows nodes to determine their preferred positions for non-linearities within a single STGCN layer, as opposed to enforcing a rigorous constraint to excise the non-linearity across all nodes. This less restrictive approach yields a more detailed replacement outcome compared to the layer-wise non-linear reduction. To demonstrate the impact of this method, we kept all other parameters constant for of STGCN-3-256 model, altering\n\n\\begin{table}\n\\begin{tabular}{c|c|c|c} \\hline \\multirow{2}{*}{Model} & \\multicolumn{3}{c}{STGCN-6-256} \\\\ \\hline \\multirow{2}{*}{Methods} & Non-linear & Test acc & Latency \\\\  & layers & (\\%) & (s) \\\\ \\hline LinGCN & 12 & 85.47 & 21171.80 \\\\ \\hline LinGCN & 11 & 86.24 & 19553.96 \\\\ \\hline LinGCN & 7 & 85.08 & 8186.35 \\\\ \\hline LinGCN & 5 & 83.64 & 7063.51 \\\\ \\hline\n**LinGCN** & **4** & **85.78** & **6371.39** \\\\ \\hline LinGCN & 3 & 84.28 & 5944.81 \\\\ \\hline LinGCN & 2 & 82.27 & 5456.12 \\\\ \\hline LinGCN & 1 & 75.93 & 4927.26 \\\\ \\hline \\end{tabular}\n\\end{table}\nTable 4: LinGCN for STGCN-6-256 model.\n\nFigure 5: STGCN-3-256 structural linearization\n\nonly the structural linearization to layer-wise linearization, and subsequently evaluated the model's accuracy, as shown in Figure 5(b). The results indicate that layer-wise linearization results in a 1.5% decrease in accuracy compared to structural linearization within the range of 2 to 5 effective non-linear layers. This demonstrates the effectiveness of the proposed structural linearization methodology.\n\n**Distillation Hyperparameter Study.** In prior experiments, we kept \\(\\eta\\) and \\(\\varphi\\) at 0.2 and 200 respectively. To assess their influence during distillation, we tested \\(\\eta\\) and \\(\\varphi\\) across a range of values while distilling the STGCN-3-256 model with 6 non-linear layers. We test \\(\\eta\\in[0.1,0.2,0.3,0.4,0.5]\\) and \\(\\varphi\\in[100,200,300,400,500]\\). Notably, we maintained \\(\\varphi=200\\) and \\(\\eta=0.2\\) for the respective \\(\\eta\\) and \\(\\varphi\\) ablations. The study, shown in Figure 5(c) and Figure 5(d), confirms that \\(\\eta=0.2\\) and \\(\\varphi=200\\) yield optimal accuracy, and larger penalties might lower accuracy due to mismatch with the teacher model. This study thus justifies our selection of the \\(\\eta\\) and \\(\\varphi\\).\n\n**LinGCN Generalize to Other Dataset.** Without loss of generality, we extended our evaluation on Flickr [57] dataset, which is a representative node classification dataset widely used in GNN tasks.It consists of 89,250 nodes, 989,006 edges, and 500 feature dimensions. This dataset's task involves categorizing images based on descriptions and common online properties. For the security setting, we assume that node features are user-owned and the graph adjacency list is public. The Flickr dataset has a larger adjacent list but smaller feature dimension compared to the NTU-XVIEW dataset. We utilize three GCN layers with 256 hidden dimensions. Each GCN layer has similar structure as STGCN backbone architecture and has two linear and nonlinear layers. We conduct full-batch GCN training to obtain ReLU-based baseline model accuracies of 0.5492/0.5521 for validation/test dataset. We obtain the accuracy/latency tradeoff detailed in the Table 5. LinGCN framework substantially diminishes the number of effective nonlinear layers, which leads to 1.7 times speedup without much accuracy loss.\n\n## 5 Discussion and Conclusion\n\nOur LinGCN optimizes HE-based GCN inference by reducing multiplication levels through a differentiable structural linearization algorithm and a compact node-wise polynomial replacement policy, both guided by a two-level distillation from an all-ReLU teacher model. Additionally, we improve HE solutions for GCN private inference, enabling finer operator fusion for node-wise activation functions. LinGCN outperforms CryptoGCN by 5% in accuracy and reducing latency by 14.2 times at ~75% accuracy, LinGCN sets a new state-of-the-art performance for private STGCN model inference. In the future, we will expand the proposed algorithm on other neural networks.\n\n## Acknowledgement\n\nThis work was in part supported by the NSF Grants CNS-2348733, 2247892, 2247893, OAC-2319962, Semiconductor Research Corporation (SRC) Artificial Intelligence Hardware program, and UConn CACC center. Any opinions, findings and conclusions, or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the funding agencies.\n\n\\begin{table}\n\\begin{tabular}{c|c|c} \\hline Num. of nonlinear & Accuracy & Latency \\\\ layers in GCN layers & (val/test, \\%) & (s) \\\\ \\hline\n6 & 0.52810.5275 & 4290.93 \\\\ \\hline\n2 & 0.5247/0.5266 & 2740.94 \\\\ \\hline\n1 & 0.5269/0.5283 & 2525.80 \\\\ \\hline \\end{tabular}\n\\end{table}\nTable 5: LinGCN for Flickr dataset.\n\nFigure 6: Ablation study for (a) Replacement sequence, (b) node-wise vs. layer-wise linearization. (c) \\(\\eta\\): KL divergence distillation parameter (d) \\(\\varphi\\): feature map distance distillation parameter.\n\n## References\n\n* [1] Weijing Shi and Raj Rajkumar. Point-gnn: Graph neural network for 3d object detection in a point cloud. In _CVPR_, pages 1711-1719, 2020.\n* [2] Weiwei Jiang and Jiayun Luo. Graph neural network for traffic forecasting: A survey. _Expert Systems with Applications_, page 117921, 2022.\n* [3] Chenyang Si, Ya Jing, Wei Wang, Liang Wang, and Tieniu Tan. Skeleton-based action recognition with spatial reasoning and temporal stack learning. In _Proceedings of the European conference on computer vision (ECCV)_, pages 103-118, 2018.\n* [4] Sijie Yan, Yuanjun Xiong, and Dahua Lin. Spatial temporal graph convolutional networks for skeleton-based action recognition. In _Thirty-second AAAI conference on artificial intelligence_, 2018.\n* [5] Shiwen Wu, Fei Sun, Wentao Zhang, Xu Xie, and Bin Cui. Graph neural networks in recommender systems: a survey. _ACM Computing Surveys (CSUR)_, 2020.\n* [6] Pietro Bongini et al. Molecular generative graph neural networks for drug discovery. _Neurocomputing_, 450:242-252, 2021.\n* [7] Anuroop Sriram et al. Towards training billion parameter graph neural networks for atomic simulations. _ArXiv_, abs/2203.09697, 2022.\n* [8] Daniel Manu et al. Co-exploration of graph neural network and network-on-chip design using automl. In _Proceedings of the 2021 on Great Lakes Symposium on VLSI_, pages 175-180, 2021.\n* [9] Rex Ying et al. Graph convolutional neural networks for web-scale recommender systems. _KDD '18_, 2018.\n* [10] Rong Zhu et al. Aligraph: A comprehensive graph neural network platform. _KDD '19_, 2019.\n* [11] Ling Liang, Jilan Lin, Zheng Qu, Ishtiyaque Ahmad, Fengbin Tu, Trinabh Gupta, Yufei Ding, and Yuan Xie. Spg: Structure-private graph database via squeezepir. _Proceedings of the VLDB Endowment_, 16(7):1615-1628, 2023.\n* [12] Ran Ran, Wei Wang, Quan Gang, Jieming Yin, Nuo Xu, and Wujie Wen. Cryptogcn: Fast and scalable homomorphically encrypted graph convolutional network inference. _Advances in Neural Information Processing Systems_, 35:37676-37689, 2022.\n* [13] Le Trieu Phong, Yoshinori Aono, Takuya Hayashi, Lihua Wang, and Shiho Moriai. Privacy-preserving deep learning: Revisited and enhanced. In _Applications and Techniques in Information Security: 8th International Conference, ATIS 2017, Auckland, New Zealand, July 6-7, 2017, Proceedings_, pages 100-110. Springer, 2017.\n* [14] Ligeng Zhu, Zhijian Liu, and Song Han. Deep leakage from gradients. _Advances in neural information processing systems_, 32, 2019.\n* [15] Jung Hee Cheon, Andrey Kim, Miran Kim, and Yongsoo Song. Homomorphic encryption for arithmetic of approximate numbers. In _International Conference on the Theory and Application of Cryptology and Information Security_, pages 409-437. Springer, 2017.\n* [16] Zvika Brakerski, Craig Gentry, and Vinod Vaikuntanathan. (leveled) fully homomorphic encryption without bootstrapping. _ACM Transactions on Computation Theory (TOCT)_, 6(3):1-36, 2014.\n* [17] Amartya Sanyal, Matt Kusner, Adria Gascon, and Varun Kanade. Tapas: Tricks to accelerate (encrypted) prediction as a service. In _International Conference on Machine Learning_, pages 4490-4499. PMLR, 2018.\n* [18] Florian Bourse, Michele Minelli, Matthias Minihold, and Pascal Paillier. Fast homomorphic evaluation of deep discretized neural networks. In _Advances in Cryptology-CRYPTO 2018: 38th Annual International Cryptology Conference, Santa Barbara, CA, USA, August 19-23, 2018, Proceedings, Part III 38_, pages 483-512. Springer, 2018.\n* [19] Ran Gilad-Bachrach, Nathan Dowlin, Kim Laine, Kristin Lauter, Michael Naehrig, and John Wernsing. Cryptonets: Applying neural networks to encrypted data with high throughput and accuracy. In _International conference on machine learning_, pages 201-210. PMLR, 2016.\n* [20] Alon Brutzkus, Ran Gilad-Bachrach, and Oren Elisha. Low latency privacy preserving inference. In _International Conference on Machine Learning_, pages 812-821. PMLR, 2019.\n\n* [21] Amir Shahroudy, Jun Liu, Tian-Tsong Ng, and Gang Wang. Ntu rgb+d: A large scale dataset for 3d human activity analysis. In _Proceedings of the IEEE conference on computer vision and pattern recognition_, pages 1010-1019, 2016.\n* [22] Ke Sun, Bin Xiao, Dong Liu, and Jingdong Wang. Deep high-resolution representation learning for human pose estimation. In _Proceedings of the IEEE/CVF conference on computer vision and pattern recognition_, pages 5693-5703, 2019.\n* [23] Kaiming He, Georgia Gkioxari, Piotr Dollar, and Ross Girshick. Mask r-cnn. In _Proceedings of the IEEE international conference on computer vision_, pages 2961-2969, 2017.\n* [24] Zhe Cao, Tomas Simon, Shih-En Wei, and Yaser Sheikh. Realtime multi-person 2d pose estimation using part affinity fields. In _Proceedings of the IEEE conference on computer vision and pattern recognition_, pages 7291-7299, 2017.\n* [25] Kim Laine. Simple encrypted arithmetic library (seal) manual, 2017.\n* [26] Melissa Chase, Hao Chen, Jintai Ding, Shafi Goldwasser, Sergey Gorbunov, Jeffrey Hoffstein, Kristin Lauter, Satya Lokam, Dustin Moody, Travis Morrison, et al. Security of homomorphic encryption. _HomomorphicEncryption. org, Redmond WA, Tech. Rep_, 2017.\n* [27] Craig Gentry. _A fully homomorphic encryption scheme_. Stanford university, 2009.\n* [28] Ilaria Chillotti, Nicolas Gama, Mariya Georgieva, and Malika Izabachene. Tfhe: fast fully homomorphic encryption over the torus. _Journal of Cryptology_, 33(1):34-91, 2020.\n* [29] Pratyush Mishra, Ryan Lehmkuhl, Akshayaram Srinivasan, Wenting Zheng, and Raluca Ada Popa. Delphi: A cryptographic inference service for neural networks. In _29th USENIX Security Symposium (USENIX Security 20)_, pages 2505-2522, 2020.\n* [30] Zahra Ghodsi, Akshaj Kumar Veldanda, Brandon Reagen, and Siddharth Garg. Cryptonas: Private inference on a relu budget. _Advances in Neural Information Processing Systems_, 33:16961-16971, 2020.\n* [31] Brian Knott, Shobha Venkataraman, Awni Hannun, Shubho Sengupta, Mark Ibrahim, and Laurens van der Maaten. Crypten: Secure multi-party computation meets machine learning. _Advances in Neural Information Processing Systems_, 34:4961-4973, 2021.\n* [32] M Sadegh Riazi, Mohammad Samragh, Hao Chen, Kim Laine, Kristin Lauter, and Farinaz Koushanfar. {XONN}:{XNOR-based} oblivious deep neural network inference. In _28th USENIX Security Symposium (USENIX Security 19)_, pages 1501-1518, 2019.\n* [33] Minsu Cho, Zahra Ghodsi, Brandon Reagen, Siddharth Garg, and Chinmay Hegde. Sphynx: Relu-efficient network design for private inference. _arXiv preprint arXiv:2106.11755_, 2021.\n* [34] Qian Lou, Yilin Shen, Hongxia Jin, and Lei Jiang. Safenet: A secure, accurate and fast neural network inference. In _International Conference on Learning Representations_, 2020.\n* [35] Minsu Cho, Ameya Joshi, Brandon Reagen, Siddharth Garg, and Chinmay Hegde. Selective network linearization for efficient private inference. In _International Conference on Machine Learning_, pages 3947-3961. PMLR, 2022.\n* [36] Nandan Kumar Jha, Zahra Ghodsi, Siddharth Garg, and Brandon Reagen. Deepreduce: Relu reduction for fast private inference. In _International Conference on Machine Learning_, pages 4839-4849. PMLR, 2021.\n* [37] Roshan Dathathri, Olli Saarikivi, Hao Chen, Kim Laine, Kristin Lauter, Saeed Maleki, Madanlal Musuvathi, and Todd Mytkowicz. Chet: an optimizing compiler for fully-homomorphic neural-network inferencing. In _Proceedings of the 40th ACM SIGPLAN Conference on Programming Language Design and Implementation_, pages 142-156, 2019.\n* [38] Miran Kim, Xiaoqian Jiang, Kristin Lauter, Elkhan Ismayilzada, and Shayan Shams. Secure human action recognition by encrypted neural network inference. _Nature communications_, 13(1):1-13, 2022.\n* [39] Miran Kim, Xiaoqian Jiang, Kristin Lauter, Elkhan Ismayilzada, and Shayan Shams. Hear: Human action recognition via neural networks on homomorphically encrypted data. _arXiv preprint arXiv:2104.09164_, 2021.\n* [40] Hongwu Peng, Shaoyi Huang, Tong Zhou, Yukui Luo, Chenghong Wang, Zigeng Wang, Jiahui Zhao, Xi Xie, Ang Li, Tony Geng, et al. Autorep: Automatic relu replacement for fast private network inference. _arXiv preprint arXiv:2308.10134_, 2023.\n\n* [41] Hongwu Peng, Shanglin Zhou, Yukui Luo, Nuo Xu, Shijin Duan, Ran Ran, Jiahui Zhao, Chenghong Wang, Tong Geng, Wujie Wen, et al. Pasnet: Polynomial architecture search framework for two-party computation-based secure neural network deployment. In _2023 60th ACM/IEEE Design Automation Conference (DAC)_, pages 1-6. IEEE, 2023.\n* [42] Hongwu Peng, Shanglin Zhou, Yukui Luo, Nuo Xu, Shijin Duan, Ran Ran, Jiahui Zhao, Shaoyi Huang, Xi Xie, Chenghong Wang, et al. Rrnet: Towards relu-reduced neural network for two-party computation based private inference. _arXiv preprint arXiv:2302.02292_, 2023.\n* [43] Qian Lou, Yilin Shen, Hongxia Jin, and Lei Jiang. Safenet: A secure, accurate and fast neural network inference. In _International Conference on Learning Representations_, 2021.\n* [44] Mingbao Lin, Yuxin Zhang, Yuchao Li, Bohong Chen, Fei Chao, Mengdi Wang, Shen Li, Yonghong Tian, and Rongrong Ji. 1xn pattern for pruning convolutional neural networks. _IEEE Transactions on Pattern Analysis and Machine Intelligence_, 2022.\n* [45] Tao Zhuang, Zhixuan Zhang, Yuheng Huang, Xiaoyi Zeng, Kai Shuang, and Xiang Li. Neuron-level structured pruning using polarization regularizer. _Advances in neural information processing systems_, 33:9865-9877, 2020.\n* [46] Itay Hubara, Matthieu Courbariaux, Daniel Soudry, Ran El-Yaniv, and Yoshua Bengio. Binarized neural networks. _Advances in neural information processing systems_, 29, 2016.\n* [47] Suraj Srinivas, Akshayvarun Subramanya, and R Venkatesh Babu. Training sparse neural networks. In _Proceedings of the IEEE conference on computer vision and pattern recognition workshops_, pages 138-145, 2017.\n* [48] Penghang Yin, Jiancheng Lyu, Shuai Zhang, Stanley Osher, Yingyong Qi, and Jack Xin. Understanding straight-through estimator in training activation quantized neural nets. _arXiv preprint arXiv:1903.05662_, 2019.\n* [49] Xia Xiao, Zigeng Wang, and Sanguthevar Rajasekaran. Autoprune: Automatic network pruning by regularizing auxiliary parameters. _Advances in neural information processing systems_, 32, 2019.\n* [50] Ramy E Ali, Jinhyun So, and A Salman Avestimehr. On polynomial approximations for privacy-preserving and verifiable relu networks. _arXiv preprint arXiv:2011.05530_, 2020.\n* [51] Geoffrey Hinton, Oriol Vinyals, and Jeff Dean. Distilling the knowledge in a neural network. _arXiv preprint arXiv:1503.02531_, 2015.\n* [52] Sergey Zagoruyko and Nikos Komodakis. Paying more attention to attention: Improving the performance of convolutional neural networks via attention transfer. In _International Conference on Learning Representations_, 2017.\n* [53] Jung Hee Cheon, Kyoohyung Han, Andrey Kim, Miran Kim, and Yongsoo Song. A full rms variant of approximate homomorphic encryption. In _International Conference on Selected Areas in Cryptography_, pages 347-368. Springer, 2018.\n* [54] Microsoft SEAL (release 3.7). [https://github.com/Microsoft/SEAL](https://github.com/Microsoft/SEAL), September 2021. Microsoft Research, Redmond, WA.\n* [55] Eunsang Lee, Joon-Woo Lee, Junghyun Lee, Young-Sik Kim, Yongjune Kim, Jong-Seon No, and Woosuk Choi. Low-complexity deep convolutional neural networks on fully homomorphic encryption using multiplexed parallel convolutions. In _International Conference on Machine Learning_, pages 12403-12422. PMLR, 2022.\n* [56] Qimai Li, Zhichao Han, and Xiao-Ming Wu. Deeper insights into graph convolutional networks for semi-supervised learning. In _Proceedings of the AAAI conference on artificial intelligence_, volume 32, 2018.\n* [57] Xiao Huang, Jundong Li, and Xia Hu. Label informed attributed network embedding. In _Proceedings of the tenth ACM international conference on web search and data mining_, pages 731-739, 2017.\n\nAppendix\n\n### HE encoding with AMA format\n\nPrior to encoding input data into polynomials, it is necessary to map the four-dimensional tensor \\(X\\in R^{B\\times C\\times T\\times J}\\) to a one-dimensional vector in \\(R^{N/2}\\) using the AMA format, as proposed in [12]. This transformation allows for more efficient execution of STGCN forward-computation in the HE domain. Below, we present the definition of the \\(Vec\\) function employed to map tensor \\(X\\) to a vector in \\(R^{N/2}\\):\n\n\\[Vec(X)=y_{j} =(y_{0,j},\\ldots,y_{i,j},\\ldots,y_{N/2,j})\\in R^{N/2} \\tag{6}\\] \\[\\text{s.t. }y_{i,j} =X_{((i\\text{\\,mod\\,}T)\\%B)\\times(i\\text{\\,mod\\,}B\\cdot T)\\times (i\\text{\\,\\%\\,}T)\\times j}\\] \\[j \\in J\\]\n\nFollowing the mapping process, the vectors \\(y_{j}\\) are encoded into polynomials with degree \\(N\\) and subsequently encrypted into ciphertext \\(ct_{j}\\), as detailed in [53]. In this study, when N is set to \\(2^{16}\\), all tensors, including intermediate tensors, can be encrypted and packed into 25 ciphertexts, which corresponds to the number of nodes. For cases where \\(N=2^{15}\\) (\\(2^{14}\\)) the number of ciphertexts is 50(100). By selecting an appropriate value for N, the encryption and packing processes can be optimized to maintain performance and efficiency.\n\n### HE Setting Details\n\nIn Table 6, we furnish comprehensive details regarding the HE inference parameters. Specifically, i-STGCN-3 denotes a 3-layer STGCN model with i effective non-linear layers, while i-STGCN-6 signifies a 6-layer STGCN model with i effective non-linear layers. In this context, N represents the polynomial degree, and Q corresponds to the coefficient modulus.\n\nTo guarantee computation precision utilizing a one-time rescale operation, we assign the scale factor \\(p\\) for both ciphertext and plaintext to \\(2^{33}\\). This allocation results in a reduction of the current Q of ciphertext by p bits. This setup ensures that the overall performance and accuracy conform to the desired criteria while capitalizing on the security and resilience advantages conferred by HE.\n\n### HE inference on GCNConv and Temporal-Conv Layer\n\nUpon obtaining the AMA-packed ciphertexts \\(ct_{j}\\), the adjacency matrix multiplication \\(A\\cdot X\\) can be decomposed into a series of plaintext multiplications, _PMult_, in the HE domain. This decomposition accelerates HE-inference without necessitating rotations. Furthermore, the subsequent temporal convolution is performed solely on the temporal dimension \\(T\\), utilizing \\(1\\times 9\\) kernels.\n\n\\begin{table}\n\\begin{tabular}{c|c c c c|c} \\hline \\multirow{2}{*}{Model} & \\multicolumn{3}{c|}{Encryption Parameters} & \\multicolumn{1}{c}{Mult} \\\\  & N & Q & p & \\(q_{0}\\) & Level \\\\ \\hline\n6-STGCN-3 & 32768 & 509 & 33 & 47 & 14 \\\\\n5-STGCN-3 & 32768 & 476 & 33 & 47 & 13 \\\\\n4-STGCN-3 & 32768 & 443 & 33 & 47 & 12 \\\\\n3-STGCN-3 & 16384 & 410 & 33 & 47 & 11 \\\\\n2-STGCN-3 & 16384 & 377 & 33 & 47 & 10 \\\\\n1-STGCN-3 & 16384 & 344 & 33 & 47 & 9 \\\\\n12-STGCN-6 & 65536 & 932 & 33 & 41 & 27 \\\\\n11-STGCN-6 & 65536 & 899 & 33 & 41 & 26 \\\\\n7-STGCN-6 & 32768 & 767 & 33 & 41 & 22 \\\\\n5-STGCN-6 & 32768 & 701 & 33 & 41 & 20 \\\\\n4-STGCN-6 & 32768 & 668 & 33 & 41 & 19 \\\\\n3-STGCN-6 & 32768 & 635 & 33 & 41 & 18 \\\\\n2-STGCN-6 & 32768 & 602 & 33 & 41 & 17 \\\\\n1-STGCN-6 & 32768 & 569 & 33 & 41 & 16 \\\\ \\hline \\end{tabular}\n\\end{table}\nTable 6: HE parameter settings in detail.\n\n\\[ct_{k}^{\\prime}=A\\cdot X=\\sum_{i=1}^{m}ct_{k}A_{i}=\\sum_{i=1}^{m}\\sum_{k=1}^{J}PMult (ct_{i_{k}},a_{i_{k}k}) \\tag{7}\\]\n\nThe AMA-packed ciphertexts allow for natural temporal convolution by single-node ciphertext \\(ct_{j}\\), facilitating independent computation. This approach results in a ReLU-reduction design through structural pruning of ReLUs. The primary constraint to consider in this context is ensuring that the level consumption of each ciphertext remains equal prior to the GCNConv layer (node aggregation).\n\n### Further Detail of Operator Fusion\n\nDuring the HE-inference process, employing weight fusion conserves the multiplicative depth, consequently reducing the ciphertext level budget. For instance, batch normalization, defined by an affine transformation \\(a^{\\prime}x+b^{\\prime}\\), and a polynomial activation function, defined by \\((ax+b)^{2}+c\\), can be readily fused into the corresponding temporal convolution layer \\(wx+b^{\\prime\\prime}\\) with the function \\(w(a(a^{\\prime}x+b^{\\prime})+b)+b^{\\prime\\prime}=(w\\cdot a\\cdot a^{\\prime})x+ab ^{\\prime}+wb+b^{\\prime\\prime}\\). As a result, three consecutive multiplications are consolidated into a single multiplication (pre-computing \\(w\\cdot a\\cdot a^{\\prime}\\)), thereby reducing the level consumption of ciphertext from 4 to 2 (\\(1\\times 9\\) convolution, batch normalization, and polynomial activation).\n\nAnalogous to the temporal-convolutional layer, the same fusion strategies can be applied to the polynomial activation and batch normalization of the GCNConv layer. Utilizing AMA-packed ciphertexts, the node aggregation in GCNConv is translated as depicted in Equation 7, where each ciphertext carries out scalar multiplication with the plaintext of matrix elements \\(a_{i_{k},k}\\). Consequently, these plaintext matrix elements \\(a_{i_{k},k}\\) are fused with the primary \\(1\\times 1\\) convolutional kernels to conserve the multiplicative level, reducing the total level consumption of the GCNConv layer from 4 to 2 (\\(1\\times 1\\) convolution, adjacency matrix multiplication, batch normalization, and polynomial activation).\n\n### Operator latency breakdown\n\nTable 7 presents a comprehensive operator latency breakdown encompassing Rot, PMult, Add, and CMult operations. The designation i-STGCN-3-128 refers to an STGCN-3-128 model with i residual non-linear layers. As indicated in the table, the non-linear reduction contributes to a significant reduction in latency. By leveraging a smaller polynomial degree N, the overall latency experiences substantial improvement.\n\n### More Training Details and Insight\n\nIn this section, we present the training curves for the STGCN-3-256 model, which employs 6 to 1 effective second-order polynomial (non-linear) layers. Figures 7(a) through Figure 7(f) depict the training curve progression. During the training process, we utilized mixed-precision training for the polynomial model, which led to occasional instability in some iterations, as evidenced by spikes in the loss values. Nevertheless, the training process demonstrated rapid recovery following such loss spikes.\n\nAs demonstrated in training curve, a smaller number of second-order polynomial (non-linear) layers contribute to a more stable training process and facilitate smoother convergence. This finding\n\n\\begin{table}\n\\begin{tabular}{c|c c c c|c|c} \\hline \\hline \\multirow{2}{*}{Model} & \\multicolumn{3}{c|}{HE Operators latency (s)} & \\multicolumn{2}{c|}{Total Latency} & Speedup \\\\  & Rot & PMult & Add & CMult & (s) & (\\(\\times\\)) \\\\ \\hline\n6-STGCN-3-128 & 1336.25 & 378.25 & 99.65 & 37.45 & 1851.60 & - \\\\\n2-STGCN-3-128 & 392.21 & 266.13 & 68.90 & 14.31 & 741.55 & 2.50 \\\\\n6-STGCN-3-256 & 2641.09 & 1508.19 & 397.17 & 74.90 & 4621.36 & - \\\\\n2-STGCN-3-256 & 777.68 & 1062.21 & 274.96 & 28.63 & 2143.47 & 2.16 \\\\\n12-STGCN-6-256 & 18955.09 & 1545.09 & 396.23 & 275.39 & 21171.80 & - \\\\\n2-STGCN-6-256 & 4090.08 & 1006.79 & 244.19 & 115.05 & 5456.12 & 3.88 \\\\ \\hline \\hline \\end{tabular}\n\\end{table}\nTable 7: Comparison of latency breakdown between the non-reduced model with optimized model.\n\nexplains the enhanced performance of the STGCN-6-256 model, which features a reduced number of non-linear layers, as compared to the full-polynomial model baseline.\n\nTo substantiate our hypothesis, we plot the polynomial replacement training curve for the STGCN-6-256 model in Figure 8. The training curves for 12 effective non-linear layers (12-STGCN-6-256), four effective non-linear layers (4-STGCN-6-256), and two effective non-linear layers (2-STGCN-6-256) are presented. As the number of non-linear layers increases, the model achieves greater expressivity; however, the polynomial replacement process becomes increasingly unstable. Consequently, for the STGCN-6-256 model with only 4 non-linear layers, a more stable replacement process facilitates better convergence, ultimately leading to improved accuracy performance.\n\nFigure 8: Polynomial replacement training curves for (a) 12-STGCN-6-256 (b) 4-STGCN-6-256 (c) 2-STGCN-6-256\n\nFigure 7: Polynomial replacement training curves for (a) 6-STGCN-3-128 (b) 5-STGCN-3-128 (c) 4-STGCN-3-128 (d) 3-STGCN-3-128 (e) 2-STGCN-3-128 (f) 1-STGCN-3-128", "response": "## Review\n\n### Summary\n\nThis study introduces LinGCN, a framework designed to enhance the efficiency of private inference using homomorphic encryption (HE) in spatiotemporal graph convolutional networks. LinGCN employs a differentiable structural linearization algorithm alongside a compact node-wise polynomial replacement strategy, guided by a two-level distillation from an all-ReLU teacher model. The results indicate significant improvements in both latency and accuracy over existing methods such as CryptoGCN. While the paper provides a clear and structured presentation of its methodology and results, concerns regarding the novelty of the approach and its generalizability across datasets remain.\n\n### Strengths\n\n- The idea of optimizing ReLUs with a fine-grained and structured approach is promising.\n- The paper is well-structured and easy to understand, with detailed explanations.\n- Experiments demonstrate clear improvements in latency and accuracy compared to prior work.\n- The motivation behind the framework is well explained and illustrated.\n- Comprehensive ablation experiments support the claims of improved performance.\n\n### Weaknesses\n\n- Limited novelty compared to existing works like CryptoGCN, as similar concepts have been previously explored.\n- Only one dataset is evaluated, raising questions about generalizability to other datasets.\n- The clarity of the results regarding LinGCN's advantages over CryptoGCN is insufficient, needing deeper analysis.\n- The impact of the proposed modifications on the security of the HE-based inference process is not discussed.\n- Analytical results on computational cost and utility loss are lacking.\n\n### Questions\n\n- What implications do the findings have for feature learning within CNNs compared to GCNs?\n- Is the accuracy improvement over CryptoGCN solely due to the two-level knowledge distillation?\n- How do the proposed methods differ from established techniques in CNNs?\n- What can be inferred from the evaluation regarding the generalizability of the proposed techniques?\n- What security implications arise from the modified designs in the HE-based GCN inference process?\n\n### Soundness\n\n**Score:** 3\n\n**Description:** 3 = good: The methodology is sound and well-supported by experimental results, although some aspects lack clarity and depth.\n\n### Presentation\n\n**Score:** 3\n\n**Description:** 3 = good: The paper is well-written and organized, though some parts could benefit from more detailed explanations.\n\n### Contribution\n\n**Score:** 3\n\n**Description:** 3 = good: While the contribution is notable, the degree of novelty is modest due to existing related works.\n\n### Rating\n\n**Score:** 6\n\n**Description:** 6 = Weak Accept: The paper is technically solid and has moderate-to-high impact potential, despite some concerns about novelty and generalizability.\n\n### Paper Decision\n\n**Decision:** Accept (poster)\n\n**Reasons:** The paper presents a well-founded method that addresses a relevant problem in the field of private inference for GCNs. While there are concerns regarding novelty and the depth of analysis, the overall contribution and improvements in performance are significant. The authors have effectively addressed reviewer comments, and the paper is well-written and structured, making it suitable for acceptance.\n"}
{"query": "You are a highly experienced, conscientious, and fair academic reviewer, please help me review this paper. The review should be organized into nine sections: \n1. Summary: A summary of the paper in 100-150 words.\n2. Strengths/Weaknesses/Questions: The Strengths/Weaknesses/Questions of paper, which should be listed in bullet points, with each point supported by specific examples from the article where possible.\n3. Soundness/Contribution/Presentation: Rate the paper's Soundness/Contribution/Presentation, and match this score to the corresponding description from the list below and provide the result. The possible scores and their descriptions are: \n    1 poor\n    2 fair\n    3 good\n    4 excellent\n4. Rating: Give this paper an appropriate rating, match this rating to the corresponding description from the list below and provide the result. The possible Ratings and their descriptions are: \n    1 strong reject\n    2 reject, significant issues present\n    3 reject, not good enough\n    4 possibly reject, but has redeeming facets\n    5 marginally below the acceptance threshold\n    6 marginally above the acceptance threshold\n    7 accept, but needs minor improvements \n    8 accept, good paper\n    9 strong accept, excellent work\n    10 strong accept, should be highlighted at the conference   \n5. Paper Decision: It must include the Decision itself(Accept or Reject) and the reasons for this decision, based on the criteria of originality, methodological soundness, significance of results, and clarity and logic of presentation.\n\nHere is the template for a review format, you must follow this format to output your review result:\n**Summary:**\nSummary content\n\n**Strengths:**\n- Strength 1\n- Strength 2\n- ...\n\n**Weaknesses:**\n- Weakness 1\n- Weakness 2\n- ...\n\n**Questions:**\n- Question 1\n- Question 2\n- ...\n\n**Soundness:**\nSoundness result\n\n**Presentation:**\nPresentation result\n\n**Contribution:**\nContribution result\n\n**Rating:**\nRating result\n\n**Paper Decision:**\n- Decision: Accept/Reject\n- Reasons: reasons content\n\n\nPlease ensure your feedback is objective and constructive. The paper is as follows:\n\n# Module-wise Training of Neural Networks via the Minimizing Movement Scheme\n\nSkander Karkar\n\nCriteo, Sorbonne Universite\n\n&Ibrahim Ayed\n\nSorbonne Universite, Thales\n\nEmmanuel de Bezenac\n\nETH Zurich\n\n&Patrick Gallinari\n\nCriteo, Sorbonne Universite\n\n###### Abstract\n\nGreedy layer-wise or module-wise training of neural networks is compelling in constrained and on-device settings where memory is limited, as it circumvents a number of problems of end-to-end back-propagation. However, it suffers from a stagnation problem, whereby early layers overfit and deeper layers stop increasing the test accuracy after a certain depth. We propose to solve this issue by introducing a module-wise regularization inspired by the minimizing movement scheme for gradient flows in distribution space. We call the method TRGL for Transport Regularized Greedy Learning and study it theoretically, proving that it leads to greedy modules that are regular and that progressively solve the task. Experimentally, we show improved accuracy of module-wise training of various architectures such as ResNets, Transformers and VGG, when our regularization is added, superior to that of other module-wise training methods and often to end-to-end training, with as much as \\(60\\%\\) less memory usage.\n\n## 1 Introduction\n\nEnd-to-end backpropagation is the standard training method of neural networks. However, it requires storing the whole model and computational graph during training, which requires large memory consumption. It also prohibits training the layers in parallel. Dividing the network into modules, a module being made up of one or more layers, accompanied by auxiliary classifiers, and greedily solving module-wise optimization problems sequentially (i.e. one after the other fully) or in parallel (i.e. at the same time batch-wise), consumes much less memory than end-to-end training as it does not need to store as many activations, and when done sequentially, only requires loading and training one module (so possibly one layer) at a time. Module-wise training has therefore been used in constrained settings in which end-to-end training can be impossible such as training on mobile devices [58, 57] and dealing with very large whole slide images [65]. When combined with batch buffers, parallel module-wise training also allows for parallel training of the modules [8]. Despite its simplicity, module-wise training has been recently shown to scale well [8, 47, 60, 45], outperforming more complicated alternatives to end-to-end training such as synthetic [33, 14] and delayed [32, 31] gradients, while having superior memory savings.\n\nIn a classification task, module-wise training splits the network into successive modules, a module being made up of one or more layers. Each module takes as input the output of the previous module, and each module has an auxiliary classifier so that a local loss can be computed, with backpropagation happening only inside the modules and not between them (see Figure 1 below).\n\nThe main drawback of module-wise training is the well-documented _stagnation problem_ observed in [43, 7, 60, 47], whereby early modules overfit and learn more discriminative features than end-to-endtraining, destroying task-relevant information, and deeper modules don't improve the test accuracy significantly, or even degrade it, which limits the deployment of module-wise training. We further highlight this phenomenon in Figures 2 and 3 in Section 4.4. To tackle this issue, InfoPro [60] propose to maximize the mutual information that each module keeps with the input, in addition to minimizing the loss. [7] make the auxiliary classifier deeper and Sedona [47] make the first module deeper. These last two methods lack a theoretical grounding, while InfoPro requires a second auxiliary network for each module besides the classifier. We propose a different perspective, leveraging the analogy between residual connections and the Euler scheme for ODEs [61]. To preserve input information, we minimize the kinetic energy of the modules along with the training loss. Intuitively, this forces the modules to change their input as little as possible. We leverage connections with the theories of gradient flows in distribution space and optimal transport to analyze our method theoretically.\n\nOur approach is particularly well-adapted to networks that use residual connections such as ResNets [27; 28], their variants (e.g. ResNeXt [62], Wide ResNet [63], EfficientNet [56] and MobileNetV2 [48]) and vision transformers that are made up essentially of residual connections [39; 17], but is immediately usable on any network where many layers have the same input and output dimension such as VGG [52]. Our contributions are the following:\n\n* We propose a new method for module-wise training. Being a regularization, it is lighter than many recent state-of-the-art methods (PredSim [45], InfoPro [60]) that train another auxiliary network besides the auxiliary classifier for each module.\n* We theoretically justify our method, proving that it is a transport regularization that forces the module to be an optimal transport map making it more regular and stable. We also show that it amounts to a discretization of the gradient flow of the loss in probability space, which means that the modules progressively minimize the loss and explains why the method avoids the accuracy collapse observed in module-wise training.\n* Experimentally, we consistently improve the test accuracy of module-wise trained networks (ResNets, VGG and Swin-Transformer) beating 8 other methods, in sequential and parallel module-wise training, and also in _multi-lap sequential_ training, a variant of sequential module-wise training that we introduce and that performs better in many cases. In particular, our regularization makes parallel module-wise training superior or comparable in accuracy to end-to-end training, while consuming \\(10\\%\\) to \\(60\\%\\) less memory.\n\n## 2 Transport-regularized module-wise training\n\nThe typical setting of (sequential) module-wise training for minimizing a loss \\(L\\), is, given a dataset \\(\\mathcal{D}\\), to solve one after the other, for \\(1{\\leq}k{\\leq}K\\), Problems\n\n\\[(T_{k},F_{k})\\in\\arg\\min_{T,F}\\sum_{x\\in\\mathcal{D}}L(F,T\\circ G_{k-1}(x)) \\tag{1}\\]\n\nFigure 1: Module-wise training.\n\nwhere \\(G_{k}=T_{k}\\circ\\ldots\\circ T_{1}\\) for \\(1{\\leq}k{\\leq}K\\), \\(G_{0}{=}\\mathsf{id}\\), \\(T_{k}\\) is the module (one or many layers) and \\(F_{k}\\) is an auxiliary classifier. Module \\(T_{k}\\) receives the output of module \\(T_{k-1}\\), and auxiliary classifier \\(F_{k}\\) computes the prediction from the output of \\(T_{k}\\) so the loss can be computed. The inputs are \\(x\\) and \\(L\\) has access to their labels \\(y\\) to calculate the loss. i.e. \\(L(F,T\\circ G_{k-1}(x))=l(F\\circ T\\circ G_{k-1}(x),y)\\) where \\(l\\) is a machine learning loss such as cross-entropy. See Figure 1. The final network trained this way is \\(F_{K}\\circ G_{K}\\). But, at inference, we can stop at any depth \\(k\\) and use \\(F_{k}\\circ G_{k}\\) if it performs better. Indeed, an intermediate module often performs as well or better than the last module because of the early overfitting and subsequent stagnation or collapse problem of module-wise training [43; 7; 60; 47].\n\nWe propose below in (2) a regularization that avoids the destruction of task-relevant information by the early modules by forcing them to minimally modify their input. Proposition 2.2 proves that by using our regularization (2), we are indeed making the modules build upon each other to solve the task, which is the property we desire in module-wise training, as the modules now act as successive proximal optimization steps in the _minimizing movement scheme_ optimization algorithm for maximizing the separability of the data representation. The background on optimal transport (OT), gradient flows and the minimizing movement scheme is in Appendices A and B.\n\n### Method statement\n\nTo keep greedily-trained modules from overfitting and destroying information needed later, we penalize their kinetic energy to force them to preserve the geometry of the problem as much as possible. If each module is a single residual block (that is a function \\(T{=}\\mathsf{id}{+}r\\), which includes many transformer architectures [39; 17]), its kinetic energy is simply the squared norm of its residue \\(r{=}T{-}\\mathsf{id}\\), which we add to the loss \\(L\\) in the target of the greedy problems (1). All layers that have the same input and output dimension can be rewritten as residual blocks and the analysis applies to a large variety of architectures such as VGG [52]. Given \\(\\tau{>}0\\), we now solve, for \\(1{\\leq}k{\\leq}K\\), Problems\n\n\\[(T^{\\tau}_{k},F^{\\tau}_{k})\\in\\arg\\min_{T,F}\\sum_{x\\in\\mathcal{D}}L(F,T\\circ G^ {\\tau}_{k-1}(x))+\\frac{1}{2\\tau}\\|T\\circ G^{\\tau}_{k-1}(x){-}G^{\\tau}_{k-1}(x) \\|^{2} \\tag{2}\\]\n\nwhere \\(G^{\\tau}_{k}{=}T^{\\tau}_{k}{\\circ}\\ldots\\circ T^{\\tau}_{1}\\) for \\(1{\\leq}k{\\leq}K\\) and \\(G^{\\tau}_{0}{=}\\mathsf{id}\\). The final network is \\(F^{\\tau}_{K}{\\circ}G^{\\tau}_{K}\\). Intuitively, this biases the modules towards moving the points as little as possible, thus at least keeping the performance of the previous module. Residual connections are already biased towards small displacements and this bias is desirable and should be encouraged [35; 64; 26; 15; 36]. But the method can be applied to any module where \\(T(x)\\) and \\(x\\) have the same dimension so that \\(T(x){-}x\\) can be computed.\n\nTo facilitate the theoretical analysis, we rewrite the method in a more general formulation using data distribution \\(\\rho\\), which can be discrete or continuous, and the distribution-wide loss \\(\\mathcal{L}\\) that arises from the point-wise loss \\(L\\). Then Problem (2) is equivalent to Problem\n\n\\[(T^{\\tau}_{k},F^{\\tau}_{k})\\in\\arg\\min_{T,F}\\ \\mathcal{L}(F,T_{\\sharp}\\rho^{ \\tau}_{k})+\\frac{1}{2\\tau}\\int_{\\Omega}\\|T(x)-x\\|^{2}\\,\\mathrm{d}\\rho^{\\tau}_ {k}(x) \\tag{3}\\]\n\nwith \\(\\rho^{\\tau}_{k+1}{=}(T^{\\tau}_{k})_{\\sharp}\\rho^{\\tau}_{k}\\), \\(\\rho^{\\tau}_{1}{=}\\rho\\) and \\(\\mathcal{L}(F,T_{\\sharp}\\rho^{\\tau}_{k})=\\int L(F,T(x))\\,\\mathrm{d}\\rho^{\\tau} _{k}(x)=\\int L(F,z)\\,\\mathrm{d}T_{\\sharp}\\rho^{\\tau}_{k}(x)\\).\n\n### Link with the minimizing movement scheme\n\nWe now formulate our main result: solving Problems (3) is equivalent to following a _minimizing movement scheme (MMS)_[50] in distribution space for minimizing \\(\\mathcal{Z}(\\mu):=\\min_{F}\\mathcal{L}(F,\\mu)\\), which is the loss of the best classifier. If we are limited to linear classifiers, \\(\\mathcal{Z}(\\rho^{\\tau}_{k})\\) is the linear separability of the representation \\(\\rho^{\\tau}_{k}\\) at module \\(k\\) of the data distribution \\(\\rho\\). The MMS, introduced in [24; 23], is a metric counterpart to Euclidean gradient descent for minimizing functionals over distributions. In our case, \\(\\mathcal{Z}\\) is the functional we want to minimize. We define the MMS below in Definition 2.1\n\nThe distribution space we work in is the metric Wasserstein space \\(\\mathbb{W}_{2}(\\Omega)=(\\mathcal{P}(\\Omega),W_{2})\\), where \\(\\Omega\\subset\\mathbb{R}^{d}\\) is a convex compact set, \\(\\mathcal{P}(\\Omega)\\) is the set of probability distributions over \\(\\Omega\\) and \\(W_{2}\\) is the Wasserstein distance over \\(\\mathcal{P}(\\Omega)\\) derived from the optimal transport problem with Euclidean cost:\n\n\\[W_{2}^{2}(\\alpha,\\beta)=\\min_{T\\,\\mathrm{s.t.}\\ T_{\\sharp}\\alpha=\\beta}\\int_{ \\Omega}\\|T(x)-x\\|^{2}\\,\\mathrm{d}\\alpha(x) \\tag{4}\\]\n\nwhere we assume that \\(\\partial\\Omega\\) is negligible and that the distributions are absolutely continous.\n\n**Definition 2.1**.: Given \\(\\mathcal{Z}:\\mathbb{W}_{2}(\\Omega)\\rightarrow\\mathbb{R}\\), and starting from \\(\\rho_{1}^{\\tau}\\in\\mathcal{P}(\\Omega)\\), the Minimizing Movement Scheme (MMS) takes proximal steps for minimizing \\(\\mathcal{Z}\\). It is s given by\n\n\\[\\rho_{k+1}^{\\tau}\\in\\arg\\min_{\\rho\\in\\mathcal{P}(\\Omega)}\\ \\mathcal{Z}(\\rho)+ \\frac{1}{2\\tau}W_{2}^{2}(\\rho,\\rho_{k}^{\\tau}) \\tag{5}\\]\n\nThe MMS (5) can be seen as a non-Euclidean implicit Euler step for following the gradient flow of \\(\\mathcal{Z}\\), and \\(\\rho_{k}^{\\tau}\\) converges to a minimizer of \\(\\mathcal{Z}\\) under some conditions (see the end of this section).\n\nSo under the mentioned assumptions on \\(\\Omega\\) and absolute continuity of the distributions, we have that Problems (3) are equivalent to the minimizing movement scheme (5):\n\n**Proposition 2.2**.: _The distributions \\(\\rho_{k+1}^{\\tau}=(T_{k}^{\\tau})_{t}\\rho_{k}^{\\tau}\\), where the functions \\(T_{k}^{\\tau}\\) are found by solving (3) and \\(\\rho_{1}^{\\tau}=\\rho\\) is the data distribution, coincide with the MMS (5) for \\(\\mathcal{Z}=\\min_{F}\\mathcal{L}(F,.)\\)._\n\nProof.: The minimizing movement scheme (5) is equivalent to taking \\(\\rho_{k+1}^{\\tau}=(T_{k}^{\\tau})_{\\tau}\\rho_{k}^{\\tau}\\) where\n\n\\[T_{k}^{\\tau}\\in\\arg\\min_{T:\\Omega\\rightarrow\\Omega}\\mathcal{Z}(T_{\\sharp}\\rho _{k}^{\\tau})+\\frac{1}{2\\tau}W_{2}^{2}(T_{\\sharp}\\rho_{k}^{\\tau},\\rho_{k}^{\\tau }) \\tag{6}\\]\n\nunder conditions that guarantee the existence of a transport map between \\(\\rho_{k}^{\\tau}\\) and any other measure, and absolute continuity of \\(\\rho_{k}^{\\tau}\\) suffices, and the loss can ensure that \\(\\rho_{k+1}^{\\tau}\\) is also absolutely continuous. Among the functions \\(T_{k}^{\\tau}\\) that solve problem (6), is the optimal transport map from \\(\\rho_{k}^{\\tau}\\) to \\(\\rho_{k+1}^{\\tau}\\). To solve specifically for this optimal transport map, we have to solve the equivalent Problem\n\n\\[T_{k}^{\\tau}\\in\\arg\\min_{T}\\mathcal{Z}(T_{\\sharp}\\rho_{k}^{\\tau})+\\frac{1}{2 \\tau}\\int_{\\Omega}\\|T(x)-x\\|^{2}\\,\\mathrm{d}\\rho_{k}^{\\tau}(x) \\tag{7}\\]\n\nProblems (6) and (7) have the same minimum value, but the minimizer of (7) is now an optimal transport map between \\(\\rho_{k}^{\\tau}\\) and \\(\\rho_{k+1}^{\\tau}\\). This is immediate from the definition (4) of the \\(W_{2}\\) distance. Equivalently minimizing first over \\(F\\) and then over \\(T\\) in (3), it follows from the definition of \\(\\mathcal{Z}\\) that Problems (3) and (7) are equivalent, which concludes. \n\nSince we solve Problems (3) over neural networks, their representation power shown by universal approximation theorems [13; 29] is important to get close to equivalence between (5) and (3), as we need to approximate an optimal transport map. We also know that the training of each module, if it is shallow, converges [5; 6; 34; 22; 18].\n\nIf \\(\\mathcal{Z}\\) is lower-semi continuous then Problems (5) always admit a solution because \\(\\mathcal{P}(\\Omega)\\) is compact. If \\(\\mathcal{Z}\\) is also \\(\\lambda\\)-geodesically convex for \\(\\lambda{>}0\\), we have convergence of \\(\\rho_{k}^{\\tau}\\) as \\(k{\\rightarrow}\\infty\\) and \\(\\tau{\\rightarrow}0\\) to a minimizer of \\(\\mathcal{Z}\\), potentially under more technical conditions (see Appendix B). Even though a machine learning loss will usually not satisfy these conditions, this analysis offers hints as to why our method avoids in practice the problem of stagnation or collapse in performance of module-wise training as \\(k\\) increases, as we are making proximal local steps in Wasserstein space to minimize the loss. This convergence discussion also suggests taking \\(\\tau\\) as small as possible and many modules.\n\n### Regularity result\n\nAs a secondary result, we show that Problem (3) has a solution and that the solution module \\(T_{k}^{\\tau}\\) is an optimal transport map between its input and output distributions, which means that it comes with some regularity. [36] show that these networks generalize better and overfit less in practice. We assume that the minimization in \\(F\\) is over a compact set \\(\\mathcal{F}\\), that \\(\\rho_{k}^{\\tau}\\) is absolutely continuous, that \\(\\mathcal{L}\\) is continuous and non-negative, that \\(\\Omega\\) is convex and compact and that \\(\\partial\\Omega\\) is negligible.\n\n**Proposition 2.3**.: _Problem (3) has a minimizer \\((T_{k}^{\\tau},F_{k}^{\\tau})\\) such that \\(T_{k}^{\\tau}\\) is an optimal transport map. And for any minimizer \\((T_{k}^{\\tau},F_{k}^{\\tau})\\), \\(T_{k}^{\\tau}\\) is an optimal transport map._\n\nThe proof is in Appendix C. OT maps have regularity properties under some boundedness assumptions. Given Theorem A.1 in Appendix A taken from [20], \\(T_{k}^{\\tau}\\) is \\(\\eta\\)-Holder continuous almost everywhere and if the optimization algorithm we use to solve the discretized problem (2) returns an approximate solution pair \\((\\tilde{F}_{k}^{\\tau},\\tilde{T}_{k}^{\\tau})\\) such that \\(\\tilde{T}_{k}^{\\tau}\\) is an \\(\\epsilon\\)-optimal transport map, i.e. \\(\\|\\tilde{T}_{k}^{\\tau}-T_{k}^{\\tau}\\|_{\\infty}\\leq\\epsilon\\), then we have (using the triangle inequality) the following stability property of the module \\(\\tilde{T}_{k}^{\\tau}\\):\n\n\\[\\|\\tilde{T}_{k}^{\\tau}(x)-\\tilde{T}_{k}^{\\tau}(y)\\|\\leq 2\\epsilon+C\\|x-y\\|^{\\eta} \\tag{8}\\]for almost every \\(x,y\\in\\text{supp}(\\rho_{k}^{\\tau})\\) and \\(C{>}0\\). Composing these stability bounds on \\(T_{k}^{\\tau}\\) and \\(\\tilde{T}_{k}^{\\tau}\\) allows to get bounds for the composition networks \\(G_{k}^{\\tau}\\) and \\(\\tilde{G}_{k}^{\\tau}{=}\\tilde{T}_{k}^{\\tau}\\circ\\dots\\circ\\tilde{T}_{1}^{\\tau}\\).\n\nTo summarize Section 2, the transport regularization makes each module more regular and it allows the modules to build on each other as \\(k\\) increases to solve the task, which is the property we desire.\n\n## 3 Practical implementation\n\n### Multi-block modules\n\nFor simplicity, we presented in (2) the case where each module is a single residual block. However, in practice, we often split the network into modules that are made-up of many residual blocks each. We show here that regularizing the kinetic energy of such modules still amounts to a transport regularization, which means that the theoretical results in Propositions 2.2 and 2.3 still apply.\n\nIf each module \\(T_{k}\\) is made up of \\(M\\) residual blocks, i.e. applies \\(x_{m+1}{=}x_{m}{+}r_{m}(x_{m})\\) for \\(0{\\leq}m{<}M\\), then its total discrete kinetic energy for a single data point \\(x_{0}\\) is the sum of its squared residue norms \\(\\sum\\|r_{m}(x_{m})\\|^{2}\\), since a residual network can be seen as a discrete Euler scheme for an ordinary differential equation [61] with velocity field \\(r\\):\n\n\\[x_{m+1}=x_{m}+r_{m}(x_{m})\\ \\longleftrightarrow\\ \\partial_{t}x_{t}=r_{t}(x_{t}) \\tag{9}\\]\n\nand \\(\\sum\\limits_{m}\\|r_{m}(x_{m})\\|^{2}\\) is then the discretization of the total kinetic energy \\(\\int_{0}^{1}\\|r_{t}(x)\\|^{2}\\,\\mathrm{d}t\\) of the ODE. If \\(\\psi_{m}^{x}\\) denotes the position of a point \\(x\\) after \\(m\\) residual blocks, then regularizing the kinetic energy of multi-block modules now means solving\n\n\\[(T_{k}^{\\tau},F_{k}^{\\tau})\\in\\arg\\min_{T,F}\\sum_{x\\in\\mathcal{D} }(L(F,T(G_{k-1}^{\\tau}(x))+\\frac{1}{2\\tau}\\sum_{m=0}^{M-1}\\|r_{m}(\\psi_{m}^{x} )\\|^{2}) \\tag{10}\\] \\[\\text{s.t. }T=(\\mathtt{id}+r_{M-1})\\circ...\\circ(\\mathtt{id}+r_{0 }),\\ \\psi_{0}^{x}=G_{k-1}^{\\tau}(x),\\psi_{m+1}^{x}=\\psi_{m}^{x}+r_{m}(\\psi_{m}^{x})\\]\n\nwhere \\(G_{k}^{\\tau}{=}T_{k}^{\\tau}\\circ\\dots\\circ T_{1}^{\\tau}\\) for \\(1{\\leq}k{\\leq}K\\) and \\(G_{0}^{\\tau}{=}\\mathtt{id}\\). We also minimize this sum of squared residue norms instead of \\(\\|T(x)-x\\|^{2}\\) (the two no longer coincide) as it works better in practice, which we assume is because it offers a more localized control of the transport. As expressed in (9), a residual network can be seen as an Euler scheme of an ODE and Problem (10) is then the discretization of\n\n\\[(T_{k}^{\\tau},F_{k}^{\\tau})\\in\\arg\\min_{T,F}\\ \\mathcal{L}(F,T_{ \\sharp}\\rho_{k}^{\\tau})+\\frac{1}{2\\tau}\\int_{0}^{1}\\|v_{t}\\|_{L^{2}((\\phi_{t}) \\rho_{k}^{\\tau})}^{2}\\,\\mathrm{d}t \\tag{11}\\] \\[\\text{s.t. }T=\\phi_{1},\\ \\partial_{t}\\phi_{t}^{x}=v_{t}(\\phi_{t}^ {x}),\\ \\phi_{0}^{\\cdot}=\\mathtt{id}\\]\n\nwhere \\(\\rho_{k+1}^{\\tau}=(T_{k}^{\\tau})_{\\sharp}\\rho_{k}^{\\tau}\\) and \\(r_{m}\\) is the discretization of vector field \\(v_{t}\\) at time \\(t=m/M\\). Here, distributions \\(\\rho_{k}^{\\tau}\\) are pushed forward through the maps \\(T_{k}^{\\tau}\\) which correspond to the flow \\(\\phi\\) at time \\(t=1\\) of the kinetically-regularized velocity field \\(v_{t}\\). We recognize in the second term in the target of (11) the optimal transport problem in its dynamic formulation (15) from [9], and given the equivalence between the Monge OT problem (4) and the dynamic OT problem (15) in Appendix A, Problem (11) is in fact equivalent to the original continuous formulation (3), and the theoretical results in Section 2 follow immediately (see also the proof of Proposition 2.3 in Appendix C).\n\n### Solving the module-wise problems\n\nThe module-wise problems can be solved in two ways. One can completely train each module with its auxiliary classifier for \\(N\\) epochs before training the next module, which receives as input the output of the previous trained module. We call this _sequential_ module-wise training. But we can also do this batch-wise, i.e. do a complete forward pass on each batch but without a full backward pass, rather a backward pass that only updates the current module \\(T_{k}^{\\tau}\\) and its auxiliary classifier \\(F_{k}^{\\tau}\\), meaning that \\(T_{k}^{\\tau}\\) forwards its output to \\(T_{k+1}^{\\tau}\\) immediately after it computes it. We call this _parallel_ module-wise training. It is called _decoupled_ greedy training in [8], which shows that combining it with batch buffers solves all three locking problems and allows a linear training parallelization in the depth of the network. We propose a variant of sequential module-wise training that we call _multi-lap sequential_ module-wise training, in which instead of training each module for \\(N\\) epochs, we train each module from the first to the last sequentially for \\(N/R\\) epochs, then go back and train from the first module to the last for \\(N/R\\) epochs again, and we do this for \\(R\\) laps. For the same total number of epochs and training time, and the same advantages (loading and training one module at a time) this provides a non-negligible improvement in accuracy over normal sequential module-wise training in most cases, as shown in Section 4. Despite our theoretical framework being that of sequential module-wise training, our method improves the test accuracy of all three module-wise training regimes.\n\n### Varying the regularization weight\n\nThe discussion in Section 2.2 suggests taking a fixed weight \\(\\tau\\) for the transport cost that is as small as possible. However, instead of using a fixed \\(\\tau\\), we might want to vary it along the depth \\(k\\) to further constrain with a smaller \\(\\tau_{k}\\) the earlier modules to avoid that they overfit or the later modules to maintain the accuracy of earlier modules. We might also want to regularize the network further in earlier epochs when the data is more entangled. We propose in Appendix D to formalize this varying weight \\(\\tau_{k,i}\\) across modules \\(k\\) and SGD iterations \\(i\\) by using a scheme inspired by the method of multipliers to solve Problems (2) and (10). However, it works best in only one experiment in practice. The observed dynamics of \\(\\tau_{k,i}\\) suggest simply finding a fixed value of \\(\\tau\\) that is multiplied by 2 for the second half of the network, which works best in all the other experiments (see Appendix E).\n\n## 4 Experiments\n\nWe call our method TRGL for Transport-Regularized Greedy Learning. For the auxiliary classifiers, we use the architecture from DGL [7; 8], that is a convolution followed by an average pooling and a fully connected layer, which is very similar to that used by InfoPro [60], except for the Swin Transformer where we use a linear layer. We call vanilla greedy module-wise training with the same architecture but without our regularization VanGL, and we include its results in all tables for ablation study purposes. The code is available at github.com/block-wise/module-wise and implementation details are in Appendix E.\n\n### Parallel module-wise training\n\nTo compare with other methods, we focus first on parallel training, as it performs better than sequential training and has been more explored recently. The first experiment is training in parallel 3 residual architectures and a VGG-19 [52] divided into 4 modules of equal depth on TinyImageNet. We compare in Table 1 our results in this setup to three of the best recent parallel module-wise training methods: DGL [8], PredSim [45] and Sedona [47], from Table 2 in [47]. We find that our TRGL has a much better test accuracy than the three other methods, especially on the smaller architectures. It also performs better than end-to-end training on the three ResNets. Parallel TRGL in this case with 4 modules consumes \\(10\\) to \\(21\\%\\) less memory than end-to-end training (with a batch size of 256).\n\nThe second experiment is training in parallel two ResNets divided into 2 modules on CIFAR100 [37]. We compare in Table 2 our results in this setup to the two delayed gradient methods DDG [32] and FR [31] from Table 2 in [31]. Here again, parallel TRGL has a better accuracy than the other two methods and than end-to-end training. With only two modules, the memory gains from less backpropagation are neutralized by the weight of the extra classifier and there are negligible memory savings compared to end-to-end training. However, parallel TRGL has a better test accuracy by up to almost 2 percentage points.\n\n\\begin{table}\n\\begin{tabular}{c c c c c c c} \\hline \\hline Architecture & Parallel VanGL & Parallel TRGL (ours) & PredSim & DGL & Sedona & E2E \\\\ \\hline VGG-19 & 56.17 \\(\\pm\\) 0.29 (\\(\\downarrow 27\\%\\)) & **57.28 \\(\\pm\\)** 0.20 (\\(\\downarrow 21\\%\\)) & 44.70 & 51.40 & 56.56 & 58.74 \\\\ ResNet-50 & 58.43 \\(\\pm\\) 0.45 (\\(\\downarrow 26\\%\\)) & **60.30 \\(\\pm\\)** 0.58 (\\(\\downarrow 20\\%\\)) & 47.48 & 53.96 & 54.40 & 58.10 \\\\ ResNet-101 & 63.64 \\(\\pm\\) 0.30 (\\(\\downarrow 24\\%\\)) & **63.71 \\(\\pm\\)** 0.40 (\\(\\downarrow 11\\%\\)) & 53.92 & 53.80 & 59.12 & 62.01 \\\\ ResNet-152 & 63.87 \\(\\pm\\) 0.16 (\\(\\downarrow 21\\%\\)) & **64.23 \\(\\pm\\)** 0.14 (\\(\\downarrow 10\\%\\)) & 51.76 & 57.64 & 64.10 & 62.32 \\\\ \\hline \\hline \\end{tabular}\n\\end{table}\nTable 1: Test accuracy of parallel TRGL with 4 modules (average and 95\\(\\%\\) confidence interval over 5 runs) on TinyImageNet, compared to DGL, PredSim, Sedona and E2E from Table 2 in [47], with memory saved compared to E2E as a percentage of E2E memory consumption in red.\n\nThe third experiment is training in parallel a ResNet-110 divided into two, four, eight and sixteen modules on STL10 [12]. We compare in Table 3 our results in this setup to the recent methods InfoPro [60] and DGL [8] from Table 2 in [60]. TRGL largely outperforms the other methods. It also outperforms end-to-end training in all but one case (that with 16 modules). With a batch size of 64, memory savings of parallel TRGL compared to end-to-end training reach \\(48\\%\\) and \\(58.5\\%\\) with 8 and 16 modules respectively, with comparable test accuracy. With 4 modules, TRGL training weighs \\(24\\%\\) less than end-to-end-training, and has a test accuracy that is better by \\(2\\) percentage points (see Section 4.2 and Table 6 for a detailed memory usage comparison with InfoPro).\n\nThe fourth experiment is training (from scratch) in parallel a Swin-Tiny Transformer [39] divided into 4 modules on three datasets. We compare in Table 4 our results with those of InfoPro [60] and InfoProL, a variant of InfoPro proposed in [46]. TRGL outperforms the other module-wise training methods. It does not outperform end-to-end training in this case, but consumes \\(29\\%\\) less memory on CIFAR10 and CIFAR100 and \\(50\\%\\) less on STL10, compared to \\(38\\%\\) for InfoPro and \\(45\\%\\) for InfoProL in [46].\n\nFinally, we compare our method to InfoPro, DGL and Sedona in Table 5 below on a large scale experiment on ImageNet.\n\n### Memory savings and training time\n\nAs seen above, parallel TRGL is lighter than end-to-end training by up to almost \\(60\\%\\). The extra memory consumed by our regularization compared to parallel VanGL is between 2 and \\(13\\%\\) of\n\n\\begin{table}\n\\begin{tabular}{c c c c c c} \\hline \\hline Dataset & Parallel VanGL & Parallel TRGL (ours) & InfoPro & InfoProL & E2E \\\\ \\hline STL10 & 67.00 \\(\\pm\\) 1.36 (\\(\\downarrow 55\\%\\)) & **67.92**\\(\\pm\\) 1.12 (\\(\\downarrow 50\\%\\)) & 64.61 (\\(\\downarrow 38\\%\\)) & 66.89 (\\(\\downarrow 45\\%\\)) & 72.19 \\\\ CIFAR10 & 83.94 \\(\\pm\\) 0.42 (\\(\\downarrow 33\\%\\)) & **86.48**\\(\\pm\\) 0.54 (\\(\\downarrow 29\\%\\)) & 83.38 (\\(\\downarrow 38\\%\\)) & 86.28 (\\(\\downarrow 45\\%\\)) & 91.37 \\\\ CIFAR100 & 69.34 \\(\\pm\\) 0.91 (\\(\\downarrow 33\\%\\)) & **74.11**\\(\\pm\\) 0.31 (\\(\\downarrow 29\\%\\)) & 68.36 (\\(\\downarrow 38\\%\\)) & 73.00 (\\(\\downarrow 45\\%\\)) & 75.03 \\\\ \\hline \\hline \\end{tabular}\n\\end{table}\nTable 4: Test accuracy of parallel TRGL with 4 modules (average and 95\\(\\%\\) confidence interval over 5 runs) on a Swin-Tiny Transformer, compared to InfoPro, InfoProL and E2E from Table 3 in [46], with memory saved compared to E2E as a percentage of E2E memory consumption in red.\n\n\\begin{table}\n\\begin{tabular}{c c c c c c} \\hline \\hline Architecture & Parallel VanGL & Parallel TRGL (ours) & DDG & FR & E2E \\\\ \\hline ResNet-101 & 77.31 \\(\\pm\\) 0.27 & **77.87**\\(\\pm\\) 0.44 & 75.75 & 76.90 & 76.52 \\\\ ResNet-152 & 75.40 \\(\\pm\\) 0.75 & **76.55**\\(\\pm\\) 1.90 & 73.61 & 76.39 & 74.80 \\\\ \\hline \\hline \\end{tabular}\n\\end{table}\nTable 2: Test accuracy of parallel TRGL with 2 modules (average and 95\\(\\%\\) confidence interval over 3 runs) on CIFAR100, compared to DDG, FR and E2E from Table 2 in [31].\n\n\\begin{table}\n\\begin{tabular}{c c c c c c c} \\hline \\hline \\(K\\) & Par VanGL & Par TRGL (ours) & DGL & InfoPro S & InfoPro C & E2E \\\\ \\hline\n2 & 79.85 \\(\\pm\\) 0.93 & **80.04**\\(\\pm\\) 0.85 & 75.03 \\(\\pm\\) 1.18 & 78.98 \\(\\pm\\) 0.51 & 79.01 \\(\\pm\\) 0.64 & 77.73 \\(\\pm\\) 1.61 \\\\\n4 & 77.11 \\(\\pm\\) 2.31 & **79.72**\\(\\pm\\) 0.81 & 73.23 \\(\\pm\\) 0.64 & 78.72 \\(\\pm\\) 0.27 & 77.27 \\(\\pm\\) 0.40 & 77.73 \\(\\pm\\) 1.61 \\\\\n8 & 75.71 \\(\\pm\\) 0.55 & **77.82**\\(\\pm\\) 0.73 & 72.67 \\(\\pm\\) 0.24 & 76.40 \\(\\pm\\) 0.49 & 74.85 \\(\\pm\\) 0.52 & 77.73 \\(\\pm\\) 1.61 \\\\\n16 & 73.57 \\(\\pm\\) 0.95 & **77.22**\\(\\pm\\) 1.20 & 72.27 \\(\\pm\\) 0.58 & 73.95 \\(\\pm\\) 0.71 & 73.73 \\(\\pm\\) 0.48 & 77.73 \\(\\pm\\) 1.61 \\\\ \\hline \\hline \\end{tabular}\n\\end{table}\nTable 3: Test accuracy of Parallel (Par) TRGL with \\(K\\) modules (average and 95\\(\\%\\) confidence interval over 5 runs) using a ResNet-110 on STL10, compared to DGL, two variants of InfoPro and E2E from Table 2 in [60].\n\nend-to-end memory. Memory savings depend then mainly on the size of the auxiliary classifier, which can easily be adjusted. Note that delayed gradients method DDG and FR increase memory usage [31], and Sedona does not claim to save memory, but rather to speed up training [47]. DGL is architecture-wise essentially identical to VanGL and consumes the same memory.\n\nWe compare in Table 6 the memory consumption of our method to that of InfoPro [60] on a ResNet-110 on STL10 with a batch size of 64 (so the same setting as in Table 3). InfoPro [60] also propose to split the network into modules that have the same weight but not necessarily the same number of layers. They only implement this for \\(K{\\leq}4\\) modules. When the modules are even in weight and not in depth, we call the training methods VanGL*, TRGL* and InfoPro*. In practice, this leads to shallower early modules which slightly hurts performance according to [47], and as seen below. However, TRGL* still outperforms InfoPro and end-to-end training, and it leads to even bigger memory savings than InfoPro*. We see in Table 6 below that TRGL saves more memory than InfoPro in two out of three cases (4 and 8 modules), and about the same in the third case (16 modules), with much better test accuracy in all cases. Likewise, TRGL* is lighter than InfoPro*, with better accuracy.\n\nHowever, parallel module-wise training does slightly slow down training. Epoch time increases by \\(6\\%\\) with 2 modules and by \\(16\\%\\) with 16 modules. TRGL is only slower than VanGL by \\(2\\%\\) for all number of modules due to the additional regularization term. This is comparable to InfoPro which reports a time overhead between 1 and \\(27\\%\\) compared to end-to-end training. See Appendix F for details.\n\n### Sequential full block-wise training\n\nBlock-wise sequential training, meaning that each module is a single residual block and that the blocks are trained sequentially, therefore requiring only enough memory to train one block and its classifier. Even though it has been less explored in recent module-wise training methods, it has been used in practice in very constrained settings such as on-device training [58; 57]. We therefore test our regularization in this section in this setting, with more details in Appendix G.\n\nWe propose here to use shallower ResNets that are initially wider. These architectures are well-adapted to layer-wise training as seen in [7]. We check first in Table 10 in Appendix G that this architecture works well with parallel module-wise training with 2 modules by comparing it favorably on CIFAR10 [37] with methods DGL [8], InfoPro [60] and DDG [32] that use a ResNet-110 with the same number of parameters.\n\nWe then train a 10-block ResNet block-wise on CIFAR100. In Tables 11 and 12 in Appendix G, we see that MLS training improves the accuracy of sequential training by 0.8 percentage points when the trainset is full, but works less well on small train sets. Of the two, the regularization mainly improves the test accuracy of MLS training. The improvement increases as the training set gets smaller and reaches 1 percentage point. While parallel module-wise training performs quite close to end-to-end training in the full data regime and much better in the small data regime, sequential and multi-lap sequential training are competitive with end-to-end training in the small data regime. Combining the multi-lap trick and the regularization improves the accuracy of sequential training by 1.2 percentage points when using the entire trainset. We report further results for full block-wise training on MNIST [38] and CIFAR10 [37] in Tables 13 and 14 in Appendix G.\n\n\\begin{table}\n\\begin{tabular}{c c c c c c c} \\hline \\hline  & \\multicolumn{4}{c}{Equally deep modules} & \\multicolumn{4}{c}{Equally heavy modules} \\\\ \\cline{2-7} \\(K\\) & Par VanGL & Par TRGL (ours) & InfoPro & Par VanGL* & Par TRGL* (ours) & InfoPro* \\\\ \\hline\n4 & 27\\% (77.11) & 24\\% (**79.72**) & 18\\% (78.72) & 41\\% (77.14) & 39\\% (**78.94**) & 33\\% (78.78) \\\\\n8 & 50\\% (75.71) & 48\\% (**77.82**) & 37\\% (76.40) & & & \\\\\n16 & 61\\% (73.57) & 58\\% (**77.22**) & 59\\% (73.95) & & & \\\\ \\hline \\hline \\end{tabular}\n\\end{table}\nTable 6: Memory savings using a ResNet-110 on STL10 split into \\(K\\) modules trained in parallel with a batch size of 64, as a percentage of the weight of end-to-end training. Average test accuracy over 5 runs is between brackets. Test accuracy of end-to-end training is 77.73%.\n\nThe \\(88\\%\\) accuracy of sequential training on CIFAR10 in Table 13 is the same as in Table 2 of [7], which is the best method for layer-wise sequential training available, with VGG networks of comparable depth and width.\n\n### Accuracy after each module\n\nFinally, we verify that our method avoids the stagnation or collapse in accuracy with depth. In Figure 2 below, we show the accuracy of each module with and without the regularization.\n\nOn the left, from parallel module-wise training experiments from Table 3, TRGL performs worse than vanilla greedy learning early, but surpasses it in later modules, indicating that it does avoid early overfitting. On the right, from sequential block-wise training experiments from Table 13, we see a large decline in performance that the regularization avoids. We see similar patterns in Figure 3 in Appendix G with parallel and MLS block-wise training.\n\n## 5 Limitations\n\nThe results in Appendix G show a few limitations of our method, as the improvements from the regularization are sometimes minimal on sequential training. However, the results show that our approach works in all settings (parallel and sequential with many or few modules), whereas other papers don't test their methods in all settings, and some show problems in other settings than the original one in subsequent papers (e.g. delayed gradients methods when the number of modules increases [31] and PredSim in [47]). Also, for parallel training in Section 4.1, the improvement from the regularization compared to VanGL is larger and increases with the number of modules (so with the memory savings) and reaches almost 5 percentage points. We show in Appendix H that our method is not very sensitive to the choice of hyperparameter \\(\\tau\\) over a large scale.\n\n## 6 Related work\n\nLayer-wise training was initially considered as a pre-training and initialization method [10; 43] and was recently shown to be competitive with end-to-end training [7; 45]. Many papers consider using a different auxiliary loss, instead of or in addition to the classification loss: kernel similarity [42], information-theory-inspired losses [53; 44; 41; 60] and biologically plausible losses [53; 45; 25; 11]. Methods [7], PredSim [45], DGL [8], Sedona [47] and InfoPro [60] report the best module-wise training results. [7; 8] do it simply through the architecture choice of the auxiliary networks. Sedona applies architecture search to decide on where to split the network into modules and what auxiliary classifier to use before module-wise training. Only BoostResNet [30] also proposes a block-wise training idea geared for ResNets. However, their results only show better early performance and end-to-end fine-tuning is required to be competitive. A method called ResIST [19] that is similar to block-wise training of ResNets randomly assigns ResBlocks to one of up to 16 modules that\n\nFigure 2: Test accuracy after each module averaged over 10 runs with \\(95\\%\\) confidence intervals. Left: parallel vanilla (VanGL, in blue) and regularized (TRGL, in red) module-wise training of a ResNet-110 with 16 modules on STL10 (Table 3). Right: sequential vanilla (VanGL, in blue) and regularized (TRGL, in red) block-wise training of a 10-block ResNet on \\(2\\%\\) of CIFAR10 (Table 13).\n\nare trained independently and reassembled before another random partition. More of a distributed training method, it is only compared to local SGD [54]. These methods can all be combined with our regularization, and we do use the auxiliary classifier from [7, 8].\n\nBesides module-wise training, methods such as DNI [33, 14], DDG [32] and FR [31], solve the update and backward locking problems with an eye towards parallelization by using delayed or predicted gradients, or even predicted inputs to address forward locking, which is what [55] do. But they observe training issues with more than 5 modules [31]. This makes them compare unfavorably to module-wise training [8]. The high dimension of the predicted gradient, which scales with the size of the network, makes [33, 14] challenging in practice. Therefore, despite its simplicity, greedy module-wise training is more appealing when working in a constrained setting.\n\nViewing ResNets as dynamic transport systems [16, 36] followed from their view as a discretization of ODEs [61]. Transport regularization of ResNets in particular is motivated by the observation that they are naturally biased towards minimally modifying their input [35, 26]. We further linked this transport viewpoint with gradient flows in the Wasserstein space to apply it in a principled way to module-wise training. Gradient flows on the data distribution appeared recently in deep learning. In [1], the focus is on functionals of measures whose first variations are known in closed form and used, through their gradients, in the algorithm. This limits the scope of their applications to transfer learning and similar tasks. Likewise, [21, 40, 4, 3] use the explicit gradient flow of \\(f\\)-divergences and other distances between measures for generation and generator refinement. In contrast, we use the discrete minimizing movement scheme which does not require computation of the first variation and allows to consider classification.\n\n## 7 Conclusion\n\nWe introduced a transport regularization for module-wise training that theoretically links module-wise training to gradient flows of the loss in probability space. Our method provably leads to more regular modules and experimentally improves the test accuracy of module-wise parallel, sequential and multi-lap sequential (a variant of sequential training that we introduce) training. Through this simple method that does not complexify the architecture, we make module-wise training competitive with end-to-end training while benefiting from its lower memory usage. Being a regularization, the method can easily be combined with other layer-wise training methods. Future work can experiment with working in Wasserstein space \\(W_{p}\\) for \\(p{\\neq}2\\), i.e. regularizing with a norm \\(\\|\\cdot\\|_{p}\\) with \\(p{\\neq}2\\).\n\n## References\n\n* [1]Alvarez-Melis, D., and Fusi, N. Dataset dynamics via gradient flows in probability space. _ICML_ (2021).\n* [2]Ambrosio, L., Gigli, N., and Savare, G. _Gradient Flows in Metric Spaces and in the Space of Probability Measures_. Birkhauser Basel, 2005.\n* [3]Ansari, A. F., Ang, M. L., and Soh, H. Refining deep generative models via discriminator gradient flow. In _ICLR_ (2021).\n* [4]Arbel, M., Korba, A., Salim, A., and Gretton, A. Maximum mean discrepancy gradient flow. In _NeurIPS_ (2019).\n* [5]Arora, R., Basu, A., Mianjy, P., and Mukherjee, A. Understanding deep neural networks with rectified linear units. In _International Conference on Learning Representations_ (2018).\n* [6]Bach, F. Breaking the curse of dimensionality with convex neural networks. _Journal of Machine Learning Research 18_, 19 (2017), 1-53.\n* [7]Bellovsky, E., E., Eickenberg, M., and Oyallon, E. Greedy layerwise learning can scale to imagenet. In _ICML_ (2019).\n* [8]Bellovsky, E., Eickenberg, M., and Oyallon, E. Decoupled greedy learning of cnns. In _ICML_ (2020).\n* [9]Benamou, J., and Brenier, Y. A computational fluid mechanics solution to the monge-kantorovich mass transfer problem. _Numerische Mathematik_ (2000).\n\n* [10]Bengio, Y., Lamblin, P., Popovici, D., and Larochelle, H. Greedy layer-wise training of deep networks. In _NeurIPS_ (2006).\n* [11]Bernd Illing, Wulfram Gerstner, G. B. Towards truly local gradients with clapp: Contrastive, local and predictive plasticity. _arXiv_ (2020).\n* [12]Coate, A., Lee, H., and Ng, A. Y. An analysis of single layer networks in unsupervised feature learning. In _AISTATS_ (2011).\n* [13]Cybenko, G. Approximation by superpositions of a sigmoidal function. _Mathematics of Control, Signals and Systems 2_ (1989), 303-314.\n* [14]Czarnecki, W. M., Swirszcz, G., Jaderberg, M., Osindero, S., Vinyals, O., and Kavukcuoglu, K. Understanding synthetic gradients and decoupled neural interfaces. In _ICML_ (2017).\n* [15]De, S., and Smith, S. L. Batch normalization biases residual blocks towards the identity function in deep networks. In _NeurIPS_ (2020).\n* [16]de Bezenac, E., Ayed, I., and Gallinari, P. Optimal unsupervised domain translation. _arXiv_ (2019).\n* [17]Dosovitskiy, A., Beyer, L., Kolesnikov, A., Weissenborn, D., Zhai, X., Unterthiner, T., Dehghani, M., Minderer, M., Heigold, G., Gelly, S., Uszkoreit, J., and Houlsby, N. An image is worth 16x16 words: Transformers for image recognition at scale. In _International Conference on Learning Representations_ (2021).\n* [18]Du, S. S., and Goel, S. Improved learning of one-hidden-layer convolutional neural networks with overlaps. _arXiv_ (2018).\n* [19]Dun, C., Wolfe, C. R., Jermaine, C. M., and Kyrillidis, A. Resist: Layer-wise decomposition of resnets for distributed training. _arXiv_ (2021).\n* [20]Figalli, A. _The Monge-Ampere Equation and Its Applications_. Zurich lectures in advanced mathematics. European Mathematical Society, 2017.\n* [21]Gao, Y., Jiao, Y., Wang, Y., Wang, Y., Yang, C., and Zhang, S. Deep generative learning via variational gradient flow. In _ICML_ (2019).\n* [22]Ge, R., Lee, J. D., and Ma, T. Learning one-hidden-layer neural networks with landscape design. In _International Conference on Learning Representations_ (2018).\n* [23]Gianazza, U., Gobbino, M., and Savare, G. Evolution problems and minimizing movements. _Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni 5_ (1994), 289-296.\n* [24]Giorgi, E. D., Marino, A., and Tosques, M. Problemi di evoluzione in spazi metrici e curve di massima pendenza. _Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti 68_, 3 (3 1980), 180-187.\n* [25]Gupta, S. K. A more biologically plausible local learning rule for anns. _arXiv_ (2020).\n* [26]Hauser, M. On residual networks learning a perturbation from identity. _arXiv_ (2019).\n* [27]He, K., Zhan, X., Ren, S., and Sun, J. Identity mappings in deep residual networks. In _ECCV_ (2016).\n* [28]He, K., Zhang, X., Ren, S., and Sun, J. Deep residual learning for image recognition. In _CVPR_ (2016).\n* [29]Hornik, K., Stinchcombe, M., and White, H. Multilayer feedforward networks are universal approximators. _Neural Networks 2_, 5 (1989), 359-366.\n* [30]Huang, F., Ash, J., Langford, J., and Schapire, R. Learning deep resnet blocks sequentially using boosting theory. In _ICML_ (2018).\n* [31]Huo, Z., Gu, B., and Huang, H. Training neural networks using features replay. In _NeurIPS_ (2018).\n* [32]Huo, Z., Gu, B., Yang, Q., and Huang, H. Decoupled parallel backpropagation with convergence guarantee. In _ICML_ (2018).\n* [33]Jaderberg, M., Czarnecki, W. M., Osindero, S., Vinyals, O., Graves, A., Silver, D., and Kavukcuoglu, K. Decoupled neural interfaces using synthetic gradients. In _ICML_ (2017).\n\n* [34]Janzamin, M., Sedghi, H., and Anandkumar, A. Beating the perils of non-convexity: Guaranteed training of neural networks using tensor methods. _arXiv_ (2016).\n* [35]Jastrzebski, S., et al. Residual connections encourage iterative inference. In _ICLR_ (2018).\n* [36]Karkar, S., Ayed, I., de Bezenac, E., and Gallinari, P. A principle of least action for the training of neural networks. In _ECML-PKDD_ (2020).\n* [37]Krizhevsky, A. Learning multiple layers of features from tiny images. _University of Toronto Technical Report_ (2009).\n* [38]LeCun, Y., Cortes, C., and Burges, C. J. MNIST handwritten digit database. _yann.lecun.com/exdb/mnist_ (2010).\n* [39]Liu, Z., Lin, Y., Cao, Y., Hu, H., Wei, Y., Zhang, Z., Lin, S., and Guo, B. Swin transformer: Hierarchical vision transformer using shifted windows. In _Proceedings of the IEEE/CVF International Conference on Computer Vision (ICCV)_ (2021).\n* [40]Liutkus, A., Ilsekli, U., Majewski, S., Durmus, A., and Stoter, F.-R. Sliced-wasserstein flows: Nonparametric generative modeling via optimal transport and diffusions. In _ICML_ (2019).\n* [41]Ma, W.-D. K., Lewis, J., and Kleijn, W. B. The hsic bottleneck: Deep learning without back-propagation. In _AAAI_ (2020).\n* [42]Mandar Kulkarni, S. K. Layer-wise training of deep networks using kernel similarity. In _DLPR workshop, ICPR_ (2016).\n* [43]Marquez, E. S., Hare, J. S., and Niranjan, M. Deep cascade learning. _IEEE Transactions on Neural Networks and Learning Systems_ (2018).\n* [44]Nguyen, T. T., and Choi, J. Layer-wise learning of stochastic neural networks with information bottleneck. _Entropy 21_ (2019).\n* [45]Nokland, A., and Eidnes, L. H. Training neural networks with local error signals. In _ICML_ (2019).\n* [46]Pathak, P., Zhang, J., and Samaras, D. Local learning on transformers via feature reconstruction, 2022.\n* [47]Pyeon, M., Moon, J., Hahn, T., and Kim, G. Sedona: Search for decoupled neural networks toward greedy block-wise learning. In _ICLR_ (2021).\n* [48]Sandler, M., Howard, A. G., Zhu, M., Zhmoginov, A., and Chen, L. Mobilenetv2: Inverted residuals and linear bottlenecks. In _2018 IEEE Conference on Computer Vision and Pattern Recognition, CVPR 2018, Salt Lake City, UT, USA, June 18-22, 2018_ (2018), Computer Vision Foundation / IEEE Computer Society, pp. 4510-4520.\n* [49]Santambrogio, F. _Optimal Transport for Applied Mathematicians_. Birkhauser, 2015.\n* [50]Santambrogio, F. Euclidean, metric, and wasserstein gradient flows: an overview. _arXiv_ (2016).\n* [51]Saxe, A. M., Mcclelland, J. L., and Ganguli, S. Exact solutions to the nonlinear dynamics of learning in deep linear neural network. In _ICLR_ (2014).\n* [52]Simonyan, K., and Zisserman, A. Very deep convolutional networks for large-scale image recognition. In _International Conference on Learning Representations (ICLR 2014)_ (2014).\n* [53]Sindy Lowe, Peter O'Connor, B. V. Putting an end to end-to-end: Gradient-isolated learning of representations. In _NeurIPS_ (2019).\n* [54]Stich, S. U. Local sgd converges fast and communicates little. In _ICLR_ (2019).\n* [55]Sun, Q., Dong, H., Chen, Z., Sun, J., Li, Z., and Dong, B. Layer-parallel training of residual networks with auxiliary-variable networks. _arXiv_ (2021).\n* [56]Tan, M., and Le, Q. EfficientNet: Rethinking model scaling for convolutional neural networks. In _Proceedings of the 36th International Conference on Machine Learning_ (09-15 Jun 2019), K. Chaudhuri and R. Salakhutdinov, Eds., vol. 97 of _Proceedings of Machine Learning Research_, PMLR, pp. 6105-6114.\n* [57]Tang, Y., Teng, Q., Zhang, L., Min, F., and He, J. Layer-wise training convolutional neural networks with smaller filters for human activity recognition using wearable sensors. _IEEE Sensors Journal_ (2021).\n\n* [58]Teng, Q., Wang, K., Zhang, L., and He, J. The layer-wise training convolutional neural networks using local loss for sensor-based human activity recognition. _IEEE Sensors Journal_ (2020).\n* [59]Villani, C. _Optimal Transport: Old and New_. Springer-Verlag, 2008.\n* [60]Wang, Y., Ni, Z., Song, S., and Le Yang, G. H. Revisiting locally supervised learning: an alternative to end-to-end training. In _ICLR_ (2021).\n* [61]Weinan, E. A proposal on machine learning via dynamical systems. _Commun. Math. Stat_ (2017).\n* [62]Xie, S., et al. Aggregated residual transformations for deep neural networks. In _CVPR_ (2017).\n* [63]Zagoruyko, S., and Komodakis, N. Wide residual networks. In _BMVC_ (2016).\n* [64]Zhang, J., et al. Towards robust resnet: A small step but a giant leap. In _IJCAI_ (2019).\n* MICCAI 2022_ (Cham, 2022), Springer Nature Switzerland, pp. 192-201.\nBackground on optimal transport\n\nThe Wasserstein space \\(\\mathbb{W}_{2}(\\Omega)\\) with \\(\\Omega\\) a convex and compact subset of \\(\\mathbb{R}^{d}\\) is the space \\(\\mathcal{P}(\\Omega)\\) of probability measures over \\(\\Omega\\), equipped with the distance \\(W_{2}\\) given by the solution to the optimal transport problem\n\n\\[W_{2}^{2}(\\alpha,\\beta)=\\min_{\\gamma\\in\\Pi(\\alpha,\\beta)}\\int_{\\Omega\\times \\Omega}\\|x-y\\|^{2}\\,\\mathrm{d}\\gamma(x,y) \\tag{12}\\]\n\nwhere \\(\\Pi(\\alpha,\\beta)\\) is the set of probability distribution over \\(\\Omega\\times\\Omega\\) with first marginal \\(\\alpha\\) and second marginal \\(\\beta\\), i.e. \\(\\Pi(\\alpha,\\beta)=\\{\\gamma\\in\\mathcal{P}(\\Omega\\times\\Omega)\\mid\\pi_{1\\sharp} \\gamma=\\alpha,\\;\\pi_{2\\sharp}\\gamma=\\beta\\}\\) where \\(\\pi_{1}(x,y)=x\\) and \\(\\pi_{2}(x,y)=y\\). The optimal transport problem can be seen as looking for a transportation plan minimizing the cost of displacing some distribution of mass from one configuration to another. This problem indeed has a solution in our setting and \\(W_{2}\\) can be shown to be a geodesic distance (see for example [49, 59]). If \\(\\alpha\\) is absolutely continuous and \\(\\partial\\Omega\\) is \\(\\alpha\\)-negligible then the problem in (12) (called the Kantorovich problem) has a unique solution and is equivalent to the Monge problem, i.e.\n\n\\[W_{2}^{2}(\\alpha,\\beta)=\\min_{T\\,s.t.\\;T_{\\sharp}\\alpha=\\beta}\\int_{\\Omega}\\|T (x)-x\\|^{2}\\,\\mathrm{d}\\alpha(x) \\tag{13}\\]\n\nand this problem has a unique solution \\(T^{\\star}\\) linked to the solution \\(\\gamma^{\\star}\\) of (12) through \\(\\gamma^{\\star}=(\\mathrm{id},T^{\\star})_{\\sharp}\\alpha\\). Another equivalent formulation of the optimal transport problem in this setting is the dynamical formulation [9]. Here, instead of directly pushing samples of \\(\\alpha\\) to \\(\\beta\\) using \\(T\\), we can equivalently displace mass, according to a continuous flow with velocity \\(v_{t}:\\mathbb{R}^{d}\\to\\mathbb{R}^{d}\\). This implies that the density \\(\\alpha_{t}\\) at time \\(t\\) satisfies the _continuity equation_\\(\\partial_{t}\\alpha_{t}+\\nabla\\cdot(\\alpha_{t}v_{t})=0\\), assuming that initial and final conditions are given by \\(\\alpha_{0}=\\alpha\\) and \\(\\alpha_{1}=\\beta\\) respectively. In this case, the optimal displacement is the one that minimizes the total action caused by \\(v\\) :\n\n\\[W_{2}^{2}(\\alpha,\\beta)= \\min_{v}\\int_{0}^{1}\\|v_{t}\\|_{L^{2}(\\alpha_{t})}^{2}\\,\\mathrm{d}t \\tag{14}\\] \\[\\mathrm{s.t.}\\;\\partial_{t}\\alpha_{t}+\\nabla\\cdot(\\alpha_{t}v_{t })=0,\\;\\alpha_{0}=\\alpha,\\alpha_{1}=\\beta\\]\n\nInstead of describing the density's evolution through the continuity equation, we can describe the paths \\(\\phi_{t}^{x}\\) taken by particles at position \\(x\\) from \\(\\alpha\\) when displaced along the flow \\(v\\). Here \\(\\phi_{t}^{x}\\) is the position at time \\(t\\) of the particle that was at \\(x\\sim\\alpha\\) at time 0. The continuity equation is then equivalent to \\(\\partial_{t}\\phi_{t}^{x}=v_{t}(\\phi_{t}^{x})\\). See chapters 4 and 5 of [49] for details. Rewriting the conditions as necessary, Problem (14) becomes\n\n\\[W_{2}^{2}(\\alpha,\\beta)= \\min_{v}\\int_{0}^{1}\\|v_{t}\\|_{L^{2}((\\phi_{t})_{\\sharp}\\alpha)} ^{2}\\,\\mathrm{d}t \\tag{15}\\] \\[\\mathrm{s.t.}\\;\\partial_{t}\\phi_{t}^{x}=v_{t}(\\phi_{t}^{x}),\\; \\phi_{0}=\\mathrm{id},(\\phi_{1})_{\\sharp}\\alpha=\\beta\\]\n\nand the optimal transport map \\(T^{\\star}\\) that solves (13) is in fact \\(T^{\\star}(x)=\\phi_{1}^{x}\\) for \\(\\phi\\) that solves the continuity equation together with the optimal \\(v^{\\star}\\) from (15). We refer to [49, 59] for these results on optimal transport.\n\nOptimal transport maps have some regularity properties under some boundedness assumptions. We mention the following result from [20]:\n\n**Theorem A.1**.: _Let \\(\\alpha\\) and \\(\\beta\\) be absolutely continuous measures on \\(\\mathbb{R}^{d}\\) and \\(T\\) the optimal transport map between \\(\\alpha\\) and \\(\\beta\\) for the Euclidean cost. Suppose there are bounded open sets \\(X\\) and \\(Y\\), such that the density of \\(\\alpha\\) (respectively of \\(\\beta\\)) is null on \\(X^{\\mathrm{c}}\\) (respectively \\(Y^{\\mathrm{c}}\\)) and bounded away from zero and infinity on \\(X\\) (respectively \\(Y\\))._\n\n_Then there exists two relatively closed sets of null measure \\(A\\subset X\\) and \\(B\\subset Y\\), such that \\(T\\) is \\(\\eta\\)-Holder continuous from \\(X\\setminus A\\) to \\(Y\\setminus B\\), i.e. \\(\\forall\\;x,y\\in X\\setminus A\\) we have_\n\n\\[\\|T(x)-T(y)\\|\\leq C\\|x-y\\|^{\\eta}\\text{ for constants }\\eta,C>0\\]\n\n## Appendix B Background on gradient flows\n\nWe follow [50, 2] for this background on gradient flows. Given a function \\(\\mathcal{L}:\\mathbb{R}^{d}\\to\\mathbb{R}\\) and an initial point \\(x_{0}\\in\\mathbb{R}^{d}\\), a _gradient flow_ is a curve \\(x:[0,\\infty[\\to\\mathbb{R}^{d}\\) that solves the Cauchy problem\n\n\\[\\begin{cases}x^{\\prime}(t)=-\\nabla\\mathcal{L}(x(t))\\\\ x(0)=x_{0}\\end{cases} \\tag{16}\\]A solution exists and is unique if \\(\\nabla\\mathcal{L}\\) is Lipschitz or \\(\\mathcal{L}\\) is convex. Given \\(\\tau>0\\) and \\(x_{0}^{\\tau}=x_{0}\\) define a sequence \\((x_{k}^{\\tau})_{k}\\) through the _minimizing movement scheme_:\n\n\\[x_{k+1}^{\\tau}\\in\\arg\\min_{x\\in\\mathbb{R}^{d}}\\ \\mathcal{L}(x)+\\frac{1}{2\\tau} \\|x-x_{k}^{\\tau}\\|^{2} \\tag{17}\\]\n\n\\(\\mathcal{L}\\) lower semi-continous and \\(\\mathcal{L}(x)\\geq C_{1}-C_{2}\\|x\\|^{2}\\) guarantees existence of a solution of (17) for \\(\\tau\\) small enough. \\(\\mathcal{L}\\)\\(\\lambda\\)-convex meets these conditions and also provides uniqueness of the solution because of strict convexity of the target. See [49, 50, 2].\n\nWe interpret the point \\(x_{k}^{\\tau}\\) as the value of a curve \\(x\\) at time \\(k\\tau\\). We can then construct a curve \\(x^{\\tau}\\) as the piecewise constant interpolation of the points \\(x_{k}^{\\tau}\\). We can also construct a curve \\(\\tilde{x}^{\\tau}\\) as the affine interpolation of the points \\(x_{k}^{\\tau}\\).\n\nIf \\(\\mathcal{L}(x_{0})<\\infty\\) and \\(\\inf\\mathcal{L}>-\\infty\\) then \\((x^{\\tau})\\) and \\((\\tilde{x}^{\\tau})\\) converge uniformly to the same curve \\(x\\) as \\(\\tau\\) goes to zero (up to extracting a subsequence). If \\(\\mathcal{L}\\) is \\(\\mathcal{C}^{1}\\), then the limit curve \\(x\\) is a solution of (16) (i.e. a gradient flow of \\(\\mathcal{L}\\)). If \\(\\mathcal{L}\\) is not differentiable then \\(x\\) is solution of the problem defined using the subdifferential of \\(\\mathcal{L}\\), i.e. \\(x\\) satisfies \\(x^{\\prime}(t)\\in-\\partial\\mathcal{L}(x(t))\\) for almost every \\(t\\).\n\nIf \\(\\mathcal{L}\\) is \\(\\lambda\\)-convex with \\(\\lambda>0\\), then the solution to (16) converges exponentially to the unique minimizer of \\(\\mathcal{L}\\) (which exists by coercivity). So taking \\(\\tau\\to 0\\) and \\(k\\to\\infty\\), we tend towards the minimizer of \\(\\mathcal{L}\\).\n\nThe advantage of the minimizing movement scheme (17) is that it can be adapted to metric spaces by replacing the Euclidean distance by the metric space's distance. In the (geodesic) metric space \\(\\mathbb{W}_{2}(\\Omega)\\) with \\(\\Omega\\) convex and compact, for \\(\\mathcal{L}:\\mathbb{W}_{2}(\\Omega)\\to\\mathbb{R}\\cup\\{\\infty\\}\\) lower semi-continuous for the weak convergence of measures in duality with \\(\\mathcal{C}(\\Omega)\\) (equivalent to lower semi-continuous with respect to the distance \\(W_{2}\\)) and \\(\\rho_{0}^{\\tau}=\\rho_{0}\\in\\mathcal{P}(\\Omega)\\), the minimizing movement scheme (17) becomes\n\n\\[\\rho_{k+1}^{\\tau}\\in\\arg\\min_{\\rho\\in\\mathcal{P}(\\Omega)}\\ \\mathcal{L}(\\rho)+ \\frac{1}{2\\tau}W_{2}^{2}(\\rho,\\rho_{k}^{\\tau}) \\tag{18}\\]\n\nThis problem has a solution because the objective is lower semi-continuous and the minimization is over \\(\\mathcal{P}(\\Omega)\\) which is compact by Banach-Alaoglu.\n\nWe can construct a piecewise constant interpolation between the measures \\(\\rho_{k}^{\\tau}\\), or a geodesic interpolation where we travel along a geodesic between \\(\\rho_{k}^{\\tau}\\) and \\(\\rho_{k+1}^{\\tau}\\) in \\(\\mathbb{W}_{2}(\\Omega)\\), constructed using the optimal transport map between these measures. Again, if \\(\\mathcal{L}(x_{0})<\\infty\\) and \\(\\inf\\mathcal{L}>-\\infty\\) then both interpolations converge uniformly to a limit curve \\(\\tilde{\\rho}\\) as \\(\\tau\\) goes to zero. Under further conditions on \\(\\mathcal{L}\\), mainly \\(\\lambda\\)-geodesic convexity (i.e. \\(\\lambda\\)-convexity along geodesics) for \\(\\lambda>0\\), we can prove stability and convergence of \\(\\tilde{\\rho}(t)\\) to a minimizer of \\(\\mathcal{L}\\) as \\(t\\to\\infty\\), see [49, 50, 2].\n\n## Appendix C Proof of Proposition 2.3\n\nProof.: Take a minimizing sequence \\((\\tilde{F}_{i},\\tilde{T}_{i})\\), i.e. such that \\(\\mathcal{C}(\\tilde{F}_{i},\\tilde{T}_{i})\\to\\min\\mathcal{C}\\), where \\(\\mathcal{C}\\geq 0\\) is the target function in (3) and denote \\(\\beta_{i}=\\tilde{T}_{i*}\\rho_{k}^{\\tau}\\). Then by compacity \\(\\tilde{F}_{i}\\to F^{\\star}\\) and \\(\\beta_{i}\\rightharpoonup\\beta^{\\star}\\) in duality with \\(\\mathcal{C}_{b}(\\Omega)\\) by Banach-Alaoglu. There exists \\(T^{\\star}\\) an optimal transport map between \\(\\rho_{k}^{\\tau}\\) and \\(\\beta^{\\star}\\). Then \\(\\mathcal{C}(F^{\\star},T^{\\star})\\leq\\lim\\mathcal{C}(\\tilde{F}_{i},\\tilde{T}_{i })=\\min\\mathcal{C}\\) by continuity of \\(\\mathcal{L}\\) and because\n\n\\[\\int_{\\Omega}\\|T^{\\star}(x)-x\\|^{2}\\,\\mathrm{d}\\rho_{k}^{\\tau}(x) =W_{2}^{2}(\\rho_{k}^{\\tau},\\beta^{\\star})\\] \\[=\\lim W_{2}^{2}(\\rho_{k}^{\\tau},\\beta_{i})\\] \\[\\leq\\lim\\int_{\\Omega}\\|\\tilde{T}_{i}(x)-x\\|^{2}\\,\\mathrm{d}\\rho_{ k}^{\\tau}(x)\\]\n\nas \\(W_{2}\\) metrizes weak convergence of measures. We take \\((F_{k}^{\\tau},T_{k}^{\\tau})=(F^{\\star},T^{\\star})\\). It is also immediate that for any minimizing pair, the transport map has to be optimal. Taking a minimizing sequence \\((\\tilde{F}_{i},\\tilde{v}^{i})\\) and the corresponding induced maps \\(\\tilde{T}_{i}\\) we get the same result for Problem (11). Problems (3) and (11) are equivalent by the equivalence between Problems (13) and (15). \n\n## Appendix D Varying the regularization weight\n\nThe discussion in Section 2.2 suggests taking a fixed weight \\(\\tau\\) for the transport cost that is as small as possible. However, instead of using a fixed \\(\\tau\\), we might want to vary it along the depth \\(k\\) to further constrain with a smaller \\(\\tau_{k}\\) the earlier modules to avoid that they overfit or the later modules to maintain the accuracy of earlier modules. We might also want to regularize the network further in earlier epochs when the data is more entangled. To unify and formalize this varying weight \\(\\tau_{k,i}\\) across modules \\(k\\) and SGD iterations \\(i\\), we use a scheme inspired by the method of multipliers to solve Problems (2) and (10). To simplify the notations, we will instead consider the weight \\(\\lambda_{k,i}{:=}2\\tau_{k,i}\\) given to the loss. We denote \\(\\theta_{k,i}\\) the parameters of both \\(T_{k}\\) and \\(F_{k}\\) at SGD iteration \\(i\\). We also denote \\(L(\\theta,x)\\) and \\(W(\\theta,x)\\) respectively the loss and the transport regularization as functions of parameters \\(\\theta\\) and data \\(x\\). We now increase the weight \\(\\lambda_{k,i}\\) of the loss every \\(s\\) iterations of SGD by a value that is proportional to the current loss. Given increase factor \\(h{>}0\\), initial parameters \\(\\theta_{k,1}\\), initial weights \\(\\lambda_{k,1}{\\geq}0\\), learning rates (\\(\\eta_{i}\\)) and batches (\\(x_{i}\\)), we apply for module \\(k\\) and \\(i{\\geq}1\\):\n\n\\[\\begin{cases}\\theta_{k,i+1}&=\\theta_{k,i}-\\eta_{i}\\nabla_{\\theta}(\\lambda_{k,i }\\,L(\\theta_{k,i},x_{i})+W(\\theta_{k,i},x_{i}))\\\\ \\lambda_{k,i+1}&=\\lambda_{k,i}+hL(\\theta_{k,i+1},x_{i+1})\\text{ if }i\\text{ mod }s=0 \\text{ else }\\lambda_{k,i}\\end{cases}\\]\n\nThe weights \\(\\lambda_{k,i}\\) will vary along modules \\(k\\) because they will evolve differently with iterations \\(i\\) for each \\(k\\). They will increase more slowly with \\(i\\) for larger \\(k\\) because deeper modules will have smaller loss. This method can be seen as a method of multipliers for the problem of minimizing the transport under the constraint of zero loss. Therefore it is immediate by slightly adapting the proof of Proposition 2.3 or from [36] that we are still solving a problem that admits a solution whose non-auxiliary part is an optimal transport map with the same regularity as stated above. We use the same initial value \\(\\lambda_{1}=\\lambda_{k,1}\\) for all modules so that this method requires choosing three hyper-parameters (\\(h\\), \\(s\\) and \\(\\lambda_{1}\\)). In practice (see Section 4.1 and Appendix E), it works best in only one experiment. Simply manually finding a value of \\(\\tau\\) that is multiplied by 2 for the second half of the network works best in all the other experiments.\n\n## Appendix E Implementation details\n\nWe use standard data augmentation and standard implementations for VGG-19, ResNet-50, ResNet-101, ResNet-110, ResNet-152 and Swin-Tiny Transformer (the same as for the other methods in Section 4.1). We use NVIDIA Tesla V100 16GB GPUs for the experiments. Training a Resnet-152 on TinyImageNet in Table 1 takes about 36 hours. Training a Resnet-152 on CIFAR100 in Table 1 takes about 11 hours. Training a ResNet-110 on STL10 in Table 3 takes about 3 hours. Training a Swin-Tiny Transformer in Table 4 take between 40 minutes and 1 hour.\n\nFor sequential and multi-lap sequential training, we use SGD with a learning rate of \\(0.007\\). With the exception of the Swin Transformer in Table 4, we use SGD for parallel training with learning rate of \\(0.003\\) in all Tables but Table 3 where the learning rate is \\(0.002\\). For the Swin Transformer in Table 4, we use the AdamW optimizer with a learning rate of \\(0.007\\) and a CosineLR scheduler.\n\nFor end-to-end training we use a learning rate of 0.1 that is divided by five at epochs 120, 160 and 200. Momentum is always 0.9. For parallel and end-to-end training, we train for 300 epochs. For sequential and multi-lap sequential training, the number of epochs varies per module (see Section G).\n\nFor experiments in Section 4.1, we use a batch size of 256, orthogonal initialization [51] with a gain of 0.1, label smoothing of 0.1 and weight decay of 0.0002. The batch size changes to 64 for Table 3 and to 1024 for Table 4.\n\nFor experiments in Section 4.3, we use a batch size of 128, orthogonal initialization with a gain of 0.05, no label smoothing and weight decay of 0.0001.\n\nIn Table 1, we use \\(\\tau=500000\\) for the first two modules and then double it for the last two modules for TRGL. In Table 2, we use \\(\\lambda_{k,1}=1,h=1\\) and \\(s=50\\) for TRGL. In Table 3, we use \\(\\tau=0.5\\) and double it at the midpoint, expect for the first line where \\(\\tau=50\\).\n\n## Appendix F Memory savings and training time\n\nWe compare in Table 7 the memory consumption of our method to that of InfoPro [60] on a ResNet-110 split into \\(K\\) modules trained in parallel on STL10 with a batch size of 64 (so the same setting as in Table 3 in Section 4.1). We report in Table 7 the memory saved as a percentage of the 6230 MiB memory required by end-to-end training with the same batch size. VanGL refers to our architecture trained without the regularization. InfoPro [60] also propose to split the network into \\(K\\) modules that have the same weight but not necessarily the same number of layers. They only implement this for \\(K{\\leq}4\\) modules. When the modules are even in weight and not in depth, we call the training methods VanGL*, TRGL* and InfoPro*. In practice, this leads to shallower early modules which slightly hurts performance according to [47]. We verify this in Table 8 (to be compared with Table 3 in Section 4.1). However, TRGL* still outperforms InfoPro and end-to-end training, and it leads to even bigger memory savings. We see in Table 7 that TRGL saves more memory than InfoPro in two out of three cases (4 and 8 modules), and about the same in the third case (16 modules), with much better test accuracy in all cases. Likewise, TRGL* is lighter than InfoPro*, with better accuracy. We also see that the added memory cost of the regularization compared to vanilla greedy learning is small. However, parallel module-wise training does slightly slow down training. Epoch time increases by \\(6\\%\\) with 2 modules and by \\(16\\%\\) with 16 modules. TRGL is only slower than VanGL by \\(2\\%\\) for for all number of modules due to the additional regularization term. This is comparable to InfoPro which report a time overhead between 1 and \\(27\\%\\) compared to end-to-end training.\n\nNote that methods DDG [32] and FR [31], being delayed gradient methods and not module-wise training methods, do no save memory (they actually increase memory usage, see FR [31]). Sedona [47] also does not claim to save memory, as their first module (the heaviest) is deeper than the others, but rather to speed up computation. Finally, DGL [8] is architecture-wise essentially identical to VanGL and consumes the same memory.\n\n## Appendix G Sequential full block-wise training\n\nTo show that our method works well with all types of module-wise training when using few modules, we train a ResNet-101 split in 2 modules on CIFAR100, sequentially and multi-lap sequentially. Results are in Table 9. We see that our idea of multi-lap sequential training adds one percentage point of accuracy to sequential training, and that the regularization further improves the accuracy by about half a percentage point. As only one module has to be trained at a time, these two training methods require only around half the memory end-to-end training requires (the size of the heaviest module and its classifier more exactly).\n\n\\begin{table}\n\\begin{tabular}{c c c c c c c} \\hline \\hline  & \\multicolumn{4}{c}{Equally deep modules} & \\multicolumn{4}{c}{Equally heavy modules} \\\\ \\cline{2-7} \\(K\\) & Par VanGL & Par TRGL (ours) & InfoPro & Par VanGL* & Par TRGL* (ours) & InfoPro* \\\\ \\hline\n4 & 27\\% (77.11) & 24\\% (**79.72**) & 18\\% (78.72) & 41\\% (77.14) & 39\\% (**78.94**) & 33\\% (78.78) \\\\\n8 & 50\\% (75.71) & 48\\% (**77.82**) & 37\\% (76.40) & & & \\\\\n16 & 61\\% (73.57) & 58\\% (**77.22**) & 59\\% (73.95) & & & \\\\ \\hline \\hline \\end{tabular}\n\\end{table}\nTable 7: Memory savings using a ResNet-110 on STL10 split into \\(K\\) modules trained in parallel with a batch size of 64, as a percentage of the weight of end-to-end training. Average test accuracy over 5 runs is between brackets. Test accuracy of end-to-end training is 77.73\\(\\%\\).\n\n\\begin{table}\n\\begin{tabular}{c c c c} \\hline \\hline \\(K\\) & Par VanGL* & Par TRGL* (ours) & InfoPro* \\\\ \\hline\n2 & 79.05 \\(\\pm\\) 1.33 & **79.47**\\(\\pm\\) 1.36 & 79.05 \\(\\pm\\) 0.57 \\\\\n4 & 77.14 \\(\\pm\\) 1.23 & **78.94**\\(\\pm\\) 1.13 & 78.78 \\(\\pm\\) 0.72 \\\\ \\hline \\hline \\end{tabular}\n\\end{table}\nTable 8: Test accuracy of parallel (Par) TRGL* with \\(K\\) modules (average and 95\\(\\%\\) confidence interval over 5 runs) on a ResNet-110 trained on STL10, compared to InfoPro* and E2E training from Table 3 in [60]\n\n\\begin{table}\n\\begin{tabular}{c c c c} \\hline \\hline Seq VanGL & Seq TRGL & MLS VanGL & MLS TRGL \\\\ \\hline\n73.31 & 73.61 & 74.34 & **74.78** \\\\ \\hline \\hline \\end{tabular}\n\\end{table}\nTable 9: Test accuracy of sequential (Seq) and multi-lap sequential (MLS) TRGL and VanGL with 2 modules on CIFAR100 using ResNet-101 (average of 2 runs).\n\nWe now focus on full block-wise training, meaning that each module is a single ResBlock, mostly sequentially. We propose here to use shallower and initially wider ResNets with a downsampling and 256 filters initially and a further downsampling and doubling of the number of filters at the midpoint, no matter the depth. In these ResNets, we use the ResBlock from [27] with two convolutional layers. If such a network is divided in \\(K\\) modules of \\(M\\) ResBlocks each, we call the network a \\(K\\)\\(-\\)\\(M\\) ResNet. These wider shallower architectures are well-adapted to layer-wise training as seen in [7]. We check in Table 10 that this architecture works well with parallel module-wise training by comparing favorably on CIFAR10 ([37]) a 2-7 ResNet with DGL, InfoPro ([60]) and DDG [32]. The 2-7 ResNet has 45 millions parameters, which is about the same as the ResNet-110 divided in two used by the other methods, and performs better when trained in parallel.\n\nWe now train a 10-block ResNet block-wise on CIFAR100 (a 10-1 ResNet in our notations). We report even the small improvements in accuracy to show that our method works in all settings (parallel or sequential with many or few splits), which other methods don't do. For sequential training, block \\(k\\) is trained for \\(50\\)\\(+\\)\\(10k\\) epochs where \\(0\\)\\(\\leq\\)\\(k\\)\\(\\leq\\)\\(10\\), block 0 being the encoder. This idea of increasing the number of epochs along with the depth is found in [43]. For MLS training, block \\(k\\) is trained for \\(10\\)\\(+\\)\\(2k\\) epochs, and this is repeated for 5 laps. In block-wise training, the last block does not always perform the best and we report the accuracy of the best block. In Table 11, we see that MLS training improves the test accuracy of sequential training by around \\(0.8\\) percentage points when the training dataset is full, but works less well on small training sets. Of the two, the regularization mainly improves the test accuracy of MLS training. The improvement increases as the training set gets smaller and reaches 1 percentage point. That is also the case for parallel module-wise training in Table 12, which already performs quite close to end-to-end training in the full data regime and much better in the small data regime. Combining the multi-lap trick and the regularization improves the performance of sequential training by 1.2 percentage points.\n\n\\begin{table}\n\\begin{tabular}{c c c c c} \\hline \\hline Parallel VanGL (ours) & Parallel TRGL (ours) & DGL & DDG & InfoPro \\\\ \\hline\n94.01 \\(\\pm\\).17 & **94.05**\\(\\pm\\).18 & 93.50 & 93.41 & 93.58 \\\\ \\hline \\hline \\end{tabular}\n\\end{table}\nTable 10: Average test accuracy and 95\\(\\%\\) confidence interval of 2-7 ResNet over 10 runs on CIFAR10 with parallel TRGL and VanGL, compared to DGL and DDG from [8] and InfoPro from [60] that split a ResNet-110 in 2 module-wise-parallel-trained modules.\n\n\\begin{table}\n\\begin{tabular}{c c c c c} \\hline \\hline Train size & \\multicolumn{1}{c}{Seq VanGL} & Seq TRGL & MLS VanGL & MLS TRGL & E2E \\\\ \\hline\n50000 & 68.74 \\(\\pm\\) 0.45 & **68.79**\\(\\pm\\) 0.56 & 69.48 \\(\\pm\\) 0.53 & **69.95**\\(\\pm\\) 0.50 & 75.85 \\(\\pm\\) 0.70 \\\\\n25000 & 60.48 \\(\\pm\\) 0.15 & **60.59**\\(\\pm\\) 0.14 & 61.33 \\(\\pm\\) 0.23 & **61.71**\\(\\pm\\) 0.32 & 65.36 \\(\\pm\\) 0.31 \\\\\n12500 & 51.64 \\(\\pm\\) 0.33 & **51.74**\\(\\pm\\) 0.26 & 51.30 \\(\\pm\\) 0.22 & **51.89**\\(\\pm\\) 0.30 & 52.39 \\(\\pm\\) 0.97 \\\\\n5000 & 36.37 \\(\\pm\\) 0.33 & **36.40**\\(\\pm\\) 0.40 & 33.68 \\(\\pm\\) 0.48 & **34.61**\\(\\pm\\) 0.59 & 36.38 \\(\\pm\\) 0.31 \\\\ \\hline \\hline \\end{tabular}\n\\end{table}\nTable 11: Average highest test accuracy and 95\\(\\%\\) confidence interval of 10-1 ResNet over 10 runs on CIFAR100 with different train sizes and sequential (Seq), multi-lap sequential (MLS) and parallel (Par) TRGL and VanGL, compared to E2E.\n\n\\begin{table}\n\\begin{tabular}{c c c c} \\hline \\hline Train size & Par VanGL & Par TRGL & E2E \\\\ \\hline\n50000 & 72.59 \\(\\pm\\) 0.40 & **72.63**\\(\\pm\\) 0.40 & 75.85 \\(\\pm\\) 0.70 \\\\\n25000 & 64.84 \\(\\pm\\) 0.19 & **65.01**\\(\\pm\\) 0.27 & 65.36 \\(\\pm\\) 0.31 \\\\\n12500 & 55.13 \\(\\pm\\) 0.24 & **55.40**\\(\\pm\\) 0.35 & 52.39 \\(\\pm\\) 0.97 \\\\\n5000 & 39.45 \\(\\pm\\) 0.23 & **40.36**\\(\\pm\\) 0.23 & 36.38 \\(\\pm\\) 0.31 \\\\ \\hline \\hline \\end{tabular}\n\\end{table}\nTable 12: Average highest test accuracy and 95\\(\\%\\) confidence interval of 10-1 ResNet over 10 runs on CIFAR100 with different train sizes and sequential (Seq), multi-lap sequential (MLS) and parallel (Par) TRGL and VanGL, compared to E2E.\n\nWe report further results of block-wise training on CIFAR10 in Table 13 and on MNIST [38] in Table 14, but now we report the accuracy of the last block. We see again greater improvement due to the regularization as the training set gets smaller, gaining up to 6 percentage points.\n\nThe \\(88\\%\\) accuracy of sequential training on CIFAR10 in Table 13 is the same as for sequential training in table 2 of [7], which is the best method for layer-wise sequential training available, with VGG networks of comparable depth and width.\n\n\\begin{table}\n\\begin{tabular}{c c c c} \\hline \\hline Train size & Seq VanGL & Seq TRGL & E2E \\\\ \\hline\n50000 & 88.02 \\(\\pm\\).18 & **88.20 \\(\\pm\\).24 & 91.88 \\(\\pm\\).18 \\\\\n25000 & 83.95 \\(\\pm\\).13 & **84.28 \\(\\pm\\).22 & 88.75 \\(\\pm\\).27 \\\\\n10000 & 76.00 \\(\\pm\\).39 & **77.18 \\(\\pm\\).34 & 82.61 \\(\\pm\\).35 \\\\\n5000 & 67.74 \\(\\pm\\).49 & **69.67 \\(\\pm\\)**.44 & 73.93 \\(\\pm\\).67 \\\\\n1000 & 45.67 \\(\\pm\\).88 & **51.34 \\(\\pm\\)**.90 & 50.63 \\(\\pm\\).98 \\\\ \\hline \\hline \\end{tabular}\n\\end{table}\nTable 13: Average last block test accuracy and 95\\(\\%\\) confidence interval of 10-1 ResNet over 10 runs on CIFAR10 with different train sizes and sequential TRGL and VanGL, compared to E2E.\n\n\\begin{table}\n\\begin{tabular}{c c c c} \\hline \\hline Train size & Par VanGL & Par TRGL & E2E \\\\ \\hline\n60000 & 99.07 \\(\\pm\\).04 & **99.08 \\(\\pm\\)**.04 & 99.30 \\(\\pm\\).03 \\\\\n30000 & 98.90 \\(\\pm\\).05 & **98.93 \\(\\pm\\)**.06 & 99.22 \\(\\pm\\).03 \\\\\n12000 & 98.52 \\(\\pm\\).06 & **98.59 \\(\\pm\\)**.06 & 98.96 \\(\\pm\\).06 \\\\\n6000 & 98.05 \\(\\pm\\).09 & **98.16 \\(\\pm\\)**.07 & 98.62 \\(\\pm\\).06 \\\\\n1500 & 96.34 \\(\\pm\\).12 & **96.91 \\(\\pm\\)**.07 & 97.19 \\(\\pm\\).08 \\\\\n1200 & 95.80 \\(\\pm\\).12 & **96.58 \\(\\pm\\)**.09 & 96.88 \\(\\pm\\).09 \\\\\n600 & 91.35 \\(\\pm\\).99 & **95.16 \\(\\pm\\)**.15 & 95.30 \\(\\pm\\).17 \\\\\n300 & 89.81 \\(\\pm\\).73 & **92.86 \\(\\pm\\)**.24 & 92.87 \\(\\pm\\).28 \\\\\n150 & 81.84 \\(\\pm\\) 1.22 & **87.48 \\(\\pm\\)**.42 & 87.82 \\(\\pm\\).59 \\\\ \\hline \\hline \\end{tabular}\n\\end{table}\nTable 14: Average last block test accuracy and 95\\(\\%\\) confidence interval of 20-1 ResNet (32 filters, fixed encoder, same classifier) over 20/50 runs on MNIST with different train sizes and parallel TRGL and VanGL, compared to E2E.\n\nFigure 3: Test accuracy after each block of 10-1 ResNet averaged over 10 runs with \\(95\\%\\) confidence intervals. Left: multi-lap sequential vanilla (VanGL, in blue) and regularized (TRGL, in red) block-wise training on \\(10\\%\\) of the CIFAR100 training set. Right: parallel vanilla (VanGL, in blue) and regularized (TRGL, in red) block-wise training on \\(10\\%\\) of CIFAR100 training set.\n\nSensitivity to hyper-parameters\n\nWe show in Figure 4 below that TRGL still performs better than VanGL (in the same setting as in Table 3 in Section 4.1) for values of \\(\\tau\\) from 0.03 to 100 and is still roughly equivalent to it for values up to 5000.\n\n## Appendix I Broader impact\n\nLess memory usage has a positive environmental impact and allows organizations with less resources to use deep learning.\n\nFigure 4: Average test accuracy over 5 runs of parallel TRGL using a ResNet110 on STL10 with 16 modules with different values of \\(\\tau\\) (in red), and of VanGL (blue line).", "response": "## Review\n\n### Summary\n\nThis paper presents a new training method for greedy layer-wise or module-wise training of neural networks, specifically targeting the stagnation problem often encountered in traditional training approaches. By introducing a module-wise regularization inspired by the minimizing movement scheme, the method significantly improves accuracy while reducing memory usage, making it suitable for constrained and on-device settings. The authors provide theoretical analyses and experimental results to support their claims, demonstrating the effectiveness of their approach across various architectures, including ResNets and Transformers. Overall, the submission shows promise but raises some questions regarding convergence analysis and the scalability of the method on larger datasets.\n\n### Strengths\n\n- The proposed method significantly reduces memory usage compared to end-to-end training, making it particularly suitable for on-device training.\n- The authors provide a theoretical analysis, showing that their approach leads to more regular and stable greedy modules.\n- The methodology effectively addresses the stagnation issue associated with layer-wise training, resulting in improved accuracy.\n- The link between their training method and optimal transport theory provides intriguing theoretical insights.\n- The paper is well-structured and generally easy to follow.\n\n### Weaknesses\n\n- The introduction of an additional regularization term increases the computational complexity and may lead to longer training times.\n- There is a lack of detailed convergence analysis, which is crucial for establishing the reliability of the proposed approach.\n- The performance gain over previous methods appears marginal, especially on smaller datasets like STL-10.\n- Concerns were raised regarding the strength of the regularization and its potential negative impact on early layers.\n- The experimental results do not sufficiently support claims regarding early overfitting and the effectiveness of the method on large-scale datasets.\n\n### Questions\n\n- What is VanGL? Is it just a module-wise training method without the regularization terms?\n- Why does the proposed method demonstrate less memory savings compared to other methods?\n- How does the proposed regularization control each module's strength, particularly with varying numbers of modules?\n- What is the wall-time training cost of the proposed method compared to existing methods?\n- Some notations in the paper are not introduced properly; for example, what is the difference between $T_k$ and $T_k^{\\tau}$?\n\n### Soundness\n\n**Score:** 3\n\n**Description:** 3 = good; the methodology is theoretically sound and shows promise, but there are concerns regarding convergence and the robustness of the results.\n\n### Presentation\n\n**Score:** 3\n\n**Description:** 3 = good; the paper is generally well-written and structured, though some convoluted notations could be improved for better clarity.\n\n### Contribution\n\n**Score:** 3\n\n**Description:** 3 = good; the paper contributes a new approach to module-wise training which could have significant implications, although further validation on larger datasets is needed.\n\n### Rating\n\n**Score:** 5\n\n**Description:** 5 = borderline accept; the paper is technically solid with reasons to accept outweighing reasons to reject, though it requires further evaluation on larger datasets.\n\n### Paper Decision\n\n**Decision:** Accept (poster)\n\n**Reasons:** The paper presents an innovative training approach that addresses significant issues in neural network training. While there are some weaknesses, particularly in the convergence analysis and the scalability of the results, the overall contributions are valuable and the authors have effectively addressed several reviewer concerns during the rebuttal phase. The strengths of the paper, including its theoretical foundation and practical applications, support the decision to accept it for presentation.\n"}
{"query": "You are a highly experienced, conscientious, and fair academic reviewer, please help me review this paper. The review should be organized into nine sections: \n1. Summary: A summary of the paper in 100-150 words.\n2. Strengths/Weaknesses/Questions: The Strengths/Weaknesses/Questions of paper, which should be listed in bullet points, with each point supported by specific examples from the article where possible.\n3. Soundness/Contribution/Presentation: Rate the paper's Soundness/Contribution/Presentation, and match this score to the corresponding description from the list below and provide the result. The possible scores and their descriptions are: \n    1 poor\n    2 fair\n    3 good\n    4 excellent\n4. Rating: Give this paper an appropriate rating, match this rating to the corresponding description from the list below and provide the result. The possible Ratings and their descriptions are: \n    1 strong reject\n    2 reject, significant issues present\n    3 reject, not good enough\n    4 possibly reject, but has redeeming facets\n    5 marginally below the acceptance threshold\n    6 marginally above the acceptance threshold\n    7 accept, but needs minor improvements \n    8 accept, good paper\n    9 strong accept, excellent work\n    10 strong accept, should be highlighted at the conference   \n5. Paper Decision: It must include the Decision itself(Accept or Reject) and the reasons for this decision, based on the criteria of originality, methodological soundness, significance of results, and clarity and logic of presentation.\n\nHere is the template for a review format, you must follow this format to output your review result:\n**Summary:**\nSummary content\n\n**Strengths:**\n- Strength 1\n- Strength 2\n- ...\n\n**Weaknesses:**\n- Weakness 1\n- Weakness 2\n- ...\n\n**Questions:**\n- Question 1\n- Question 2\n- ...\n\n**Soundness:**\nSoundness result\n\n**Presentation:**\nPresentation result\n\n**Contribution:**\nContribution result\n\n**Rating:**\nRating result\n\n**Paper Decision:**\n- Decision: Accept/Reject\n- Reasons: reasons content\n\n\nPlease ensure your feedback is objective and constructive. The paper is as follows:\n\n# Computational Complexity of Learning Neural Networks: Smoothness and Degeneracy\n\nAmit Daniely\n\nHebrew University and Google\n\namit.daniely@mail.huji.ac.il\n\n&Nathan Srebro\n\nTTI-Chicago\n\nnati@ttic.edu\n\n&Gal Vardi\n\nTTI-Chicago and Hebrew University\n\ngalvardi@ttic.edu\n\n###### Abstract\n\nUnderstanding when neural networks can be learned efficiently is a fundamental question in learning theory. Existing hardness results suggest that assumptions on both the input distribution and the network's weights are necessary for obtaining efficient algorithms. Moreover, it was previously shown that depth-\\(2\\) networks can be efficiently learned under the assumptions that the input distribution is Gaussian, and the weight matrix is non-degenerate. In this work, we study whether such assumptions may suffice for learning deeper networks and prove negative results. We show that learning depth-\\(3\\) ReLU networks under the Gaussian input distribution is hard even in the smoothed-analysis framework, where a random noise is added to the network's parameters. It implies that learning depth-\\(3\\) ReLU networks under the Gaussian distribution is hard even if the weight matrices are non-degenerate. Moreover, we consider depth-\\(2\\) networks, and show hardness of learning in the smoothed-analysis framework, where both the network parameters and the input distribution are smoothed. Our hardness results are under a well-studied assumption on the existence of local pseudorandom generators.\n\n## 1 Introduction\n\nThe computational complexity of learning neural networks has been extensively studied in recent years, and there has been much effort in obtaining both upper bounds and hardness results. Nevertheless, it is still unclear when neural networks can be learned in polynomial time, namely, under what assumptions provably efficient algorithms exist.\n\nExisting results imply hardness already for learning depth-\\(2\\) ReLU networks in the standard PAC learning framework (e.g., [31; 14]). Thus, without any assumptions on the input distribution or the network's weights, efficient learning algorithms might not be achievable. Even when assuming that the input distribution is Gaussian, strong hardness results were obtained for depth-\\(3\\) ReLU networks [16; 11], suggesting that assumptions merely on the input distribution might not suffice. Also, a hardness result by Daniely and Vardi [15] shows that strong assumptions merely on the network's weights (without restricting the input distribution) might not suffice even for efficiently learning depth-\\(2\\) networks. The aforementioned hardness results hold already for _improper learning_, namely, where the learning algorithm is allowed to return a hypothesis that does not belong to the considered hypothesis class. Thus, a combination of assumptions on the input distribution and the network's weights seems to be necessary for obtaining efficient algorithms.\n\nSeveral polynomial-time algorithms for learning depth-\\(2\\) neural networks have been obtained, under assumptions on the input distribution and the network's weights [6; 28; 43; 20; 7; 21]. In these works, it is assumed that the weight matrices are non-degenerate. That is, they either assume that the condition number of the weight matrix is bounded, or some similar non-degeneracy assumption. Specifically, Awasthi et al. [6] gave an efficient algorithm for learning depth-\\(2\\) (one-hidden-layer) ReLU networks, that may include bias terms in the hidden neurons, under the assumption that the input distribution is Gaussian, and the weight matrix is non-degenerate. The non-degeneracy assumption holds w.h.p. if we add a small random noise to any weight matrix, and hence their result implies efficient learning of depth-\\(2\\) ReLU networks under the Gaussian distribution in the _smoothed-analysis_ framework.\n\nThe positive results on depth-\\(2\\) networks suggest the following question:\n\n_Is there an efficient algorithm for learning ReLU networks of depth larger than \\(2\\) under the Gaussian distribution, where the weight matrices are non-degenerate, or in the smoothed-analysis framework where the network's parameters are smoothed?_\n\nIn this work, we give a negative answer to this question, already for depth-\\(3\\) networks1. We show that learning depth-\\(3\\) ReLU networks under the Gaussian distribution is hard even in the smoothed-analysis framework, where a random noise is added to the network's parameters. As a corollary, we show that learning depth-\\(3\\) ReLU networks under the Gaussian distribution is hard even if the weight matrices are non-degenerate. Our hardness results are under a well-studied cryptographic assumption on the existence of _local pseudorandom generators (PRG)_ with polynomial stretch.\n\nFootnote 1: We note that in our results the neural networks have the ReLU activation also in the output neuron.\n\nMotivated by the existing positive results on smoothed-analysis in depth-\\(2\\) networks, we also study whether learning depth-\\(2\\) networks with smoothed parameters can be done under weak assumptions on the input distribution. Specifically, we consider the following question:\n\n_Is there an efficient algorithm for learning depth-\\(2\\) ReLU networks in the smoothed-analysis framework, where both the network's parameters and the input distribution are smoothed?_\n\nWe give a negative answer to this question, by showing hardness of learning depth-\\(2\\) ReLU networks where a random noise is added to the network's parameters, and the input distribution on \\(\\mathbb{R}^{d}\\) is obtained by smoothing an i.i.d. Bernoulli distribution on \\(\\{0,1\\}^{d}\\). This hardness result is also under the assumption on the existence of local PRGs.\n\nRelated workHardness of learning neural networksHardness of improperly learning depth-\\(2\\) neural networks follows from hardness of improperly learning DNFs or intersections of halfspaces, since these classes can be expressed by depth-\\(2\\) networks. Klivans and Sherstov [31] showed, assuming the hardness of the shortest vector problem, that learning intersections of halfspaces is hard. Hardness of learning DNF formulas is implied by Applebaum et al. [4] under a combination of two assumptions: the first is related to the planted dense subgraph problem in hypergraphs, and the second is related to local PRGs. Daniely and Shalev-Shwartz [14] showed hardness of learning DNFs under a common assumption, namely, that refuting a random \\(K\\)-SAT formula is hard. All of the above results are distribution-free, namely, they do not imply hardness of learning neural networks under some specific distribution.\n\nApplebaum and Raykov [3] showed, under an assumption on a specific candidate for Goldreich's PRG (i.e., based on a predicate called \\(\\mathrm{XOR}\\)-\\(\\mathrm{MAJ}\\)), that learning depth-\\(3\\) Boolean circuits under the uniform distribution on the hypercube is hard. Daniely and Vardi [16] proved distribution-specific hardness of learning Boolean circuits of depth-\\(2\\) (namely, DNFs) and depth-\\(3\\), under the assumpion on the existence of local PRGs that we also use in this work. For DNF formulas, they showed hardness of learning under a distribution where each component is drawn i.i.d. from a Bernoulli distribution (which is non-uniform). For depth-\\(3\\) Boolean circuits, they showed hardness of learning under the uniform distribution on the hypercube. Since the Boolean circuits can be expressed by ReLU networks of the same depth, these results readily translate to distribution-specific hardness of learning neural networks. Chen et al. [11] showed hardness of learning depth-\\(2\\) neural networks under the uniform distribution on the hypercube, based on an assumption on the hardness of the _Learning with Rounding (LWR)_ problem. Note that the input distributions in the above results are supported on the hypercube, and they do not immediately imply hardness of learning neural networks under continuous distributions.\n\nWhen considering the computational complexity of learning neural networks, perhaps the most natural choice of an input distribution is the standard Gaussian distribution. Daniely and Vardi [16] established hardness of learning depth-\\(3\\) networks under this distribution, based on the assumption on the existence of local PRGs. Chen et al. [11] also showed hardness of learning depth-\\(3\\) networks under the Gaussian distribution, but their result holds already for networks that do not have an activation function in the output neuron, and it is based on the LWR assumption. They also showed hardness of learning constant depth ReLU networks from label queries (i.e., where the learner has the ability to query the value of the target network at any desired input) under the Gaussian distribution, based either on the decisional Diffie-Hellman or the Learning with Errors assumptions.\n\nThe above results suggest that assumptions on the input distribution might not suffice for achieving an efficient algorithm for learning depth-\\(3\\) neural networks. A natural question is whether assumptions on the network weights may suffice. Daniely and Vardi [15] showed (under the assumption that refuting a random \\(K\\)-SAT formula is hard) that distribution-free learning of depth-\\(2\\) neural networks is hard already if the weights are drawn from some \"natural\" distribution or satisfy some \"natural\" properties. Thus, if we do not impose any assumptions on the input distribution, then even very strong assumptions on the network's weights do not suffice for efficient learning.\n\nSeveral works in recent years have shown hardness of distribution-specific learning shallow neural networks using gradient-methods or statistical query (SQ) algorithms [38, 39, 41, 24, 18, 11]. It is worth noting that while the SQ framework captures some variants of the gradient-descent algorithm, it does not capture, for example, stochastic gradient-descent (SGD), which examines training points individually (see a discussion in [24]).\n\nWe emphasize that none of the above distribution-specific hardness results for neural networks (either for improper learning or SQ learning) holds in the smoothed analysis framework or for non-degenerate weights.\n\nLearning neural networks in polynomial time.Awasthi et al. [6] gave a polynomial-time algorithm for learning depth-\\(2\\) (one-hidden-layer) ReLU networks, under the assumption that the input distribution is Gaussian, and the weight matrix of the target network is non-degenerate. Their algorithm is based on tensor decomposition, and it can handle bias terms in the hidden layer. Their result also implies that depth-\\(2\\) ReLU networks with Gaussian inputs can be learned efficiently under the smoothed-analysis framework. Our work implies that such a result might not be possible in depth-\\(3\\) networks (with activation in the output neuron). Prior to [6], several works gave polynomial time algorithms for learning depth-\\(2\\) neural networks where the input distribution is Gaussian and the weight matrix is non-degenerate [28, 43, 20, 21, 7], but these works either do not handle the presence of bias terms or do not handle the ReLU activation. Some of the aforementioned works consider networks with multiple outputs, and allow certain non-Gaussian input distributions. Provable guarantees for learning neural networks in super-polynomial time were given in [18, 12, 17, 22, 41, 42, 23].\n\n## 2 Preliminaries\n\nNotations.We use bold-face letters to denote vectors, e.g., \\(\\mathbf{x}=(x_{1},\\ldots,x_{d})\\). For a vector \\(\\mathbf{x}\\) and a sequence \\(S=(i_{1},\\ldots,i_{k})\\) of \\(k\\) indices, we let \\(\\mathbf{x}_{S}=(x_{i_{1}},\\ldots,x_{i_{k}})\\), i.e., the restriction of \\(\\mathbf{x}\\) to the indices \\(S\\). We denote by \\(\\mathbb{1}[\\cdot]\\) the indicator function, for example \\(\\mathbb{1}[t\\geq 5]\\) equals \\(1\\) if \\(t\\geq 5\\) and \\(0\\) otherwise. For an integer \\(d\\geq 1\\) we denote \\([d]=\\{1,\\ldots,d\\}\\). We denote by \\(\\mathcal{N}(0,\\sigma^{2})\\) the normal distribution with mean \\(0\\) and variance \\(\\sigma^{2}\\), and by \\(\\mathcal{N}(\\mathbf{0},\\Sigma)\\) the multivariate normal distribution with mean \\(\\mathbf{0}\\) and covariance matrix \\(\\Sigma\\). The identity matrix of size \\(d\\) is denoted by \\(I_{d}\\). For \\(\\mathbf{x}\\in\\mathbb{R}^{d}\\) we denote by \\(\\|\\mathbf{x}\\|\\) the Euclidean norm. We use \\(\\mathrm{poly}(a_{1},\\ldots,a_{n})\\) to denote a polynomial in \\(a_{1},\\ldots,a_{n}\\).\n\nLocal pseudorandom generators.An \\((n,m,k)\\)-hypergraph is a hypergraph over \\(n\\) vertices \\([n]\\) with \\(m\\) hyperedges \\(S_{1},\\ldots,S_{m}\\), each of cardinality \\(k\\). Each hyperedge \\(S=(i_{1},\\ldots,i_{k})\\) is ordered, and all the \\(k\\) members of a hyperedge are distinct. We let \\(\\mathcal{G}_{n,m,k}\\) be the distribution over such hypergraphs in which a hypergraph is chosen by picking each hyperedge uniformly and independently at random among all the possible \\(n\\cdot(n-1)\\cdot\\ldots\\cdot(n-k+1)\\) ordered hyperedges. Let\\(\\{0,1\\}\\) be a predicate, and let \\(G\\) be a \\((n,m,k)\\)-hypergraph. We call _Goldreich's pseudorandom generator (PRG)_[25] the function \\(f_{P,G}:\\{0,1\\}^{n}\\to\\{0,1\\}^{m}\\) such that for \\(\\mathbf{x}\\in\\{0,1\\}^{n}\\), we have \\(f_{P,G}(\\mathbf{x})=(P(\\mathbf{x}_{\\mathcal{S}_{1}}),\\ldots,P(\\mathbf{x}_{S_{m} }))\\). The integer \\(k\\) is called the _locality_ of the PRG. If \\(k\\) is a constant then the PRG and the predicate \\(P\\) are called _local_. We say that the PRG has _polynomial stretch_ if \\(m=n^{s}\\) for some constant \\(s>1\\). We let \\(\\mathcal{F}_{P,n,m}\\) denote the collection of functions \\(f_{P,G}\\) where \\(G\\) is an \\((n,m,k)\\)-hypergraph. We sample a function from \\(\\mathcal{F}_{P,n,m}\\) by choosing a random hypergraph \\(G\\) from \\(\\mathcal{G}_{n,m,k}\\).\n\nWe denote by \\(G\\stackrel{{ R}}{{\\leftarrow}}\\mathcal{G}_{n,m,k}\\) the operation of sampling a hypergraph \\(G\\) from \\(\\mathcal{G}_{n,m,k}\\), and by \\(\\mathbf{x}\\stackrel{{ R}}{{\\leftarrow}}\\{0,1\\}^{n}\\) the operation of sampling \\(\\mathbf{x}\\) from the uniform distribution on \\(\\{0,1\\}^{n}\\). We say that \\(\\mathcal{F}_{P,n,m}\\) is \\(\\varepsilon\\)-pseudorandom generator (\\(\\varepsilon\\)-PRG) if for every polynomial-time probabilistic algorithm \\(\\mathcal{A}\\) the _distinguishing advantage_\n\n\\[\\left|\\Pr_{G\\stackrel{{ R}}{{\\leftarrow}}\\mathcal{G}_{n,m,k}, \\mathbf{x}\\stackrel{{ R}}{{\\leftarrow}}\\{0,1\\}^{n}}[\\mathcal{A}( G,f_{P,G}(\\mathbf{x}))=1]-\\Pr_{G\\stackrel{{ R}}{{\\leftarrow}}\\mathcal{G}_{n,m,k},\\mathbf{y}\\stackrel{{ R}}{{\\leftarrow}}\\{0,1\\}^{m}}[\\mathcal{A}(G,\\mathbf{y})=1]\\right|\\]\n\nis at most \\(\\varepsilon\\). Thus, the distinguisher \\(\\mathcal{A}\\) is given a random hypergraph \\(G\\) and a string \\(\\mathbf{y}\\in\\{0,1\\}^{m}\\), and its goal is to distinguish between the case where \\(\\mathbf{y}\\) is chosen at random, and the case where \\(\\mathbf{y}\\) is a random image of \\(f_{P,G}\\).\n\nOur assumption is that local PRGs with polynomial stretch and constant distinguishing advantage exist:\n\n**Assumption 2.1**.: _For every constant \\(s>1\\), there exists a constant \\(k\\) and a predicate \\(P:\\{0,1\\}^{k}\\to\\{0,1\\}\\), such that \\(\\mathcal{F}_{P,n,n^{s}}\\) is \\(\\frac{1}{3}\\)-PRG._\n\nWe remark that the same assumption was used by Daniely and Vardi [16] to show hardness-of-learning results. Local PRGs have been extensively studied in the last two decades. In particular, local PRGs with polynomial stretch have shown to have remarkable applications, such as secure-computation with constant computational overhead [27, 5], and general-purpose obfuscation based on constant degree multilinear maps (cf. [33, 34]). Significant evidence for Assumption 2.1 was shown in Applebaum [1]. Moreover, a concrete candidate for a local PRG, based on the XOR-MAJ predicate, was shown to be secure against all known attacks [2, 13, 35, 3]. See [16] for further discussion on the assumption, and on prior work regarding the relation between Goldreich's PRG and hardness of learning.\n\nNeural networks.We consider feedforward ReLU networks. Starting from an input \\(\\mathbf{x}\\in\\mathbb{R}^{d}\\), each layer in the network is of the form \\(\\mathbf{z}\\mapsto\\sigma(W_{i}\\mathbf{z}+\\mathbf{b}_{i})\\), where \\(\\sigma(a)=[a]_{+}=\\max\\{0,a\\}\\) is the ReLU activation which applies to vectors entry-wise, \\(W_{i}\\) is the weight matrix, and \\(\\mathbf{b}_{i}\\) are the bias terms. The _weights vector_ of the \\(j\\)-th neuron in the \\(i\\)-th layer is the \\(j\\)-th row of \\(W_{i}\\), and its _outgoing-weights vector_ is the \\(j\\)-th column of \\(W_{i+1}\\). We define the _depth_ of the network as the number of layers. Unless stated otherwise, the output neuron also has a ReLU activation function. Note that a depth-\\(k\\) network with activation in the output neuron has \\(k\\) non-linear layers. A neuron which is not an input or output neuron is called a _hidden neuron_. We sometimes consider neural networks with multiple outputs. The parameters of the neural network is the set of its weight matrices and bias vectors. We often view the parameters as a vector \\(\\boldsymbol{\\theta}\\in\\mathbb{R}^{p}\\) obtained by concatenating these matrices and vectors. For \\(B\\geq 0\\), we say that the parameters are \\(B\\)-bounded if the absolute values of all weights and biases are at most \\(B\\).\n\nLearning neural networks and the smoothed-analysis framework.We first define neural networks learning under the standard PAC framework:\n\n**Definition 2.1** (Distribution-specific PAC learning).: _Learning depth-\\(k\\) neural networks under an input distribution \\(\\mathcal{D}\\) on \\(\\mathbb{R}^{d}\\) is defined by the following framework:_\n\n1. _An adversary chooses a set of_ \\(B\\)_-bounded parameters_ \\(\\boldsymbol{\\theta}\\in\\mathbb{R}^{p}\\) _for a depth-_\\(k\\) _neural network_ \\(N_{\\boldsymbol{\\theta}}:\\mathbb{R}^{d}\\to\\mathbb{R}\\)_, as well as some_ \\(\\epsilon>0\\)_._\n2. _Consider an examples oracle, such that each example_ \\((\\mathbf{x},y)\\in\\mathbb{R}^{d}\\times\\mathbb{R}\\) _is drawn i.i.d. with_ \\(\\mathbf{x}\\sim\\mathcal{D}\\) _and_ \\(y=N_{\\boldsymbol{\\theta}}(\\mathbf{x})\\)_. Then, given access to the examples oracle, the goal of the learning algorithm_ \\(\\mathcal{L}\\) _is to return with probability at least_ \\(\\frac{3}{4}\\) _a hypothesis_ \\(h:\\mathbb{R}^{d}\\to\\mathbb{R}\\) _such that_ \\(\\mathbb{E}_{\\mathbf{x}\\sim\\mathcal{D}}\\left[\\left(h(\\mathbf{x})-N_{ \\boldsymbol{\\theta}}(\\mathbf{x})\\right)^{2}\\right]\\leq\\epsilon\\)_. We say that_ \\(\\mathcal{L}\\) _is_ efficient _if it runs in time_ \\(\\mathrm{poly}(d,p,B,1/\\epsilon)\\)We consider learning in the _smoothed-analysis_ framework [40], which is a popular paradigm for analyzing non-worst-case computational complexity [36]. The smoothed-analysis framework has been successfully applied to many learning problems (e.g., [6; 21; 30; 8; 9; 26; 19; 29; 10]). In the smoothed-analysis setting, the target network is not purely controlled by an adversary. Instead, the adversary can first generate an arbitrary network, and the parameters for this network (i.e., the weight matrices and bias terms) will be randomly perturbed to yield a perturbed network. The algorithm only needs to work with high probability on the perturbed network. This limits the power of the adversary and prevents it from creating highly degenerate cases. Formally, we consider the following framework (we note that a similar model was considered in [6; 21]):\n\n**Definition 2.2** (Learning with smoothed parameters).: _Learning depth-\\(k\\) neural networks with smoothed parameters under an input distribution \\(\\mathcal{D}\\) is defined as follows:_\n\n1. _An adversary chooses a set of_ \\(B\\)_-bounded parameters_ \\(\\boldsymbol{\\theta}\\in\\mathbb{R}^{p}\\) _for a depth-_\\(k\\) _neural network_ \\(N_{\\boldsymbol{\\theta}}:\\mathbb{R}^{d}\\rightarrow\\mathbb{R}\\)_, as well as some_ \\(\\tau,\\epsilon>0\\)_._\n2. _A perturbed set of parameters_ \\(\\hat{\\boldsymbol{\\theta}}\\) _is obtained by a random perturbation to_ \\(\\boldsymbol{\\theta}\\)_, namely,_ \\(\\hat{\\boldsymbol{\\theta}}=\\boldsymbol{\\theta}+\\boldsymbol{\\xi}\\) _for_ \\(\\boldsymbol{\\xi}\\sim\\mathcal{N}(\\mathbf{0},\\tau^{2}I_{p})\\)_._\n3. _Consider an examples oracle, such that each example_ \\((\\mathbf{x},y)\\in\\mathbb{R}^{d}\\times\\mathbb{R}\\) _is drawn i.i.d. with_ \\(\\mathbf{x}\\sim\\mathcal{D}\\) _and_ \\(y=N_{\\boldsymbol{\\hat{\\theta}}}(\\mathbf{x})\\)_. Then, given access to the examples oracle, the goal of the learning algorithm_ \\(\\mathcal{L}\\) _is to return with probability at least_ \\(\\frac{3}{4}\\) _(over the random perturbation_ \\(\\boldsymbol{\\xi}\\) _and the internal randomness of_ \\(\\mathcal{L}\\)_) a hypothesis_ \\(h:\\mathbb{R}^{d}\\rightarrow\\mathbb{R}\\) _such that_ \\(\\mathbb{E}_{\\mathbf{x}\\sim\\mathcal{D}}\\left[\\left(h(\\mathbf{x})-N_{\\boldsymbol{ \\hat{\\theta}}}(\\mathbf{x})\\right)^{2}\\right]\\leq\\epsilon\\)_._\n\n_We say that \\(\\mathcal{L}\\) is efficient if it runs in time \\(\\mathrm{poly}(d,p,B,1/\\epsilon,1/\\tau)\\)._\n\nFinally, we also consider a setting where both the parameters and the input distribution are smoothed:\n\n**Definition 2.3** (Learning with smoothed parameters and inputs).: _Learning depth-\\(k\\) neural networks with smoothed parameters and inputs under an input distribution \\(\\mathcal{D}\\) is defined as follows:_\n\n1. _An adversary chooses a set of_ \\(B\\)_-bounded parameters_ \\(\\boldsymbol{\\theta}\\in\\mathbb{R}^{p}\\) _for a depth-_\\(k\\) _neural network_ \\(N_{\\boldsymbol{\\theta}}:\\mathbb{R}^{d}\\rightarrow\\mathbb{R}\\)_, as well as some_ \\(\\tau,\\omega,\\epsilon>0\\)_._\n2. _A perturbed set of parameters_ \\(\\hat{\\boldsymbol{\\theta}}\\) _is obtained by a random perturbation to_ \\(\\boldsymbol{\\theta}\\)_, namely,_ \\(\\hat{\\boldsymbol{\\theta}}=\\boldsymbol{\\theta}+\\boldsymbol{\\xi}\\) _for_ \\(\\boldsymbol{\\xi}\\sim\\mathcal{N}(\\mathbf{0},\\tau^{2}I_{p})\\)_. Moreover, a smoothed input distribution_ \\(\\hat{\\mathcal{D}}\\) _is obtained from_ \\(\\mathcal{D}\\)_, such that_ \\(\\hat{\\mathbf{x}}\\sim\\hat{\\mathcal{D}}\\) _is chosen by drawing_ \\(\\mathbf{x}\\sim\\mathcal{D}\\) _and adding a random perturbation from_ \\(\\mathcal{N}(\\mathbf{0},\\omega^{2}I_{d})\\)_._\n3. _Consider an examples oracle, such that each example_ \\((\\mathbf{x},y)\\in\\mathbb{R}^{d}\\times\\mathbb{R}\\) _is drawn i.i.d. with_ \\(\\mathbf{x}\\sim\\hat{\\mathcal{D}}\\) _and_ \\(y=N_{\\hat{\\boldsymbol{\\theta}}}(\\mathbf{x})\\)_. Then, given access to the examples oracle, the goal of the learning algorithm_ \\(\\mathcal{L}\\) _is to return with probability at least_ \\(\\frac{3}{4}\\) _(over the random perturbation_ \\(\\boldsymbol{\\xi}\\) _and the internal randomness of_ \\(\\mathcal{L}\\)_) a hypothesis_ \\(h:\\mathbb{R}^{d}\\rightarrow\\mathbb{R}\\) _such that_ \\(\\mathbb{E}_{\\mathbf{x}\\sim\\hat{\\mathcal{D}}}\\left[\\left(h(\\mathbf{x})-N_{\\hat{ \\boldsymbol{\\theta}}}(\\mathbf{x})\\right)^{2}\\right]\\leq\\epsilon\\)_._ _We say that_ \\(\\mathcal{L}\\) _is efficient if it runs in time_ \\(\\mathrm{poly}(d,p,B,1/\\epsilon,1/\\tau,1/\\omega)\\)_._\n\nWe emphasize that all of the above definitions consider learning in the distribution-specific setting. Thus, the learning algorithm may depend on the specific input distribution \\(\\mathcal{D}\\).\n\n## 3 Results\n\nAs we discussed in the introduction, there exist efficient algorithms for learning depth-\\(2\\) (one-hidden-layer) ReLU networks with smoothed parameters under the Gaussian distribution. We now show that such a result may not be achieved for depth-\\(3\\) networks:\n\n**Theorem 3.1**.: _Under Assumption 2.1, there is no efficient algorithm that learns depth-\\(3\\) networks with smoothed parameters (in the sense of Definition 2.2) under the standard Gaussian distribution._\n\nWe prove the theorem in Section 4. Next, we conclude that learning depth-\\(3\\) neural networks under the Gaussian distribution on \\(\\mathbb{R}^{d}\\) is hard in the standard PAC framework even if all weight matrices are non-degenerate, namely, when the minimal singular values of the weight matrices are lower bounded by \\(1/\\,\\mathrm{poly}(d)\\). As we discussed in the introduction, in one-hidden-layer networks with similar assumptions there exist efficient learning algorithms.\n\n**Corollary 3.1**.: _Under Assumption 2.1, there is no efficient algorithm that learns depth-\\(3\\) networks (in the sense of Definition 2.1) under the standard Gaussian distribution on \\(\\mathbb{R}^{d}\\), even if the smallest singular value of each weight matrix is at least \\(1/\\operatorname{poly}(d)\\)._\n\nThe proof of the above corollary follows from Theorem 3.1, using the fact that by adding a small random perturbation to the weight matrices, we get w.h.p. non-degenerate matrices. Hence, an efficient algorithm that learns non-degenerate networks suffices for obtaining an efficient algorithm that learns under the smoothed analysis framework. See Appendix B for the formal proof.\n\nThe above hardness results consider depth-\\(3\\) networks that include activation in the output neuron. Thus, the networks have three non-linear layers. We remark that these results readily imply hardness also for depth-\\(4\\) networks without activation in the output, and hardness for depth-\\(k\\) networks for any \\(k>3\\). We also note that the hardness results hold already for networks where all hidden layers are of the same width, e.g., where all layers are of width \\(d\\).\n\nTheorem 3.1 gives a strong limitation on learning depth-\\(3\\) networks in the smoothed-analysis framework. Motivated by existing positive results on smoothed-analysis in depth-\\(2\\) networks, we now study whether learning depth-\\(2\\) networks with smoothed parameters can be done under weak assumptions on the input distribution. Specifically, we consider the case where both the parameters and the input distribution are smoothed. We show that efficiently learning depth-\\(2\\) networks may not be possible with smoothed parameters where the input distribution on \\(\\mathbb{R}^{d}\\) is obtained by smoothing an i.i.d. Bernoulli distribution on \\(\\{0,1\\}^{d}\\).\n\n**Theorem 3.2**.: _Under Assumption 2.1, there is no efficient algorithm that learns depth-\\(2\\) networks with smoothed parameters and inputs (in the sense of Definition 2.3), under the distribution \\(\\mathcal{D}\\) on \\(\\{0,1\\}^{d}\\) where each coordinate is drawn i.i.d. from a Bernoulli distribution which takes the value \\(0\\) with probability \\(\\frac{1}{\\sqrt{d}}\\)._\n\nWe prove the theorem in Appendix C. The proof follows from similar ideas to the proof of Theorem 3.1, which we discuss in the next section.\n\n## 4 Proof of Theorem 3.1\n\nThe proof builds on a technique from Daniely and Vardi [16]. It follows by showing that an efficient algorithm for learning depth-\\(3\\) neural networks with smoothed parameters under the Gaussian distribution can be used for breaking a local PRG. Intuitively, the main challenge in our proof in comparison to [16] is that our reduction must handle the random noise which is added to the parameters. Specifically, [16] define a certain examples oracle, and show that the examples returned by the oracle are realizable by some neural network which depends on the unknown \\(\\mathbf{x}\\in\\{0,1\\}^{n}\\) used by the PRG. Since the network depends on this unknown \\(\\mathbf{x}\\), some of the parameters of the network are unknown, and hence it is non-trivial how to define an examples oracle which is realizable by a perturbed network. Moreover, we need to handle this random perturbation without increasing the network's depth. For these reasons, our reduction is significantly different from [16].\n\nWe now provide the formal proof. The proof relies on several lemmas which we prove in Appendix A. For a sufficiently large \\(n\\), let \\(\\mathcal{D}\\) be the standard Gaussian distribution on \\(\\mathbb{R}^{n^{2}}\\). Assume that there is a \\(\\operatorname{poly}(n)\\)-time algorithm \\(\\mathcal{L}\\) that learns depth-\\(3\\) neural networks with at most \\(n^{2}\\) hidden neurons and parameter magnitudes bounded by \\(n^{3}\\), with smoothed parameters, under the distribution \\(\\mathcal{D}\\), with \\(\\epsilon=\\frac{1}{n}\\), and \\(\\tau=1/\\operatorname{poly}(n)\\) that we will specify later. Let \\(m(n)\\leq\\operatorname{poly}(n)\\) be the sample complexity of \\(\\mathcal{L}\\), namely, \\(\\mathcal{L}\\) uses a sample of size at most \\(m(n)\\) and returns w.p. at least \\(\\frac{3}{4}\\) a hypothesis \\(h\\) with \\(\\mathbb{E}_{\\mathbf{z}\\sim\\mathcal{D}}\\left[\\left(h(\\mathbf{z})-N_{\\mathbf{\\hat{ \\theta}}}(\\mathbf{z})\\right)^{2}\\right]\\leq\\epsilon=\\frac{1}{n}\\), where \\(N_{\\mathbf{\\hat{\\theta}}}\\) is the perturbed network. Let \\(s>1\\) be a constant such that \\(n^{s}\\geq m(n)+n^{3}\\) for every sufficiently large \\(n\\). By Assumption 2.1, there exists a constant \\(k\\) and a predicate \\(P:\\{0,1\\}^{k}\\rightarrow\\{0,1\\}\\), such that \\(\\mathcal{F}_{P,n,n^{s}}\\) is \\(\\frac{1}{3}\\)-PRG. We will show an efficient algorithm \\(\\mathcal{A}\\) with distinguishing advantage greater than \\(\\frac{1}{3}\\) and thus reach a contradiction.\n\nThroughout this proof, we will use the following notations. For a hyperedge \\(S=(i_{1},\\ldots,i_{k})\\) we denote by \\(\\mathbf{z}^{S}\\in\\{0,1\\}^{kn}\\) the following encoding of \\(S\\): the vector \\(\\mathbf{z}^{S}\\) is a concatenation of \\(k\\) vectors in \\(\\{0,1\\}^{n}\\), such that the \\(j\\)-th vector has \\(0\\) in the \\(i_{j}\\)-th coordinate and \\(1\\) elsewhere. Thus, \\(\\mathbf{z}^{S}\\) consists of \\(k\\) size-\\(n\\) slices, each encoding a member of \\(S\\). For \\(\\mathbf{z}\\in\\{0,1\\}^{kn}\\), \\(i\\in[k]\\) and \\(j\\in[n]\\), we denote \\(z_{i,j}=z_{(i-1)n+j}\\). That is, \\(z_{i,j}\\) is the \\(j\\)-th component in the \\(i\\)-th slice in \\(\\mathbf{z}\\). For \\(\\mathbf{x}\\in\\{0,1\\}^{n}\\), let\\(P_{\\mathbf{x}}:\\{0,1\\}^{kn}\\to\\{0,1\\}\\) be such that for every hyperedge \\(S\\) we have \\(P_{\\mathbf{x}}(\\mathbf{z}^{S})=P(\\mathbf{x}_{S})\\). Let \\(c\\) be such that \\(\\Pr_{t\\sim\\mathcal{N}(0,1)}[t\\leq c]=\\frac{1}{n}\\). Let \\(\\mu\\) be the density of \\(\\mathcal{N}(0,1)\\), let \\(\\mu_{-}(t)=n\\cdot\\mathbbm{1}[t\\leq c]\\cdot\\mu(t)\\), and let \\(\\mu_{+}=\\frac{n}{n-1}\\cdot\\mathbbm{1}[t\\geq c]\\cdot\\mu(t)\\). Note that \\(\\mu_{-},\\mu_{+}\\) are the densities of the restriction of \\(\\mu\\) to the intervals \\(t\\leq c\\) and \\(t\\geq c\\) respectively. Let \\(\\Psi:\\mathbb{R}^{kn}\\to\\{0,1\\}^{kn}\\) be a mapping such that for every \\(\\mathbf{z}^{\\prime}\\in\\mathbb{R}^{kn}\\) and \\(i\\in[kn]\\) we have \\(\\Psi(\\mathbf{z}^{\\prime})_{i}=\\mathbbm{1}[z^{\\prime}_{i}\\geq c]\\). For \\(\\tilde{\\mathbf{z}}\\in\\mathbb{R}^{n^{2}}\\) we denote \\(\\tilde{\\mathbf{z}}_{[kn]}=(\\tilde{z}_{1},\\ldots,\\tilde{z}_{kn})\\), namely, the first \\(kn\\) components of \\(\\tilde{\\mathbf{z}}\\) (assuming \\(n^{2}\\geq kn\\)).\n\n### Defining the target network for \\(\\mathcal{L}\\)\n\nSince our goal is to use the algorithm \\(\\mathcal{L}\\) for breaking PRGs, in this subsection we define a neural network \\(\\tilde{N}:\\mathbb{R}^{n^{2}}\\to\\mathbb{R}\\) that we will later use as a target network for \\(\\mathcal{L}\\). The network \\(\\tilde{N}\\) contains the subnetworks \\(N_{1},N_{2},N_{3}\\) which we define below.\n\nLet \\(N_{1}\\) be a depth-\\(2\\) neural network with input dimension \\(kn\\), at most \\(n\\log(n)\\) hidden neurons, at most \\(\\log(n)\\) output neurons (with activations in the output neurons), and parameter magnitudes bounded by \\(n^{3}\\) (all bounds are for a sufficiently large \\(n\\)), which satisfies the following. We denote the set of output neurons of \\(N_{1}\\) by \\(\\mathcal{E}_{1}\\). Let \\(\\mathbf{z}^{\\prime}\\in\\mathbb{R}^{kn}\\) be an input to \\(N_{1}\\) such that \\(\\Psi(\\mathbf{z}^{\\prime})=\\mathbf{z}^{S}\\) for some hyperedge \\(S\\), and assume that for every \\(i\\in[kn]\\) we have \\(z^{\\prime}_{i}\\not\\in\\left(c,c+\\frac{1}{n^{2}}\\right)\\). Fix some \\(\\mathbf{x}\\in\\{0,1\\}^{n}\\). Then, for \\(S\\) with \\(P_{\\mathbf{x}}(\\mathbf{z}^{S})=0\\) the inputs to all output neurons \\(\\mathcal{E}_{1}\\) are at most \\(-1\\), and for \\(S\\) with \\(P_{\\mathbf{x}}(\\mathbf{z}^{S})=1\\) there exists a neuron in \\(\\mathcal{E}_{1}\\) with input at least \\(2\\). Recall that our definition of a neuron's input includes the addition of the bias term. The construction of the network \\(N_{1}\\) is given in Lemma A.3. Intuitively, the network \\(N_{1}\\) consists of a layer that transforms w.h.p. the input \\(\\mathbf{z}^{\\prime}\\) to \\(\\Psi(\\mathbf{z}^{\\prime})=\\mathbf{z}^{S}\\), followed by a layer that satisfies the following: Building on a lemma from [16] which shows that \\(P_{\\mathbf{x}}(\\mathbf{z}^{S})\\) can be computed by a DNF formula, we define a layer where each output neuron corresponds to a term in the DNF formula, such that if the term evaluates to 0 then the input to the neuron is at most \\(-1\\), and otherwise it is at least \\(2\\). Note that the network \\(N_{1}\\) depends on \\(\\mathbf{x}\\). However, only the second layer depends on \\(\\mathbf{x}\\), and thus given an input we may compute the first layer even without knowing \\(\\mathbf{x}\\). Let \\(N^{\\prime}_{1}:\\mathbb{R}^{kn}\\to\\mathbb{R}\\) be a depth-\\(3\\) network with no activation in the output neuron, obtained from \\(N_{1}\\) by summing the outputs from all neurons \\(\\mathcal{E}_{1}\\).\n\nLet \\(N_{2}\\) be a depth-\\(2\\) neural network with input dimension \\(kn\\), at most \\(n\\log(n)\\) hidden neurons, at most \\(2n\\) output neurons, and parameter magnitudes bounded by \\(n^{3}\\) (for a sufficiently large \\(n\\)), which satisfies the following. We denote the set of output neurons of \\(N_{2}\\) by \\(\\mathcal{E}_{2}\\). Let \\(\\mathbf{z}^{\\prime}\\in\\mathbb{R}^{kn}\\) be an input to \\(N_{2}\\) such that for every \\(i\\in[kn]\\) we have \\(z^{\\prime}_{i}\\not\\in\\left(c,c+\\frac{1}{n^{2}}\\right)\\). If \\(\\Psi(\\mathbf{z}^{\\prime})\\) is an encoding of a hyperedge then the inputs to all output neurons \\(\\mathcal{E}_{2}\\) are at most \\(-1\\), and otherwise there exists a neuron in \\(\\mathcal{E}_{2}\\) with input at least \\(2\\). The construction of the network \\(N_{2}\\) is given in Lemma A.5. Intuitively, each neuron in \\(\\mathcal{E}_{2}\\) is responsible for checking whether \\(\\Psi(\\mathbf{z}^{\\prime})\\) violates some requirement that must hold in an encoding of a hyperedge. Let \\(N^{\\prime}_{2}:\\mathbb{R}^{kn}\\to\\mathbb{R}\\) be a depth-\\(3\\) network with no activation in the output neuron, obtained from \\(N_{2}\\) by summing the outputs from all neurons \\(\\mathcal{E}_{2}\\).\n\nLet \\(N_{3}\\) be a depth-\\(2\\) neural network with input dimension \\(kn\\), at most \\(n\\log(n)\\) hidden neurons, \\(kn\\leq n\\log(n)\\) output neurons, and parameter magnitudes bounded by \\(n^{3}\\) (for a sufficiently large \\(n\\)), which satisfies the following. We denote the set of output neurons of \\(N_{3}\\) by \\(\\mathcal{E}_{3}\\). Let \\(\\mathbf{z}^{\\prime}\\in\\mathbb{R}^{kn}\\) be an input to \\(N_{3}\\). If there exists \\(i\\in[kn]\\) such that \\(z^{\\prime}_{i}\\in\\left(c,c+\\frac{1}{n^{2}}\\right)\\) then there exists a neuron in \\(\\mathcal{E}_{3}\\) with input at least \\(2\\). Moreover, if for all \\(i\\in[kn]\\) we have \\(z^{\\prime}_{i}\\not\\in\\left(c-\\frac{1}{n^{2}},c+\\frac{2}{n^{2}}\\right)\\) then the inputs to all neurons in \\(\\mathcal{E}_{3}\\) are at most \\(-1\\). The construction of the network \\(N_{3}\\) is straightforward and given in Lemma A.6. Let \\(N^{\\prime}_{3}:\\mathbb{R}^{kn}\\to\\mathbb{R}\\) be a depth-\\(3\\) neural network with no activation function in the output neuron, obtained from \\(N_{3}\\) by summing the outputs from all neurons \\(\\mathcal{E}_{3}\\).\n\nLet \\(N^{\\prime}:\\mathbb{R}^{kn}\\to\\mathbb{R}\\) be a depth-\\(3\\) network obtained from \\(N^{\\prime}_{1},N^{\\prime}_{2},N^{\\prime}_{3}\\) as follows. For \\(\\mathbf{z}^{\\prime}\\in\\mathbb{R}^{kn}\\) we have \\(N^{\\prime}(\\mathbf{z}^{\\prime})=\\left[1-N^{\\prime}_{1}(\\mathbf{z}^{\\prime})-N^ {\\prime}_{2}(\\mathbf{z}^{\\prime})-N^{\\prime}_{3}(\\mathbf{z}^{\\prime})\\right]_ {+}\\). The network \\(N^{\\prime}\\) has at most \\(n^{2}\\) neurons, and parameter magnitudes bounded by \\(n^{3}\\) (all bounds are for a sufficiently large \\(n\\)). Finally, let \\(\\tilde{N}:\\mathbb{R}^{n^{2}}\\to\\mathbb{R}\\) be a depth-\\(3\\) neural network such that \\(\\tilde{N}(\\tilde{\\mathbf{z}})=N^{\\prime}\\left(\\tilde{\\mathbf{z}}_{[kn]}\\right)\\).\n\n### Defining the noise magnitude \\(\\tau\\) and analyzing the perturbed network\n\nIn order to use the algorithm \\(\\mathcal{L}\\) w.r.t. some neural network with parameters \\(\\boldsymbol{\\theta}\\), we need to implement an examples oracle, such that the examples are labeled according to a neural network with parameters \\(\\boldsymbol{\\theta}+\\boldsymbol{\\xi}\\), where \\(\\boldsymbol{\\xi}\\) is a random perturbation. Specifically, we use \\(\\mathcal{L}\\) with an examples oracle where the labels correspond to a network \\(\\hat{N}:\\mathbb{R}^{n^{2}}\\to\\mathbb{R}\\), obtained from \\(\\tilde{N}\\) (w.r.t. an appropriate \\(\\mathbf{x}\\in\\{0,1\\}^{n}\\) in the construction of \\(N_{1}\\)) by adding a small perturbation to the parameters. The perturbation is such that we add i.i.d. noise to each parameter in \\(\\tilde{N}\\), where the noise is distributed according to \\(\\mathcal{N}(0,\\tau^{2})\\), and \\(\\tau=1/\\operatorname{poly}(n)\\) is small enough such that the following holds. Let \\(f_{\\mathbf{\\theta}}:\\mathbb{R}^{n^{2}}\\to\\mathbb{R}\\) be any depth-\\(3\\) neural network parameterized by \\(\\mathbf{\\theta}\\in\\mathbb{R}^{r}\\) for some \\(r>0\\) with at most \\(n^{2}\\) neurons, and parameter magnitudes bounded by \\(n^{3}\\) (note that \\(r\\) is polynomial in \\(n\\)). Then w.p. at least \\(1-\\frac{1}{n}\\) over \\(\\mathbf{\\xi}\\sim\\mathcal{N}(\\mathbf{0},\\tau^{2}I_{r})\\), we have \\(|\\xi_{i}|\\leq\\frac{1}{10}\\) for all \\(i\\in[r]\\), and the network \\(f_{\\mathbf{\\theta}+\\mathbf{\\xi}}\\) is such that for every input \\(\\tilde{\\mathbf{z}}\\in\\mathbb{R}^{n^{2}}\\) with \\(\\|\\tilde{\\mathbf{z}}\\|\\leq 2n\\) and every neuron we have: Let \\(a,b\\) be the inputs to the neuron in the computations \\(f_{\\mathbf{\\theta}}(\\tilde{\\mathbf{z}})\\) and \\(f_{\\mathbf{\\theta}+\\mathbf{\\xi}}(\\tilde{\\mathbf{z}})\\) (respectively), then \\(|a-b|\\leq\\frac{1}{2}\\). Thus, \\(\\tau\\) is sufficiently small, such that w.h.p. adding i.i.d. noise \\(\\mathcal{N}(0,\\tau^{2})\\) to each parameter does not change the inputs to the neurons by more than \\(\\frac{1}{2}\\). Note that such an inverse-polynomial \\(\\tau\\) exists, since when the network size, parameter magnitudes, and input size are bounded by some \\(\\operatorname{poly}(n)\\), then the input to each neuron in \\(f_{\\mathbf{\\theta}}(\\tilde{\\mathbf{z}})\\) is \\(\\operatorname{poly}(n)\\)-Lipschitz as a function of \\(\\mathbf{\\theta}\\), and thus it suffices to choose \\(\\tau\\) that implies w.p. at least \\(1-\\frac{1}{n}\\) that \\(\\|\\mathbf{\\xi}\\|\\leq\\frac{1}{q(n)}\\) for a sufficiently large polynomial \\(q(n)\\) (see Lemma A.7 for details).\n\nLet \\(\\tilde{\\mathbf{\\theta}}\\in\\mathbb{R}^{p}\\) be the parameters of the network \\(\\tilde{N}\\). Recall that the parameters vector \\(\\tilde{\\mathbf{\\theta}}\\) is the concatenation of all weight matrices and bias terms. Let \\(\\tilde{\\mathbf{\\theta}}\\in\\mathbb{R}^{p}\\) be the parameters of \\(\\tilde{N}\\), namely, \\(\\tilde{\\mathbf{\\theta}}=\\tilde{\\mathbf{\\theta}}+\\mathbf{\\xi}\\) where \\(\\mathbf{\\xi}\\sim\\mathcal{N}(\\mathbf{0},\\tau^{2}I_{p})\\). By our choice of \\(\\tau\\) and the construction of the networks \\(N_{1},N_{2},N_{3}\\), w.p. at least \\(1-\\frac{1}{n}\\) over \\(\\mathbf{\\xi}\\), for every \\(\\tilde{\\mathbf{z}}\\) with \\(\\|\\tilde{\\mathbf{z}}\\|\\leq 2n\\), the inputs to the neurons \\(\\mathcal{E}_{1},\\mathcal{E}_{2},\\mathcal{E}_{3}\\) in the computation \\(\\hat{N}(\\tilde{\\mathbf{z}})\\) satisfy the following properties, where we denote \\(\\mathbf{z}^{\\prime}=\\tilde{\\mathbf{z}}_{[kn]}\\):\n\n1. If \\(\\Psi(\\mathbf{z}^{\\prime})=\\mathbf{z}^{S}\\) for some hyperedge \\(S\\), and for every \\(i\\in[kn]\\) we have \\(z^{\\prime}_{i}\\not\\in\\left(c,c+\\frac{1}{n^{2}}\\right)\\), then the inputs to \\(\\mathcal{E}_{1}\\) satisfy: If \\(P_{\\mathbf{x}}(\\mathbf{z}^{S})=0\\) then the inputs to all neurons in \\(\\mathcal{E}_{1}\\) are at most \\(-\\frac{1}{2}\\), and if \\(P_{\\mathbf{x}}(\\mathbf{z}^{S})=1\\) then there exists a neuron in \\(\\mathcal{E}_{1}\\) with input at least \\(\\frac{3}{2}\\).\n2. If for every \\(i\\in[kn]\\) we have \\(z^{\\prime}_{i}\\not\\in\\left(c,c+\\frac{1}{n^{2}}\\right)\\), then the inputs to \\(\\mathcal{E}_{2}\\) satisfy: If \\(\\Psi(\\mathbf{z}^{\\prime})\\) is an encoding of a hyperedge then the inputs to all neurons \\(\\mathcal{E}_{2}\\) are at most \\(-\\frac{1}{2}\\), and otherwise there exists a neuron in \\(\\mathcal{E}_{2}\\) with input at least \\(\\frac{3}{2}\\).\n3. The inputs to \\(\\mathcal{E}_{3}\\) satisfy: If there exists \\(i\\in[kn]\\) such that \\(z^{\\prime}_{i}\\in\\left(c,c+\\frac{1}{n^{2}}\\right)\\) then there exists a neuron in \\(\\mathcal{E}_{3}\\) with input at least \\(\\frac{3}{2}\\), and if for all \\(i\\in[kn]\\) we have \\(z^{\\prime}_{i}\\not\\in\\left(c-\\frac{1}{n^{2}},c+\\frac{2}{n^{2}}\\right)\\) then the inputs to all neurons in \\(\\mathcal{E}_{3}\\) are at most \\(-\\frac{1}{2}\\).\n\n### Stating the algorithm \\(\\mathcal{A}\\)\n\nGiven a sequence \\((S_{1},y_{1}),\\ldots,(S_{n^{*}},y_{n^{*}})\\), where \\(S_{1},\\ldots,S_{n^{*}}\\) are i.i.d. random hyperedges, the algorithm \\(\\mathcal{A}\\) needs to distinguish whether \\(\\mathbf{y}=(y_{1},\\ldots,y_{n^{*}})\\) is random or that we have \\(\\mathbf{y}=(P(\\mathbf{x}_{S_{1}}),\\ldots,P(\\mathbf{x}_{S_{n^{*}}}))=(P_{ \\mathbf{x}}(\\mathbf{z}^{S_{1}}),\\ldots,P_{\\mathbf{x}}(\\mathbf{z}^{S_{n^{*}} }))\\) for a random \\(\\mathbf{x}\\in\\{0,1\\}^{n}\\). We let \\(\\mathcal{S}=((\\mathbf{z}^{S_{1}},y_{1}),\\ldots,(\\mathbf{z}^{S_{n^{*}}},y_{n^{* *}}))\\).\n\nWe use the efficient algorithm \\(\\mathcal{L}\\) in order to obtain distinguishing advantage greater than \\(\\frac{1}{3}\\) as follows. Let \\(\\mathbf{\\xi}\\) be a random perturbation, and let \\(\\hat{N}\\) be the perturbed network as defined above, w.r.t. the unknown \\(\\mathbf{x}\\in\\{0,1\\}^{n}\\). Note that given a perturbation \\(\\mathbf{\\xi}\\), only the weights in the second layer of the subnetwork \\(N_{1}\\) in \\(\\hat{N}\\) are unknown, since all other parameters do not depend on \\(\\mathbf{x}\\). The algorithm \\(\\mathcal{A}\\) runs \\(\\mathcal{L}\\) with the following examples oracle. In the \\(i\\)-th call, the oracle first draws \\(\\mathbf{z}\\in\\{0,1\\}^{kn}\\) such that each component is drawn i.i.d. from a Bernoulli distribution which takes the value \\(0\\) w.p. \\(\\frac{1}{n}\\). If \\(\\mathbf{z}\\) is an encoding of a hyperedge then the oracle replaces \\(\\mathbf{z}\\) with \\(\\mathbf{z}^{S_{i}}\\). Then, the oracle chooses \\(\\mathbf{z}^{\\prime}\\in\\mathbb{R}^{kn}\\) such that for each component \\(j\\), if \\(z_{j}=1\\) then \\(z^{\\prime}_{j}\\) is drawn from \\(\\mu_{+}\\), and otherwise \\(z^{\\prime}_{j}\\) is drawn from \\(\\mu_{-}\\). Let \\(\\tilde{\\mathbf{z}}\\in\\mathbb{R}^{n^{2}}\\) be such that \\(\\tilde{\\mathbf{z}}_{[kn]}=\\mathbf{z}^{\\prime}\\), and the other \\(n^{2}-kn\\) components of \\(\\tilde{\\mathbf{z}}\\) are drawn i.i.d. from \\(\\mathcal{N}(0,1)\\). Note that the vector \\(\\tilde{\\mathbf{z}}\\) has the distribution \\(\\mathcal{D}\\), due to the definitions of the densities \\(\\mu_{+}\\) and \\(\\mu_{-}\\), and since replacing an encoding of a random hyperedge by an encoding of another random hyperedge does not change the distribution of \\(\\mathbf{z}\\). Let \\(\\hat{b}\\in\\mathbb{R}\\) be the bias term of the output neuron of \\(\\hat{N}\\). The oracle returns \\((\\tilde{\\mathbf{z}},\\tilde{y})\\), where the labels \\(\\tilde{y}\\) are chosen as follows:\n\n* If \\(\\Psi(\\mathbf{z}^{\\prime})\\) is not an encoding of a hyperedge, then \\(\\tilde{y}=0\\).\n* If \\(\\Psi(\\mathbf{z}^{\\prime})\\) is an encoding of a hyperedge:* If \\(\\mathbf{z}^{\\prime}\\) does not have components in the interval \\((c-\\frac{1}{n^{2}},c+\\frac{2}{n^{2}})\\), then if \\(y_{i}=0\\) we set \\(\\tilde{y}=\\hat{b}\\), and if \\(y_{i}=1\\) we set \\(\\tilde{y}=0\\).\n* If \\(\\mathbf{z}^{\\prime}\\) has a component in the interval \\((c,c+\\frac{1}{n^{2}})\\), then \\(\\tilde{y}=0\\).\n* If \\(\\mathbf{z}^{\\prime}\\) does not have components in the interval \\((c,c+\\frac{1}{n^{2}})\\), but has a component in the interval \\((c-\\frac{1}{n^{2}},c+\\frac{2}{n^{2}})\\), then the label \\(\\tilde{y}\\) is determined as follows: If \\(y_{i}=1\\) then \\(\\tilde{y}=0\\). If \\(y_{i}=0\\), we have: Let \\(N_{3}\\) be the network \\(\\hat{N}\\) after omitting the neurons \\(\\mathcal{E}_{1},\\mathcal{E}_{2}\\) and their incoming and outgoing weights. Then, we set \\(\\tilde{y}=[\\hat{b}-\\hat{N}_{3}(\\tilde{\\mathbf{z}})]_{+}\\). Note that since only the second layer of \\(N_{1}\\) depends on \\(\\mathbf{x}\\), then we can compute \\(\\hat{N}_{3}(\\tilde{\\mathbf{z}})\\) without knowing \\(\\mathbf{x}\\).\n\nLet \\(h\\) be the hypothesis returned by \\(\\mathcal{L}\\). Recall that \\(\\mathcal{L}\\) uses at most \\(m(n)\\) examples, and hence \\(\\mathcal{S}\\) contains at least \\(n^{3}\\) examples that \\(\\mathcal{L}\\) cannot view. We denote the indices of these examples by \\(I=\\{m(n)+1,\\ldots,m(n)+n^{3}\\}\\), and the examples by \\(\\mathcal{S}_{I}=\\{(\\mathbf{z}^{\\mathcal{S}_{i}},y_{i})\\}_{i\\in I}\\). By \\(n^{3}\\) additional calls to the oracle, the algorithm \\(\\mathcal{A}\\) obtains the examples \\(\\tilde{\\mathcal{S}_{I}}=\\{(\\tilde{\\mathbf{z}}_{i},\\tilde{y}_{i})\\}_{i\\in I}\\) that correspond to \\(\\mathcal{S}_{I}\\). Let \\(h^{\\prime}\\) be a hypothesis such that for all \\(\\tilde{\\mathbf{z}}\\in\\mathbb{R}^{n^{2}}\\) we have \\(h^{\\prime}(\\tilde{\\mathbf{z}})=\\max\\{0,\\min\\{\\hat{b},h(\\tilde{\\mathbf{z}})\\}\\}\\), thus, for \\(\\hat{b}\\geq 0\\) the hypothesis \\(h^{\\prime}\\) is obtained from \\(h\\) by clipping the output to the interval \\([0,\\hat{b}]\\). Let \\(\\ell_{I}(h^{\\prime})=\\frac{1}{|I|}\\sum_{i\\in I}(h^{\\prime}(\\tilde{\\mathbf{z}} _{i})-\\tilde{y}_{i})^{2}\\). Now, if \\(\\ell_{I}(h^{\\prime})\\leq\\frac{2}{n}\\), then \\(\\mathcal{A}\\) returns \\(1\\), and otherwise it returns \\(0\\). We remark that the decision of our algorithm is based on \\(h^{\\prime}\\) (rather than \\(h\\)) since we need the outputs to be bounded, in order to allow using Hoeffding's inequality in our analysis, which we discuss in the next subsection.\n\n### Analyzing the algorithm \\(\\mathcal{A}\\)\n\nNote that the algorithm \\(\\mathcal{A}\\) runs in \\(\\operatorname{poly}(n)\\) time. We now show that if \\(\\mathcal{S}\\) is pseudorandom then \\(\\mathcal{A}\\) returns \\(1\\) w.p. greater than \\(\\frac{2}{3}\\), and if \\(\\mathcal{S}\\) is random then \\(\\mathcal{A}\\) returns \\(1\\) w.p. less than \\(\\frac{1}{3}\\).\n\nWe start with the case where \\(\\mathcal{S}\\) is pseudorandom. In Lemma A.8, we prove that if \\(\\mathcal{S}\\) is pseudorandom then w.h.p. (over \\(\\boldsymbol{\\xi}\\sim\\mathcal{N}(\\mathbf{0},\\tau^{2}I_{p})\\) and the i.i.d. inputs \\(\\tilde{\\mathbf{z}}_{i}\\sim\\mathcal{D}\\)) the examples \\((\\tilde{\\mathbf{z}}_{1},\\tilde{y}_{1}),\\ldots,\\)\\((\\tilde{\\mathbf{z}}_{m(n)+n^{3}},\\tilde{y}_{m(n)+n^{3}})\\) returned by the oracle are realized by \\(\\hat{N}\\). Thus, \\(\\tilde{y}_{i}=\\hat{N}(\\tilde{\\mathbf{z}}_{i})\\) for all \\(i\\). As we show in the lemma, this claim follows by considering the definition of the oracle for the different cases of \\((\\tilde{\\mathbf{z}}_{i})_{[kn]}\\), and using Properties (P1)-(P3) to show that \\(\\hat{N}\\) behaves similarly.\n\nRecall that the algorithm \\(\\mathcal{L}\\) is such that w.p. at least \\(\\frac{3}{4}\\) (over \\(\\boldsymbol{\\xi}\\sim\\mathcal{N}(\\mathbf{0},\\tau^{2}I_{p})\\), the i.i.d. inputs \\(\\tilde{\\mathbf{z}}_{i}\\sim\\mathcal{D}\\), and its internal randomness), given a size-\\(m(n)\\) dataset labeled by \\(\\hat{N}\\), it returns a hypothesis \\(h\\) such that \\(\\mathbb{E}_{\\tilde{\\mathbf{z}}\\sim\\mathcal{D}}\\left[(h(\\tilde{\\mathbf{z}})- \\hat{N}(\\tilde{\\mathbf{z}}))^{2}\\right]\\leq\\frac{1}{n}\\). By the definition of \\(h^{\\prime}\\) and the construction of \\(\\hat{N}\\), if \\(h\\) has small error then \\(h^{\\prime}\\) also has small error, namely, we have \\(\\mathbb{E}_{\\tilde{\\mathbf{z}}\\sim\\mathcal{D}}\\left[(h^{\\prime}(\\tilde{ \\mathbf{z}})-\\hat{N}(\\tilde{\\mathbf{z}}))^{2}\\right]\\leq\\frac{1}{n}\\). In Lemma A.9 we use the above arguments and Hoeffding's inequality over \\(\\tilde{\\mathcal{S}_{I}}\\), and prove that w.p. greater than \\(\\frac{2}{3}\\) we have \\(\\ell_{I}(h^{\\prime})\\leq\\frac{2}{n}\\).\n\nNext, we consider the case where \\(\\mathcal{S}\\) is random. Let \\(\\tilde{\\mathcal{Z}}\\subseteq\\mathbb{R}^{n^{2}}\\) be such that \\(\\tilde{\\mathbf{z}}\\in\\tilde{\\mathcal{Z}}\\) iff \\(\\tilde{\\mathbf{z}}_{[kn]}\\) does not have components in the interval \\((c-\\frac{1}{n^{2}},c+\\frac{2}{n^{2}})\\), and \\(\\Psi(\\tilde{\\mathbf{z}}_{[kn]})=\\mathbf{z}^{S}\\) for a hyperedge \\(S\\). If \\(\\mathcal{S}\\) is random, then by the definition of our examples oracle, for every \\(i\\in[m(n)+n^{3}]\\) such that \\(\\tilde{\\mathbf{z}}_{i}\\in\\tilde{\\mathcal{Z}}\\), we have \\(\\tilde{y}_{i}=\\hat{b}\\) w.p. \\(\\frac{1}{2}\\) and \\(\\tilde{y}_{i}=0\\) otherwise. Also, by the definition of the oracle, \\(\\tilde{y}_{i}\\) is independent of \\(S_{i}\\) and independent of the choice of the vector \\(\\tilde{\\mathbf{z}}_{i}\\) that corresponds to \\(\\mathbf{z}^{S_{i}}\\). Hence, for such \\(\\tilde{\\mathbf{z}}_{i}\\in\\tilde{\\mathcal{Z}}\\) with \\(i\\in I\\), any hypothesis cannot predict the label \\(\\tilde{y}_{i}\\), and the expected loss for the example is at least \\(\\left(\\frac{b}{2}\\right)^{2}\\). Moreover, in Lemma A.11 we show that \\(\\Pr\\left[\\tilde{\\mathbf{z}}_{i}\\in\\tilde{\\mathcal{Z}}\\right]\\geq\\frac{1}{2\\log (n)}\\) for a sufficiently large \\(n\\). In Lemma A.12 we use these arguments to prove a lower bound on \\(\\mathbb{E}_{\\tilde{\\mathcal{S}}_{I}}\\left[\\ell_{I}(h^{\\prime})\\right]\\), and by Hoeffding's inequality over \\(\\tilde{\\mathcal{S}_{I}}\\) we conclude that w.p. greater than \\(\\frac{2}{3}\\) we have \\(\\ell_{I}(h^{\\prime})>\\frac{2}{n}\\).\n\nOverall, if \\(\\mathcal{S}\\) is pseudorandom then w.p. greater than \\(\\frac{2}{3}\\) the algorithm \\(\\mathcal{A}\\) returns \\(1\\), and if \\(\\mathcal{S}\\) is random then w.p. greater than \\(\\frac{2}{3}\\) it returns \\(0\\). Thus, the distinguishing advantage is greater than \\(\\frac{1}{3}\\).\n\nDiscussion\n\nUnderstanding the computational complexity of learning neural networks is a central question in learning theory. Our results imply that the assumptions which allow for efficient learning in one-hidden-layer networks might not suffice in deeper networks. Also, in depth-\\(2\\) networks we show that it is not sufficient to assume that both the parameters and the inputs are smoothed. We hope that our hardness results will help focus on assumptions that may allow for efficient learning.\n\nWe emphasize that our hardness results are for neural networks that include the ReLU activation also in the output neuron. In contrast, the positive results on learning depth-\\(2\\) networks that we discussed in the introduction do not include activation in the output neuron. Therefore, as far as we are aware, there is still a small gap between the upper bounds and our hardness results: (1) Under the assumption that the input is Gaussian and the weights are non-degenerate, the cases of depth-\\(2\\) networks with activation in the output neuron and of depth-\\(3\\) networks without activation in the output are not settled; (2) In the setting where both the parameters and the input distribution are smoothed, the case of depth-\\(2\\) networks without activation in the output is not settled. These gaps are an intriguing subject for further research.\n\n## Acknowledgments and Disclosure of Funding\n\nThis work was done as part of the NSF-Simons Sponsored Collaboration on the Theoretical Foundations of Deep Learning.\n\n## References\n\n* [1] B. Applebaum. Pseudorandom generators with long stretch and low locality from random local one-way functions. _SIAM Journal on Computing_, 42(5):2008-2037, 2013.\n* [2] B. Applebaum and S. Lovett. Algebraic attacks against random local functions and their countermeasures. In _Proceedings of the forty-eighth annual ACM symposium on Theory of Computing_, pages 1087-1100, 2016.\n* [3] B. Applebaum and P. Raykov. Fast pseudorandom functions based on expander graphs. In _Theory of Cryptography Conference_, pages 27-56. Springer, 2016.\n* [4] B. Applebaum, B. Barak, and A. Wigderson. Public-key cryptography from different assumptions. In _Proceedings of the forty-second ACM symposium on Theory of computing_, pages 171-180, 2010.\n* [5] B. Applebaum, I. Damgard, Y. Ishai, M. Nielsen, and L. Zichron. Secure arithmetic computation with constant computational overhead. In _Annual International Cryptology Conference_, pages 223-254. Springer, 2017.\n* [6] P. Awasthi, A. Tang, and A. Vijayaraghavan. Efficient algorithms for learning depth-2 neural networks with general relu activations. _Advances in Neural Information Processing Systems_, 34:13485-13496, 2021.\n* [7] A. Bakshi, R. Jayaram, and D. P. Woodruff. Learning two layer rectified neural networks in polynomial time. In _Conference on Learning Theory_, pages 195-268. PMLR, 2019.\n* [8] A. Bhaskara, M. Charikar, A. Moitra, and A. Vijayaraghavan. Smoothed analysis of tensor decompositions. In _Proceedings of the forty-sixth annual ACM symposium on Theory of computing_, pages 594-603, 2014.\n* [9] A. Bhaskara, A. Chen, A. Perreault, and A. Vijayaraghavan. Smoothed analysis in unsupervised learning via decoupling. In _2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS)_, pages 582-610. IEEE, 2019.\n* [10] A. Brutzkus, A. Daniely, and E. Malach. Id3 learns juntas for smoothed product distributions. In _Conference on Learning Theory_, pages 902-915. PMLR, 2020.\n\n* [11] S. Chen, A. Gollakota, A. R. Klivans, and R. Meka. Hardness of noise-free learning for two-hidden-layer neural networks. _arXiv preprint arXiv:2202.05258_, 2022.\n* [12] S. Chen, A. R. Klivans, and R. Meka. Learning deep relu networks is fixed-parameter tractable. In _2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS)_, pages 696-707. IEEE, 2022.\n* [13] G. Couteau, A. Dupin, P. Meaux, M. Rossi, and Y. Rotella. On the concrete security of goldreich's pseudorandom generator. In _International Conference on the Theory and Application of Cryptology and Information Security_, pages 96-124. Springer, 2018.\n* [14] A. Daniely and S. Shalev-Shwartz. Complexity theoretic limitations on learning dnf's. In _Conference on Learning Theory_, pages 815-830, 2016.\n* [15] A. Daniely and G. Vardi. Hardness of learning neural networks with natural weights. _arXiv preprint arXiv:2006.03177_, 2020.\n* [16] A. Daniely and G. Vardi. From local pseudorandom generators to hardness of learning. In _Conference on Learning Theory_, pages 1358-1394. PMLR, 2021.\n* [17] I. Diakonikolas and D. M. Kane. Small covers for near-zero sets of polynomials and learning latent variable models. In _2020 IEEE 61st Annual Symposium on Foundations of Computer Science (FOCS)_, pages 184-195. IEEE, 2020.\n* [18] I. Diakonikolas, D. M. Kane, V. Kontonis, and N. Zarifis. Algorithms and sq lower bounds for pac learning one-hidden-layer relu networks. _arXiv preprint arXiv:2006.12476_, 2020.\n* [19] R. Ge, Q. Huang, and S. M. Kakade. Learning mixtures of gaussians in high dimensions. In _Proceedings of the forty-seventh annual ACM symposium on Theory of computing_, pages 761-770, 2015.\n* [20] R. Ge, J. D. Lee, and T. Ma. Learning one-hidden-layer neural networks with landscape design. _arXiv preprint arXiv:1711.00501_, 2017.\n* [21] R. Ge, R. Kuditipudi, Z. Li, and X. Wang. Learning two-layer neural networks with symmetric inputs. _arXiv preprint arXiv:1810.06793_, 2018.\n* [22] S. Goel and A. R. Klivans. Learning neural networks with two nonlinear layers in polynomial time. In _Conference on Learning Theory_, pages 1470-1499. PMLR, 2019.\n* [23] S. Goel, V. Kanade, A. Klivans, and J. Thaler. Reliably learning the relu in polynomial time. In _Conference on Learning Theory_, pages 1004-1042. PMLR, 2017.\n* [24] S. Goel, A. Gollakota, Z. Jin, S. Karmalkar, and A. Klivans. Superpolynomial lower bounds for learning one-layer neural networks using gradient descent. _arXiv preprint arXiv:2006.12011_, 2020.\n* [25] O. Goldreich. Candidate one-way functions based on expander graphs. _IACR Cryptol. ePrint Arch._, 2000:63, 2000.\n* [26] N. Haghtalab, T. Roughgarden, and A. Shetty. Smoothed analysis of online and differentially private learning. _Advances in Neural Information Processing Systems_, 33:9203-9215, 2020.\n* [27] Y. Ishai, E. Kushilevitz, R. Ostrovsky, and A. Sahai. Cryptography with constant computational overhead. In _Proceedings of the fortieth annual ACM symposium on Theory of computing_, pages 433-442, 2008.\n* [28] M. Janzamin, H. Sedghi, and A. Anandkumar. Beating the perils of non-convexity: Guaranteed training of neural networks using tensor methods. _arXiv preprint arXiv:1506.08473_, 2015.\n* [29] A. T. Kalai and S.-H. Teng. Decision trees are pac-learnable from most product distributions: a smoothed analysis. _arXiv preprint arXiv:0812.0933_, 2008.\n* [30] A. T. Kalai, A. Samorodnitsky, and S.-H. Teng. Learning and smoothed analysis. In _2009 50th Annual IEEE Symposium on Foundations of Computer Science_, pages 395-404. IEEE, 2009.\n\n* [31] A. R. Klivans and A. A. Sherstov. Cryptographic hardness for learning intersections of halfspaces. In _FOCS_, 2006.\n* [32] B. Laurent and P. Massart. Adaptive estimation of a quadratic functional by model selection. _Annals of Statistics_, pages 1302-1338, 2000.\n* [33] H. Lin. Indistinguishability obfuscation from constant-degree graded encoding schemes. In _Annual International Conference on the Theory and Applications of Cryptographic Techniques_, pages 28-57. Springer, 2016.\n* [34] H. Lin and V. Vaikuntanathan. Indistinguishability obfuscation from ddh-like assumptions on constant-degree graded encodings. In _2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS)_, pages 11-20. IEEE, 2016.\n* [35] P. Meaux, C. Carlet, A. Journault, and F.-X. Standaert. Improved filter permutators: Combining symmetric encryption design, boolean functions, low complexity cryptography, and homomorphic encryption, for private delegation of computations. _IACR Cryptol. ePrint Arch._, 2019:483, 2019.\n* [36] T. Roughgarden. _Beyond the worst-case analysis of algorithms_. Cambridge University Press, 2021.\n* [37] A. Sankar, D. A. Spielman, and S.-H. Teng. Smoothed analysis of the condition numbers and growth factors of matrices. _SIAM Journal on Matrix Analysis and Applications_, 28(2):446-476, 2006.\n* [38] O. Shamir. Distribution-specific hardness of learning neural networks. _The Journal of Machine Learning Research_, 19(1):1135-1163, 2018.\n* [39] L. Song, S. Vempala, J. Wilmes, and B. Xie. On the complexity of learning neural networks. In _Advances in neural information processing systems_, pages 5514-5522, 2017.\n* [40] D. A. Spielman and S.-H. Teng. Smoothed analysis of algorithms: Why the simplex algorithm usually takes polynomial time. _Journal of the ACM (JACM)_, 51(3):385-463, 2004.\n* [41] S. Vempala and J. Wilmes. Gradient descent for one-hidden-layer neural networks: Polynomial convergence and sq lower bounds. In _Conference on Learning Theory_, pages 3115-3117. PMLR, 2019.\n* [42] Y. Zhang, J. D. Lee, and M. I. Jordan. l1-regularized neural networks are improperly learnable in polynomial time. In _International Conference on Machine Learning_, pages 993-1001. PMLR, 2016.\n* [43] K. Zhong, Z. Song, P. Jain, P. L. Bartlett, and I. S. Dhillon. Recovery guarantees for one-hidden-layer neural networks. In _International conference on machine learning_, pages 4140-4149. PMLR, 2017.\n\nMissing lemmas for the proof of Theorem 3.1\n\n**Lemma A.1** (Daniely and Vardi [16]).: _For every predicate \\(P:\\{0,1\\}^{k}\\rightarrow\\{0,1\\}\\) and \\(\\mathbf{x}\\in\\{0,1\\}^{n}\\), there is a DNF formula \\(\\psi\\) over \\(\\{0,1\\}^{kn}\\) with at most \\(2^{k}\\) terms, such that for every hyperedge \\(S\\) we have \\(P_{\\mathbf{x}}(\\mathbf{z}^{S})=\\psi(\\mathbf{z}^{S})\\). Moreover, each term in \\(\\psi\\) is a conjunction of positive literals._\n\nProof.: The following proof is from Daniely and Vardi [16], and we give it here for completeness.\n\nWe denote by \\(\\mathcal{B}\\subseteq\\{0,1\\}^{k}\\) the set of satisfying assignments of \\(P\\). Note that the size of \\(\\mathcal{B}\\) is at most \\(2^{k}\\). Consider the following DNF formula over \\(\\{0,1\\}^{kn}\\):\n\n\\[\\psi(\\mathbf{z})=\\bigvee_{\\mathbf{b}\\in\\mathcal{B}}\\bigwedge_{j\\in[k]}\\ \\bigwedge_{ \\{l:x_{l}\\neq y_{j}\\}}z_{j,l}\\.\\]\n\nFor a hyperedge \\(S=(i_{1},\\ldots,i_{k})\\), we have\n\n\\[\\psi(\\mathbf{z}^{S})=1 \\iff\\exists\\mathbf{b}\\in\\mathcal{B}\\ \\forall j\\in[k]\\ \\forall x_{l} \\neq b_{j},\\ z_{j,l}^{S}=1\\] \\[\\iff\\exists\\mathbf{b}\\in\\mathcal{B}\\ \\forall j\\in[k]\\ \\forall x_{l} \\neq b_{j},\\ i_{j}\\neq l\\] \\[\\iff\\exists\\mathbf{b}\\in\\mathcal{B}\\ \\forall j\\in[k],\\ x_{i_{j}}=b_{j}\\] \\[\\iff\\exists\\mathbf{b}\\in\\mathcal{B},\\ \\mathbf{x}_{S}=\\mathbf{b}\\] \\[\\iff P(\\mathbf{x}_{S})=1\\] \\[\\iff P_{\\mathbf{x}}(\\mathbf{z}^{S})=1\\.\\]\n\n**Lemma A.2**.: _Let \\(\\mathbf{x}\\in\\{0,1\\}^{n}\\). There exists an affine layer with at most \\(2^{k}\\) outputs, weights bounded by a constant and bias terms bounded by \\(n\\log(n)\\) (for a sufficiently large \\(n\\)), such that given an input \\(\\mathbf{z}^{S}\\in\\{0,1\\}^{kn}\\) for some hyperedge \\(S\\), it satisfies the following: For \\(S\\) with \\(P_{\\mathbf{x}}(\\mathbf{z}^{S})=0\\) all outputs are at most \\(-1\\), and for \\(S\\) with \\(P_{\\mathbf{x}}(\\mathbf{z}^{S})=1\\) there exists an output greater or equal to \\(2\\)._\n\nProof.: By Lemma A.1, there exists a DNF formula \\(\\varphi_{\\mathbf{x}}\\) over \\(\\{0,1\\}^{kn}\\) with at most \\(2^{k}\\) terms, such that \\(\\varphi_{\\mathbf{x}}(\\mathbf{z}^{S})=P_{\\mathbf{x}}(\\mathbf{z}^{S})\\). Thus, if \\(P_{\\mathbf{x}}(\\mathbf{z}^{S})=0\\) then all terms in \\(\\varphi_{\\mathbf{x}}\\) are not satisfied for the input \\(\\mathbf{z}^{S}\\), and if \\(P_{\\mathbf{x}}(\\mathbf{z}^{S})=1\\) then there is at least one term in \\(\\varphi_{\\mathbf{x}}\\) which is satisfied for the input \\(\\mathbf{z}^{S}\\). Therefore, it suffices to construct an affine layer such that for an input \\(\\mathbf{z}^{S}\\), the \\(j\\)-th output will be at most \\(-1\\) if the \\(j\\)-th term of \\(\\varphi_{\\mathbf{x}}\\) is not satisfied, and at least \\(2\\) otherwise. Each term \\(C_{j}\\) in \\(\\varphi_{\\mathbf{x}}\\) is a conjunction of positive literals. Let \\(I_{j}\\subseteq[kn]\\) be the indices of these literals. The \\(j\\)-th output of the affine layer will be\n\n\\[\\left(\\sum_{l\\in I_{j}}3z_{l}^{S}\\right)-3|I_{j}|+2\\.\\]\n\nNote that if the conjunction \\(C_{j}\\) holds, then this expression is exactly \\(3|I_{j}|-3|I_{j}|+2=2\\), and otherwise it is at most \\(3(|I_{j}|-1)-3|I_{j}|+2=-1\\). Finally, note that all weights are bounded by \\(3\\) and all bias terms are bounded by \\(n\\log(n)\\) (for large enough \\(n\\)). \n\n**Lemma A.3**.: _Let \\(\\mathbf{x}\\in\\{0,1\\}^{n}\\). There exists a depth-\\(2\\) neural network \\(N_{1}\\) with input dimension \\(kn\\), \\(2kn\\) hidden neurons, at most \\(2^{k}\\) output neurons, and parameter magnitudes bounded by \\(n^{3}\\) (for a sufficiently large \\(n\\)), which satisfies the following. We denote the set of output neurons of \\(N_{1}\\) by \\(\\mathcal{E}_{1}\\). Let \\(\\mathbf{z}^{\\prime}\\in\\mathbb{R}^{kn}\\) be such that \\(\\Psi(\\mathbf{z}^{\\prime})=\\mathbf{z}^{S}\\) for some hyperedge \\(S\\), and assume that for every \\(i\\in[kn]\\) we have \\(z_{i}^{\\prime}\\not\\in\\left(c,c+\\frac{1}{n^{2}}\\right)\\). Then, for \\(S\\) with \\(P_{\\mathbf{x}}(\\mathbf{z}^{S})=0\\) the inputs to all neurons \\(\\mathcal{E}_{1}\\) are at most \\(-1\\), and for \\(S\\) with \\(P_{\\mathbf{x}}(\\mathbf{z}^{S})=1\\) there exists a neuron in \\(\\mathcal{E}_{1}\\) with input at least \\(2\\). Moreover, only the second layer of \\(N_{1}\\) depends on \\(\\mathbf{x}\\)._\n\nProof.: First, we construct a depth-\\(2\\) neural network \\(N_{\\Psi}:\\mathbb{R}^{kn}\\rightarrow[0,1]^{kn}\\) with a single layer of nonlinearity, such that for every \\(\\mathbf{z}^{\\prime}\\in\\mathbb{R}^{kn}\\) with \\(z_{i}^{\\prime}\\not\\in(c,c+\\frac{1}{n^{2}})\\) for every \\(i\\in[kn]\\), we have \\(N_{\\Psi}(\\mathbf{z}^{\\prime})=\\Psi(\\mathbf{z}^{\\prime})\\). The network \\(N_{\\Psi}\\) has \\(2kn\\) hidden neurons, and computes \\(N_{\\Psi}(\\mathbf{z}^{\\prime})=(f(z_{1}^{\\prime}),\\ldots,f(z_{kn}^{\\prime}))\\), where \\(f:\\mathbb{R}\\rightarrow[0,1]\\) is such that\n\n\\[f(t)=n^{2}\\cdot\\left(\\left[t-c\\right]_{+}-\\left[t-\\left(c+\\frac{1}{n^{2}} \\right)\\right]_{+}\\right)\\.\\]Note that if \\(t\\leq c\\) then \\(f(t)=0\\), if \\(t\\geq c+\\frac{1}{n^{2}}\\) then \\(f(t)=1\\), and if \\(c<t<c+\\frac{1}{n^{2}}\\) then \\(f(t)\\in(0,1)\\). Also, note that all weights and bias terms can be bounded by \\(n^{2}\\) (for large enough \\(n\\)). Moreover, the network \\(N_{\\Psi}\\) does not depend on \\(\\mathbf{x}\\).\n\nLet \\(\\mathbf{z}^{\\prime}\\in\\mathbb{R}^{kn}\\) such that \\(\\Psi(\\mathbf{z}^{\\prime})=\\mathbf{z}^{S}\\) for some hyperedge \\(S\\), and assume that for every \\(i\\in[kn]\\) we have \\(z^{\\prime}_{i}\\not\\in\\left(c,c+\\frac{1}{n^{2}}\\right)\\). For such \\(\\mathbf{z}^{\\prime}\\), we have \\(N_{\\Psi}(\\mathbf{z}^{\\prime})=\\Psi(\\mathbf{z}^{\\prime})=\\mathbf{z}^{S}\\). Hence, it suffices to show that we can construct an affine layer with at most \\(2^{k}\\) outputs, weights bounded by a constant and bias terms bounded by \\(n^{3}\\), such that given an input \\(\\mathbf{z}^{S}\\) it satisfies the following: For \\(S\\) with \\(P_{\\mathbf{x}}(\\mathbf{z}^{S})=0\\) all outputs are at most \\(-1\\), and for \\(S\\) with \\(P_{\\mathbf{x}}(\\mathbf{z}^{S})=1\\) there exists an output greater or equal to \\(2\\). We construct such an affine layer in Lemma A.2. \n\n**Lemma A.4**.: _There exists an affine layer with \\(2k+n\\) outputs, weights bounded by a constant and bias terms bounded by \\(n\\log(n)\\) (for a sufficiently large \\(n\\)), such that given an input \\(\\mathbf{z}\\in\\{0,1\\}^{kn}\\), if it is an encoding of a hyperedge then all outputs are at most \\(-1\\), and otherwise there exists an output greater or equal to \\(2\\)._\n\nProof.: Note that \\(\\mathbf{z}\\in\\{0,1\\}^{kn}\\) is not an encoding of a hyperedge iff at least one of the following holds:\n\n1. At least one of the \\(k\\) size-\\(n\\) slices in \\(\\mathbf{z}\\) contains \\(0\\) more than once.\n2. At least one of the \\(k\\) size-\\(n\\) slices in \\(\\mathbf{z}\\) does not contain \\(0\\).\n3. There are two size-\\(n\\) slices in \\(\\mathbf{z}\\) that encode the same index.\n\nWe define the outputs of our affine layer as follows. First, we have \\(k\\) outputs that correspond to (1). In order to check whether slice \\(i\\in[k]\\) contains \\(0\\) more than once, the output will be \\(3n-4-(-\\sum_{j\\in[n]}3z_{i,j})\\). Second, we have \\(k\\) outputs that correspond to (2): in order to check whether slice \\(i\\in[k]\\) does not contain \\(0\\), the output will be \\((\\sum_{j\\in[n]}3z_{i,j})-3n+2\\). Finally, we have \\(n\\) outputs that correspond to (3): in order to check whether there are two slices that encode the same index \\(j\\in[n]\\), the output will be \\(3k-4-(\\sum_{i\\in[k]}3z_{i,j})\\). Note that all weights are bounded by \\(3\\) and all bias terms are bounded by \\(n\\log(n)\\) for large enough \\(n\\). \n\n**Lemma A.5**.: _There exists a depth-\\(2\\) neural network \\(N_{2}\\) with input dimension \\(kn\\), at most \\(2kn\\) hidden neurons, \\(2k+n\\) output neurons, and parameter magnitudes bounded by \\(n^{3}\\) (for a sufficiently large \\(n\\)), which satisfies the following. We denote the set of output neurons of \\(N_{2}\\) by \\(\\mathcal{E}_{2}\\). Let \\(\\mathbf{z}^{\\prime}\\in\\mathbb{R}^{kn}\\) be such that for every \\(i\\in[kn]\\) we have \\(z^{\\prime}_{i}\\not\\in\\left(c,c+\\frac{1}{n^{2}}\\right)\\). If \\(\\Psi(\\mathbf{z}^{\\prime})\\) is an encoding of a hyperedge then the inputs to all neurons \\(\\mathcal{E}_{2}\\) are at most \\(-1\\), and otherwise there exists a neuron in \\(\\mathcal{E}_{2}\\) with input at least \\(2\\)._\n\nProof.: Let \\(N_{\\Psi}:\\mathbb{R}^{kn}\\rightarrow[0,1]^{kn}\\) be the depth-\\(2\\) neural network from the proof of Lemma A.3, with a single layer of non-linearity with \\(2kn\\) hidden neurons, and parameter magnitudes bounded by \\(n^{2}\\), such that for every \\(\\mathbf{z}^{\\prime}\\in\\mathbb{R}^{kn}\\) with \\(z^{\\prime}_{i}\\not\\in\\left(c,c+\\frac{1}{n^{2}}\\right)\\) for every \\(i\\in[kn]\\), we have \\(N_{\\Psi}(\\mathbf{z}^{\\prime})=\\Psi(\\mathbf{z}^{\\prime})\\).\n\nLet \\(\\mathbf{z}^{\\prime}\\in\\mathbb{R}^{kn}\\) be such that for every \\(i\\in[kn]\\) we have \\(z^{\\prime}_{i}\\not\\in\\left(c,c+\\frac{1}{n^{2}}\\right)\\). For such \\(\\mathbf{z}^{\\prime}\\) we have \\(N_{\\Psi}(\\mathbf{z}^{\\prime})=\\Psi(\\mathbf{z}^{\\prime})\\). Hence, it suffices to show that we can construct an affine layer with \\(2k+n\\) outputs, weights bounded by a constant and bias terms bounded by \\(n^{3}\\), such that given an input \\(\\mathbf{z}\\in\\{0,1\\}^{kn}\\), if it is an encoding of a hyperedge then all outputs are at most \\(-1\\), and otherwise there exists an output greater or equal to \\(2\\). We construct such an affine layer in Lemma A.4. \n\n**Lemma A.6**.: _There exists a depth-\\(2\\) neural network \\(N_{3}\\) with input dimension \\(kn\\), at most \\(n\\log(n)\\) hidden neurons, \\(kn\\leq n\\log(n)\\) output neurons, and parameter magnitudes bounded by \\(n^{3}\\) (for a sufficiently large \\(n\\)), which satisfies the following. We denote the set of output neurons of \\(N_{3}\\) by \\(\\mathcal{E}_{3}\\). Let \\(\\mathbf{z}^{\\prime}\\in\\mathbb{R}^{kn}\\). If there exists \\(i\\in[kn]\\) such that \\(z^{\\prime}_{i}\\in\\left(c,c+\\frac{1}{n^{2}}\\right)\\) then there exists a neuron in \\(\\mathcal{E}_{3}\\) with input at least \\(2\\). If for all \\(i\\in[kn]\\) we have \\(z^{\\prime}_{i}\\not\\in\\left(c-\\frac{1}{n^{2}},c+\\frac{2}{n^{2}}\\right)\\) then the inputs to all neurons in \\(\\mathcal{E}_{3}\\) are at most \\(-1\\)._\n\nProof.: It suffices to construct a univariate depth-\\(2\\) network \\(f:\\mathbb{R}\\rightarrow\\mathbb{R}\\) with one non-linear layer and a constant number of hidden neurons, such that for every input \\(z^{\\prime}_{i}\\in\\left(c,c+\\frac{1}{n^{2}}\\right)\\) we have \\(f(z^{\\prime}_{i})=2\\), and for every \\(z^{\\prime}_{i}\\not\\in\\left(c-\\frac{1}{n^{2}},c+\\frac{2}{n^{2}}\\right)\\) we have \\(f(z^{\\prime}_{i})=-1\\).\n\nWe construct \\(f\\) as follows:\n\n\\[f(z^{\\prime}_{i})= (3n^{2})\\left(\\left[z^{\\prime}_{i}-\\left(c-\\frac{1}{n^{2}}\\right) \\right]_{+}-\\left[z^{\\prime}_{i}-c\\right]_{+}\\right)-\\] \\[(3n^{2})\\left(\\left[z^{\\prime}_{i}-\\left(c+\\frac{1}{n^{2}}\\right) \\right]_{+}-\\left[z^{\\prime}_{i}-\\left(c+\\frac{2}{n^{2}}\\right)\\right]_{+} \\right)-1\\;.\\]\n\nNote that all weights and bias terms are bounded by \\(n^{3}\\) (for large enough \\(n\\)). \n\n**Lemma A.7**.: _Let \\(q=\\mathrm{poly}(n)\\) and \\(r=\\mathrm{poly}(n)\\). Then, there exists \\(\\tau=\\frac{1}{\\mathrm{poly}(n)}\\) such that for a sufficiently large \\(n\\), with probability at least \\(1-\\exp(-n/2)\\) a vector \\(\\boldsymbol{\\xi}\\sim\\mathcal{N}(\\mathbf{0},\\tau^{2}I_{r})\\) satisfies \\(\\|\\boldsymbol{\\xi}\\|\\leq\\frac{1}{q}\\)._\n\nProof.: Let \\(\\tau=\\frac{1}{q\\sqrt{2rn}}\\). Every component \\(\\xi_{i}\\) in \\(\\boldsymbol{\\xi}\\) has the distribution \\(\\mathcal{N}(0,\\tau^{2})\\). By a standard tail bound of the Gaussian distribution, we have for every \\(i\\in[r]\\) and \\(t\\geq 0\\) that \\(\\Pr[\\xi_{i}\\geq t]\\leq 2\\exp\\left(-\\frac{t^{2}}{2\\tau^{2}}\\right)\\). Hence, for \\(t=\\frac{1}{q\\sqrt{\\tau}}\\), we get\n\n\\[\\Pr\\left[\\xi_{i}\\geq\\frac{1}{q\\sqrt{\\tau}}\\right]\\leq 2\\exp\\left(-\\frac{1}{2\\tau^ {2}q^{2}r}\\right)=2\\exp\\left(-\\frac{2rnq^{2}}{2q^{2}r}\\right)=2\\exp\\left(-n \\right)\\;.\\]\n\nBy the union bound, with probability at least \\(1-r\\cdot 2e^{-n}\\), we have\n\n\\[\\|\\boldsymbol{\\xi}\\|^{2}\\leq r\\cdot\\frac{1}{rq^{2}}=\\frac{1}{q^{2}}\\;.\\]\n\nThus, for a sufficiently large \\(n\\), with probability at least \\(1-\\exp(-n/2)\\) we have \\(\\|\\boldsymbol{\\xi}\\|\\leq\\frac{1}{q}\\). \n\n**Lemma A.8**.: _If \\(\\mathcal{S}\\) is pseudorandom then with probability at least \\(\\frac{39}{40}\\) (over \\(\\boldsymbol{\\xi}\\sim\\mathcal{N}(\\mathbf{0},\\tau^{2}I_{p})\\) and the i.i.d. inputs \\(\\tilde{\\mathbf{z}}_{i}\\sim\\mathcal{D}\\)) the examples \\((\\tilde{\\mathbf{z}}_{1},\\tilde{y}_{1}),\\ldots,(\\tilde{\\mathbf{z}}_{m(n)+n^{3}},\\tilde{y}_{m(n)+n^{3}})\\) returned by the oracle are realized by \\(\\hat{N}\\)._\n\nProof.: By our choice of \\(\\tau\\), with probability at least \\(1-\\frac{1}{n}\\) over \\(\\boldsymbol{\\xi}\\sim\\mathcal{N}(\\mathbf{0},\\tau^{2}I_{p})\\), we have \\(|\\xi_{j}|\\leq\\frac{1}{10}\\) for all \\(j\\in[p]\\), and for every \\(\\tilde{\\mathbf{z}}\\) with \\(\\|\\tilde{\\mathbf{z}}\\|\\leq 2n\\) the inputs to the neurons \\(\\mathcal{E}_{1},\\mathcal{E}_{2},\\mathcal{E}_{3}\\) in the computation \\(\\hat{N}(\\tilde{\\mathbf{z}})\\) satisfy Properties (P1) through (P3). We first show that with probability at least \\(1-\\frac{1}{n}\\) all examples \\(\\tilde{\\mathbf{z}}_{1},\\ldots,\\tilde{\\mathbf{z}}_{m(n)+n^{3}}\\) satisfy \\(\\|\\tilde{\\mathbf{z}}_{i}\\|\\leq 2n\\). Hence, with probability at least \\(1-\\frac{2}{n}\\), Properties (P1) through (P3) hold for the computations \\(\\hat{N}(\\tilde{\\mathbf{z}}_{i})\\) for all \\(i\\in[m(n)+n^{3}]\\).\n\nNote that \\(\\left\\|\\tilde{\\mathbf{z}}_{i}\\right\\|^{2}\\) has the Chi-squared distribution. Since \\(\\tilde{\\mathbf{z}}_{i}\\) is of dimension \\(n^{2}\\), a concentration bound by Laurent and Massart [32, Lemma 1] implies that for all \\(t>0\\) we have\n\n\\[\\Pr\\left[\\left\\|\\tilde{\\mathbf{z}}_{i}\\right\\|^{2}-n^{2}\\geq 2n\\sqrt{t}+2t \\right]\\leq e^{-t}\\;.\\]\n\nPlugging-in \\(t=\\frac{n^{2}}{4}\\), we get\n\n\\[\\Pr\\left[\\left\\|\\tilde{\\mathbf{z}}_{i}\\right\\|^{2}\\geq 4n^{2}\\right] =\\Pr\\left[\\left\\|\\tilde{\\mathbf{z}}_{i}\\right\\|^{2}-n^{2}\\geq 3n^{2}\\right]\\] \\[\\leq\\Pr\\left[\\left\\|\\tilde{\\mathbf{z}}_{i}\\right\\|^{2}-n^{2}\\geq \\frac{3n^{2}}{2}\\right]\\] \\[=\\Pr\\left[\\left\\|\\tilde{\\mathbf{z}}_{i}\\right\\|^{2}-n^{2}\\geq 2n \\sqrt{\\frac{n^{2}}{4}}+2\\cdot\\frac{n^{2}}{4}\\right]\\] \\[\\leq\\exp\\left(-\\frac{n^{2}}{4}\\right)\\;.\\]\n\nThus, we have \\(\\Pr\\left[\\left\\|\\tilde{\\mathbf{z}}_{i}\\right\\|\\geq 2n\\right]\\leq\\exp\\left(-\\frac{n^{2}}{4}\\right)\\). By the union bound, with probability at least\n\n\\[1-\\left(m(n)+n^{3}\\right)\\exp\\left(-\\frac{n^{2}}{4}\\right)\\geq 1-\\frac{1}{n}\\](for a sufficiently large \\(n\\)), all examples \\((\\tilde{\\mathbf{z}}_{i},\\tilde{y}_{i})\\) satisfy \\(\\|\\tilde{\\mathbf{z}}_{i}\\|\\leq 2n\\).\n\nThus, we showed that with probability at least \\(1-\\frac{2}{n}\\geq\\frac{39}{40}\\) (for a sufficiently large \\(n\\)), we have \\(|\\xi_{j}|\\leq\\frac{1}{10}\\) for all \\(j\\in[p]\\), and Properties (P1) through (P3) hold for the computations \\(\\hat{N}(\\tilde{\\mathbf{z}}_{i})\\) for all \\(i\\in[m(n)+n^{3}]\\). It remains to show that if these properties hold, then the examples \\((\\tilde{\\mathbf{z}}_{1},\\tilde{y}_{1}),\\ldots,(\\tilde{\\mathbf{z}}_{m(n)+n^{3}},\\tilde{y}_{m(n)+n^{3}})\\) are realized by \\(\\hat{N}\\).\n\nLet \\(i\\in[m(n)+n^{3}]\\). For brevity, we denote \\(\\tilde{\\mathbf{z}}=\\tilde{\\mathbf{z}}_{i}\\), \\(\\tilde{y}=\\tilde{y}_{i}\\), and \\(\\mathbf{z}^{\\prime}=\\tilde{\\mathbf{z}}_{[kn]}\\). Since \\(|\\xi_{j}|\\leq\\frac{1}{10}\\) for all \\(j\\in[p]\\), and all incoming weights to the output neuron in \\(\\tilde{N}\\) are \\(-1\\), then in \\(\\hat{N}\\) all incoming weights to the output neuron are in \\(\\left[-\\frac{11}{10},-\\frac{9}{10}\\right]\\), and the bias term in the output neuron, denoted by \\(\\hat{b}\\), is in \\(\\left[\\frac{9}{10},\\frac{11}{10}\\right]\\). Consider the following cases:\n\n* If \\(\\Psi(\\mathbf{z}^{\\prime})\\) is not an encoding of a hyperedge then \\(\\tilde{y}=0\\), and \\(\\hat{N}(\\tilde{\\mathbf{z}})\\) satisfies: 1. If \\(\\mathbf{z}^{\\prime}\\) does not have components in \\(\\left(c,c+\\frac{1}{n}\\right)\\), then there exists a neuron in \\(\\mathcal{E}_{2}\\) with output at least \\(\\frac{3}{2}\\) (by Property (P2)).\n* If \\(\\mathbf{z}^{\\prime}\\) has a component in \\(\\left(c,c+\\frac{1}{n}\\right)\\), then there exists a neuron in \\(\\mathcal{E}_{3}\\) with output at least \\(\\frac{3}{2}\\) (by Property (P3)). In both cases, since all incoming weights to the output neuron in \\(\\hat{N}\\) are in \\(\\left[-\\frac{11}{10},-\\frac{9}{10}\\right]\\), and \\(\\hat{b}\\in\\left[\\frac{9}{10},\\frac{11}{10}\\right]\\), then the input to the output neuron (including the bias term) is at most \\(\\frac{11}{10}-\\frac{3}{2}\\cdot\\frac{9}{10}<0\\), and thus its output is \\(0\\).\n* If \\(\\Psi(\\mathbf{z}^{\\prime})\\) is an encoding of a hyperedge \\(S\\), then by the definition of the examples oracle we have \\(S=S_{i}\\). Hence:\n* If \\(\\mathbf{z}^{\\prime}\\) does not have components in \\(\\left(c-\\frac{1}{n^{2}},c+\\frac{2}{n^{2}}\\right)\\), then:\n* If \\(y_{i}=0\\) then the oracle sets \\(\\tilde{y}=\\hat{b}\\). Since \\(\\mathcal{S}\\) is pseudorandom, we have \\(P_{\\mathbf{x}}(\\mathbf{z}^{S})=P_{\\mathbf{x}}(\\mathbf{z}^{S_{i}})=y_{i}=0\\). Hence, in the computation \\(\\hat{N}(\\tilde{\\mathbf{z}})\\) the inputs to all neurons in \\(\\mathcal{E}_{1},\\mathcal{E}_{2},\\mathcal{E}_{3}\\) are at most \\(-\\frac{1}{2}\\) (by Properties (P1), (P2) and (P3)), and thus their outputs are \\(0\\). Therefore, \\(\\hat{N}(\\tilde{\\mathbf{z}})=\\hat{b}\\).\n* If \\(y_{i}=1\\) then the oracle sets \\(\\tilde{y}=0\\). Since \\(\\mathcal{S}\\) is pseudorandom, we have \\(P_{\\mathbf{x}}(\\mathbf{z}^{S})=P_{\\mathbf{x}}(\\mathbf{z}^{S_{i}})=y_{i}=1\\). Hence, in the computation \\(\\hat{N}(\\tilde{\\mathbf{z}})\\) there exists a neuron in \\(\\mathcal{E}_{1}\\) with output at least \\(\\frac{3}{2}\\) (by Property (P1)). Since all incoming weights to the output neuron in \\(\\hat{N}\\) are in \\(\\left[-\\frac{11}{10},-\\frac{9}{10}\\right]\\), and \\(\\hat{b}\\in\\left[\\frac{9}{10},\\frac{11}{10}\\right]\\), then the input to output neuron (including the bias term) is at most \\(\\frac{11}{10}-\\frac{3}{2}\\cdot\\frac{9}{10}<0\\), and thus its output is \\(0\\).\n* If \\(\\mathbf{z}^{\\prime}\\) has a component in \\(\\left(c,c+\\frac{1}{n^{2}}\\right)\\), then \\(\\tilde{y}=0\\). Also, in the computation \\(\\hat{N}(\\tilde{\\mathbf{z}})\\) there exists a neuron in \\(\\mathcal{E}_{3}\\) with output at least \\(\\frac{3}{2}\\) (by Property (P3)). Since all incoming weights to the output neuron in \\(\\hat{N}\\) are in \\(\\left[-\\frac{11}{10},-\\frac{9}{10}\\right]\\), and \\(\\hat{b}\\in\\left[\\frac{9}{10},\\frac{11}{10}\\right]\\), then the input to output neuron (including the bias term) is at most \\(\\frac{11}{10}-\\frac{3}{2}\\cdot\\frac{9}{10}<0\\), and thus its output is \\(0\\).\n* If \\(y_{i}=1\\) the oracle sets \\(\\tilde{y}=[\\hat{b}-\\hat{N}_{3}(\\tilde{\\mathbf{z}})]_{+}\\). Since \\(\\mathcal{S}\\) is pseudorandom, we have \\(P_{\\mathbf{x}}(\\mathbf{z}^{S})=P_{\\mathbf{x}}(\\mathbf{z}^{S_{i}})=y_{i}=0\\). Therefore, in the computation \\(\\hat{N}(\\tilde{\\mathbf{z}})\\) all neurons in \\(\\mathcal{E}_{1},\\mathcal{E}_{2}\\) have output \\(0\\) (by Properties (P1) and (P2)), and hence their contribution to the output of \\(\\hat{N}\\) is \\(0\\). Thus, by the definition of \\(\\hat{N}_{3}\\), we have \\(\\hat{N}(\\tilde{\\mathbf{z}})=[\\hat{b}-\\hat{N}_{3}(\\tilde{\\mathbf{z}})]_{+}\\).\n\n**Lemma A.9**.: _If \\(\\mathcal{S}\\) is pseudorandom, then for a sufficiently large \\(n\\), with probability greater than \\(\\frac{2}{3}\\) we have_\n\n\\[\\ell_{I}(h^{\\prime})\\leq\\frac{2}{n}\\;.\\]\n\nProof.: By Lemma A.8, if \\(\\mathcal{S}\\) is pseudorandom then with probability at least \\(\\frac{39}{40}\\) (over \\(\\boldsymbol{\\xi}\\sim\\mathcal{N}(\\boldsymbol{0},\\tau^{2}I_{p})\\) and the i.i.d. inputs \\(\\tilde{\\mathbf{z}}_{i}\\sim\\mathcal{D}\\)) the examples \\((\\tilde{\\mathbf{z}}_{1},\\tilde{y}_{1}),\\ldots,(\\tilde{\\mathbf{z}}_{m(n)}, \\tilde{y}_{m(n)})\\) returned by the oracle are realized by \\(\\hat{N}\\). Recall that the algorithm \\(\\mathcal{L}\\) is such that with probability at least \\(\\frac{3}{4}\\) (over \\(\\boldsymbol{\\xi}\\sim\\mathcal{N}(\\boldsymbol{0},\\tau^{2}I_{p})\\), the i.i.d. inputs \\(\\tilde{\\mathbf{z}}_{i}\\sim\\mathcal{D}\\), and possibly its internal randomness), given a size-\\(m(n)\\) dataset labeled by \\(\\hat{N}\\), it returns a hypothesis \\(h\\) such that \\(\\mathbb{E}_{\\tilde{\\mathbf{z}}\\sim\\mathcal{D}}\\left[(h(\\tilde{\\mathbf{z}})- \\hat{N}(\\tilde{\\mathbf{z}}))^{2}\\right]\\leq\\frac{1}{n}\\). Hence, with probability at least \\(\\frac{3}{4}-\\frac{1}{40}\\) the algorithm \\(\\mathcal{L}\\) returns such a good hypothesis \\(h\\), given \\(m(n)\\) examples labeled by our examples oracle. Indeed, note that \\(\\mathcal{L}\\) can return a bad hypothesis only if the random choices are either bad for \\(\\mathcal{L}\\) (when used with realizable examples) or bad for the realizability of the examples returned by our oracle. By the definition of \\(h^{\\prime}\\) and the construction of \\(\\hat{N}\\), if \\(h\\) has small error then \\(h^{\\prime}\\) also has small error, namely,\n\n\\[\\mathop{\\mathbb{E}}_{\\tilde{\\mathbf{z}}\\sim\\mathcal{D}}\\left[(h^{\\prime}( \\tilde{\\mathbf{z}})-\\hat{N}(\\tilde{\\mathbf{z}}))^{2}\\right]\\leq\\mathop{ \\mathbb{E}}_{\\tilde{\\mathbf{z}}\\sim\\mathcal{D}}\\left[(h(\\tilde{\\mathbf{z}})- \\hat{N}(\\tilde{\\mathbf{z}}))^{2}\\right]\\leq\\frac{1}{n}\\;.\\]\n\nLet \\(\\hat{\\ell}_{I}(h^{\\prime})=\\frac{1}{|I|}\\sum_{i\\in I}(h^{\\prime}(\\tilde{ \\mathbf{z}}_{i})-\\hat{N}(\\tilde{\\mathbf{z}}_{i}))^{2}\\). Recall that by our choice of \\(\\tau\\) we have \\(\\Pr[\\hat{b}>\\frac{11}{10}]\\leq\\frac{1}{n}\\). Since, \\((h^{\\prime}(\\tilde{\\mathbf{z}})-\\hat{N}(\\tilde{\\mathbf{z}}))^{2}\\in[0,\\hat{b }^{2}]\\) for all \\(\\tilde{\\mathbf{z}}\\in\\mathbb{R}^{n^{2}}\\), by Hoeffding's inequality, we have for a sufficiently large \\(n\\) that\n\n\\[\\Pr\\left[\\left|\\hat{\\ell}_{I}(h^{\\prime})-\\mathop{\\mathbb{E}}_{ \\tilde{\\mathcal{S}}_{I}}\\hat{\\ell}_{I}(h^{\\prime})\\right|\\geq\\frac{1}{n}\\right] =\\Pr\\left[\\left|\\hat{\\ell}_{I}(h^{\\prime})-\\mathop{\\mathbb{E}}_{ \\tilde{\\mathcal{S}}_{I}}\\hat{\\ell}_{I}(h^{\\prime})\\right|\\geq\\frac{1}{n} \\right|\\hat{b}\\leq\\frac{11}{10}\\right]\\cdot\\Pr\\left[\\hat{b}\\leq\\frac{11}{10}\\right]\\] \\[\\quad+\\Pr\\left[\\left|\\hat{\\ell}_{I}(h^{\\prime})-\\mathop{\\mathbb{ E}}_{\\tilde{\\mathcal{S}}_{I}}\\hat{\\ell}_{I}(h^{\\prime})\\right|\\geq\\frac{1}{n} \\right|\\hat{b}>\\frac{11}{10}\\right]\\cdot\\Pr\\left[\\hat{b}>\\frac{11}{10}\\right]\\] \\[\\leq 2\\exp\\left(-\\frac{2n^{3}}{n^{2}(11/10)^{4}}\\right)\\cdot 1+1 \\cdot\\frac{1}{n}\\] \\[\\leq\\frac{1}{40}\\;.\\]\n\nMoreover, by Lemma A.8,\n\n\\[\\Pr\\left[\\ell_{I}(h^{\\prime})\\neq\\hat{\\ell}_{I}(h^{\\prime})\\right]\\leq\\Pr \\left[\\exists i\\in I\\text{ s.t. }\\tilde{y}_{i}\\neq\\hat{N}(\\tilde{\\mathbf{z}}_{i})\\right]\\leq \\frac{1}{40}\\;.\\]\n\nOverall, by the union bound we have with probability at least \\(1-\\left(\\frac{1}{4}+\\frac{1}{40}+\\frac{1}{40}+\\frac{1}{40}\\right)>\\frac{2}{3}\\) for sufficiently large \\(n\\) that:\n\n* \\(\\mathbb{E}_{\\tilde{\\mathcal{S}}_{I}}\\hat{\\ell}_{I}(h^{\\prime})=\\mathbb{E}_{ \\tilde{\\mathbf{z}}\\sim\\mathcal{D}}\\left[(h^{\\prime}(\\tilde{\\mathbf{z}})-\\hat{ N}(\\tilde{\\mathbf{z}}))^{2}\\right]\\leq\\frac{1}{n}\\).\n* \\(\\left|\\hat{\\ell}_{I}(h^{\\prime})-\\mathbb{E}_{\\tilde{\\mathcal{S}}_{I}}\\hat{\\ell }_{I}(h^{\\prime})\\right|\\leq\\frac{1}{n}\\).\n* \\(\\ell_{I}(h^{\\prime})-\\hat{\\ell}_{I}(h^{\\prime})=0\\).\n\nCombining the above, we get that if \\(\\mathcal{S}\\) is pseudorandom, then with probability greater than \\(\\frac{2}{3}\\) we have\n\n\\[\\ell_{I}(h^{\\prime})=\\left(\\ell_{I}(h^{\\prime})-\\hat{\\ell}_{I}(h^{\\prime}) \\right)+\\left(\\hat{\\ell}_{I}(h^{\\prime})-\\mathop{\\mathbb{E}}_{\\tilde{\\mathcal{ S}}_{I}}\\hat{\\ell}_{I}(h^{\\prime})\\right)+\\mathop{\\mathbb{E}}_{\\tilde{ \\mathcal{S}}_{I}}\\hat{\\ell}_{I}(h^{\\prime})\\leq 0+\\frac{1}{n}+\\frac{1}{n}=\\frac{2}{n}\\;.\\]\n\n**Lemma A.10**.: _Let \\(\\mathbf{z}\\in\\{0,1\\}^{kn}\\) be a random vector whose components are drawn i.i.d. from a Bernoulli distribution, which takes the value \\(0\\) with probability \\(\\frac{1}{n}\\). Then, for a sufficiently large \\(n\\), the vector \\(\\mathbf{z}\\) is an encoding of a hyperedge with probability at least \\(\\frac{1}{\\log(n)}\\)._Proof.: The vector \\(\\mathbf{z}\\) represents a hyperedge iff in each of the \\(k\\) size-\\(n\\) slices in \\(\\mathbf{z}\\) there is exactly one \\(0\\)-bit and each two of the \\(k\\) slices in \\(\\mathbf{z}\\) encode different indices. Hence,\n\n\\[\\Pr\\left[\\mathbf{z}\\text{ represents a hyperedge}\\right] =n\\cdot(n-1)\\cdot\\ldots\\cdot(n-k+1)\\cdot\\left(\\frac{1}{n}\\right)^ {k}\\left(\\frac{n-1}{n}\\right)^{nk-k}\\] \\[\\geq\\left(\\frac{n-k}{n}\\right)^{k}\\left(\\frac{n-1}{n}\\right)^{k( n-1)}\\] \\[=\\left(1-\\frac{k}{n}\\right)^{k}\\left(1-\\frac{1}{n}\\right)^{k(n-1 )}\\.\\]\n\nSince for every \\(x\\in(0,1)\\) we have \\(e^{-x}<1-\\frac{x}{2}\\), then for a sufficiently large \\(n\\) the above is at least\n\n\\[\\exp\\left(-\\frac{2k^{2}}{n}\\right)\\cdot\\exp\\left(-\\frac{2k(n-1)}{n}\\right)\\geq \\exp\\left(-1\\right)\\cdot\\exp\\left(-2k\\right)\\geq\\frac{1}{\\log(n)}\\.\\]\n\n**Lemma A.11**.: _Let \\(\\tilde{\\mathbf{z}}\\in\\mathbb{R}^{n^{2}}\\) be the vector returned by the oracle. We have_\n\n\\[\\Pr\\left[\\tilde{\\mathbf{z}}\\in\\tilde{\\mathcal{Z}}\\right]\\geq\\frac{1}{2\\log(n) }\\.\\]\n\nProof.: Let \\(\\mathbf{z}^{\\prime}=\\tilde{\\mathbf{z}}_{[kn]}\\). We have\n\n\\[\\Pr\\left[\\tilde{\\mathbf{z}}\\not\\in\\tilde{\\mathcal{Z}}\\right] \\leq\\Pr\\left[\\exists j\\in[kn]\\text{ s.t. }z^{\\prime}_{j}\\in\\left(c- \\frac{1}{n^{2}},c+\\frac{2}{n^{2}}\\right)\\right]\\] \\[\\quad+\\Pr\\left[\\Psi(\\mathbf{z}^{\\prime})\\text{ does not represent a hyperedge}\\right]. \\tag{1}\\]\n\nWe now bound the terms in the above RHS. First, since \\(\\mathbf{z}^{\\prime}\\) has the Gaussian distribution, then its components are drawn i.i.d. from a density function bounded by \\(\\frac{1}{2\\pi}\\). Hence, for a sufficiently large \\(n\\) we have\n\n\\[\\Pr\\left[\\exists j\\in[kn]\\text{ s.t. }z^{\\prime}_{j}\\in\\left(c-\\frac{1}{n^{2}},c+ \\frac{2}{n^{2}}\\right)\\right]\\leq kn\\cdot\\frac{1}{2\\pi}\\cdot\\frac{3}{n^{2}}= \\frac{3k}{2\\pi n}\\leq\\frac{\\log(n)}{n}. \\tag{2}\\]\n\nLet \\(\\mathbf{z}=\\Psi(\\mathbf{z}^{\\prime})\\). Note that \\(\\mathbf{z}\\) is a random vector whose components are drawn i.i.d. from a Bernoulli distribution, where the probability to get \\(0\\) is \\(\\frac{1}{n}\\). By Lemma A.10, \\(\\mathbf{z}\\) is an encoding of a hyperedge with probability at least \\(\\frac{1}{\\log(n)}\\). Combining it with Eq. (1) and (2),, we get for a sufficiently large \\(n\\) that\n\n\\[\\Pr\\left[\\tilde{\\mathbf{z}}\\not\\in\\tilde{\\mathcal{Z}}\\right]\\leq\\frac{\\log(n) }{n}+\\left(1-\\frac{1}{\\log(n)}\\right)\\leq 1-\\frac{1}{2\\log(n)}\\,\\]\n\nas required. \n\n**Lemma A.12**.: _If \\(\\mathcal{S}\\) is random, then for a sufficiently large \\(n\\) with probability larger than \\(\\frac{2}{3}\\) we have_\n\n\\[\\ell_{I}(h^{\\prime})>\\frac{2}{n}\\.\\]\n\nProof.: Let \\(\\tilde{\\mathcal{Z}}\\subseteq\\mathbb{R}^{n^{2}}\\) be such that \\(\\tilde{\\mathbf{z}}\\in\\tilde{\\mathcal{Z}}\\) iff \\(\\tilde{\\mathbf{z}}_{[kn]}\\) does not have components in the interval \\((c-\\frac{1}{n^{2}},c+\\frac{2}{n^{2}})\\), and \\(\\Psi(\\tilde{\\mathbf{z}}_{[kn]})=\\mathbf{z}^{S}\\) for a hyperedge \\(S\\). If \\(\\mathcal{S}\\) is random, then by the definition of our examples oracle, for every \\(i\\in[m(n)+n^{3}]\\) such that \\(\\tilde{\\mathbf{z}}_{i}\\in\\tilde{\\mathcal{Z}}\\), we have \\(\\tilde{y}_{i}=\\hat{b}\\) with probability \\(\\frac{1}{2}\\) and \\(\\tilde{y}_{i}=0\\) otherwise. Also, by the definition of the oracle, \\(\\tilde{y}_{i}\\) is independent of \\(S_{i}\\) and independent of the choice of the vector \\(\\tilde{\\mathbf{z}}_{i}\\) that corresponds to \\(\\mathbf{z}^{S_{i}}\\). If \\(\\hat{b}\\geq\\frac{9}{10}\\) then for a sufficiently large \\(n\\) the hypothesis \\(h^{\\prime}\\) satisfies for each random example \\((\\tilde{\\mathbf{z}}_{i},\\tilde{y}_{i})\\in\\tilde{\\mathcal{S}}_{I}\\) the following\n\n\\[\\Pr_{(\\tilde{\\mathbf{z}}_{i},\\tilde{y}_{i})} \\left[(h^{\\prime}(\\tilde{\\mathbf{z}}_{i})-\\tilde{y}_{i})^{2}\\geq \\frac{1}{5}\\right]\\] \\[\\geq\\Pr_{(\\tilde{\\mathbf{z}}_{i},\\tilde{y}_{i})}\\left[(h^{\\prime} (\\tilde{\\mathbf{z}}_{i})-\\tilde{y}_{i})^{2}\\geq\\frac{1}{5}\\left|\\left.\\tilde{ \\mathbf{z}}_{i}\\in\\tilde{\\mathcal{Z}}\\right]\\cdot\\Pr_{\\tilde{\\mathbf{z}}_{i}} \\left[\\tilde{\\mathbf{z}}_{i}\\in\\tilde{\\mathcal{Z}}\\right]\\right.\\] \\[\\geq\\Pr_{(\\tilde{\\mathbf{z}}_{i},\\tilde{y}_{i})}\\left[(h^{\\prime} (\\tilde{\\mathbf{z}}_{i})-\\tilde{y}_{i})^{2}\\geq\\left(\\frac{\\hat{b}}{2}\\right) ^{2}\\left|\\left.\\tilde{\\mathbf{z}}_{i}\\in\\tilde{\\mathcal{Z}}\\right]\\cdot\\Pr_{ \\tilde{\\mathbf{z}}_{i}}\\left[\\tilde{\\mathbf{z}}_{i}\\in\\tilde{\\mathcal{Z}} \\right]\\right.\\] \\[\\geq\\frac{1}{2}\\cdot\\Pr_{\\tilde{\\mathbf{z}}_{i}}\\left[\\tilde{ \\mathbf{z}}_{i}\\in\\tilde{\\mathcal{Z}}\\right]\\;.\\]\n\nIn Lemma A.11, we show that \\(\\Pr_{\\tilde{\\mathbf{z}}_{i}}\\left[\\tilde{\\mathbf{z}}_{i}\\in\\tilde{\\mathcal{Z}} \\right]\\geq\\frac{1}{2\\log(n)}\\). Hence,\n\n\\[\\Pr_{(\\tilde{\\mathbf{z}}_{i},\\tilde{y}_{i})}\\left[(h^{\\prime}(\\tilde{\\mathbf{z }}_{i})-\\tilde{y}_{i})^{2}\\geq\\frac{1}{5}\\right]\\geq\\frac{1}{2}\\cdot\\frac{1}{ 2\\log(n)}\\geq\\frac{1}{4\\log(n)}\\;.\\]\n\nThus, if \\(\\hat{b}\\geq\\frac{9}{10}\\) then we have\n\n\\[\\mathop{\\mathbb{E}}_{\\tilde{\\mathcal{S}}_{I}}\\left[\\ell_{I}(h^{\\prime})\\right] \\geq\\frac{1}{5}\\cdot\\frac{1}{4\\log(n)}=\\frac{1}{20\\log(n)}\\;.\\]\n\nTherefore, for large \\(n\\) we have\n\n\\[\\Pr\\left[\\mathop{\\mathbb{E}}_{\\tilde{\\mathcal{S}}_{I}}\\left[\\ell_{I}(h^{\\prime })\\right]\\geq\\frac{1}{20\\log(n)}\\right]\\geq 1-\\frac{1}{n}\\geq\\frac{7}{8}\\;.\\]\n\nSince, \\((h^{\\prime}(\\tilde{\\mathbf{z}})-\\tilde{y})^{2}\\in[0,\\hat{b}^{2}]\\) for all \\(\\tilde{\\mathbf{z}},\\tilde{y}\\) returned by the examples oracle, and the examples \\(\\tilde{\\mathbf{z}}_{i}\\) for \\(i\\in I\\) are i.i.d., then by Hoeffding's inequality, we have for a sufficiently large \\(n\\) that\n\n\\[\\Pr\\left[\\left|\\ell_{I}(h^{\\prime})-\\mathop{\\mathbb{E}}_{\\tilde{ \\mathcal{S}}_{I}}\\ell_{I}(h^{\\prime})\\right|\\geq\\frac{1}{n}\\right] =\\Pr\\left[\\left|\\ell_{I}(h^{\\prime})-\\mathop{\\mathbb{E}}_{\\tilde{ \\mathcal{S}}_{I}}\\ell_{I}(h^{\\prime})\\right|\\geq\\frac{1}{n}\\right|\\hat{b}\\leq \\frac{11}{10}\\right]\\cdot\\Pr\\left[\\hat{b}\\leq\\frac{11}{10}\\right]\\] \\[\\leq 2\\exp\\left(-\\frac{2n^{3}}{n^{2}(11/10)^{4}}\\right)\\cdot 1+1 \\cdot\\frac{1}{n}\\] \\[\\leq\\frac{1}{8}\\;.\\]\n\nHence, for large enough \\(n\\), with probability at least \\(1-\\frac{1}{8}-\\frac{1}{8}=\\frac{3}{4}>\\frac{2}{3}\\) we have both \\(\\mathop{\\mathbb{E}}_{\\tilde{\\mathcal{S}}_{I}}\\left[\\ell_{I}(h^{\\prime})\\right] \\geq\\frac{1}{20\\log(n)}\\) and \\(\\left|\\ell_{I}(h^{\\prime})-\\mathop{\\mathbb{E}}_{\\tilde{\\mathcal{S}}_{I}}\\ell_{ I}(h^{\\prime})\\right|\\leq\\frac{1}{n}\\), and thus\n\n\\[\\ell_{I}(h^{\\prime})\\geq\\frac{1}{20\\log(n)}-\\frac{1}{n}>\\frac{2}{n}\\;.\\]\n\n## Appendix B Proof of Corollary 3.1\n\nBy the proof of Theorem 3.1, under Assumption 2.1, there is no \\(\\mathrm{poly}(d)\\)-time algorithm \\(\\mathcal{L}_{s}\\) that satisfies the following: Let \\(\\boldsymbol{\\theta}\\in\\mathbb{R}^{p}\\) be \\(B\\)-bounded parameters of a depth-\\(3\\) network \\(N_{\\boldsymbol{\\theta}}:\\mathbb{R}^{d}\\to\\mathbb{R}\\), and let \\(\\tau,\\epsilon>0\\). Assume that \\(p,B,1/\\epsilon,1/\\tau\\leq\\mathrm{poly}(d)\\), and that the widths of the hidden layers in \\(\\mathcal{N}_{\\boldsymbol{\\theta}}\\) are \\(d\\) (i.e., the weight matrices are square). Let \\(\\boldsymbol{\\xi}\\in\\mathcal{N}(\\boldsymbol{0},\\tau^{2}I_{p})\\) and let \\(\\hat{\\boldsymbol{\\theta}}=\\boldsymbol{\\theta}+\\boldsymbol{\\xi}\\). Then, with probability at least \\(\\frac{3}{4}-\\frac{1}{1000}\\), given access to an examples oracle for \\(\\mathcal{N}_{\\hat{\\boldsymbol{\\theta}}}\\), the algorithm \\(\\mathcal{L}_{s}\\) returns a hypothesis \\(h\\) with \\(\\mathop{\\mathbb{E}}_{\\mathbf{x}}\\left[(h(\\mathbf{x})-N_{\\hat{\\boldsymbol{ \\theta}}})^{2}\\right]\\leq\\epsilon\\).\n\nNote that in the above, the requirements from \\(\\mathcal{L}_{s}\\) are somewhat weaker than in our original definition of learning with smoothed parameters. Indeed, we assume that the widths of the hidden layers are \\(d\\) and the required success probability is only \\(\\frac{3}{4}-\\frac{1}{1000}\\) (rather than \\(\\frac{3}{4}\\)). We now explain why the hardness result holds already under these conditions:* Note that if we change the assumption on the learning algorithm in proof of Theorem 3.1 such that it succeeds with probability at least \\(\\frac{3}{4}-\\frac{1}{1000}\\) (rather than \\(\\frac{3}{4}\\)), then in the case where \\(\\mathcal{S}\\) is pseudorandom we get that the algorithm \\(\\mathcal{A}\\) returns \\(1\\) with probability at least \\(1-\\left(\\frac{1}{4}+\\frac{1}{1000}+\\frac{1}{40}+\\frac{1}{40}+\\frac{1}{40}+\\frac {1}{40}\\right)\\) (see the proof of Lemma A.9), which is still greater than \\(\\frac{2}{3}\\). Also, the analysis of the case where \\(\\mathcal{S}\\) is random does not change, and thus in this case \\(\\widehat{\\mathcal{A}}\\) returns \\(0\\) with probability greater than \\(\\frac{2}{3}\\). Consequently, we still get distinguishing advantage greater than \\(\\frac{1}{3}\\).\n* Regarding the requirement on the widths, we note that in the proof of Theorem 3.1 the layers satisfy the following. The input dimension is \\(d=n^{2}\\), the width of the first hidden layer is at most \\(3n\\log(n)\\leq d\\), and the width of the second hidden layer is at most \\(\\log(n)+2n+n\\log(n)\\leq d\\) (all bounds are for a sufficiently large \\(d\\)). In order to get a network where all layers are of width \\(d\\), we add new neurons to the hidden layers, with incoming weights \\(0\\), outgoing weights \\(0\\), and bias terms \\(-1\\). Then, for an appropriate choice of \\(\\tau=1/\\operatorname{poly}(n)\\), even in the perturbed network the outputs of these new neurons will be \\(0\\) w.h.p. for every input \\(\\tilde{\\mathbf{z}}_{1},\\ldots,\\tilde{\\mathbf{z}}_{m(n)+n^{3}}\\), and thus they will not affect the network's output. Thus, using the same argument as in the proof of Theorem 3.1, we conclude that the hardness results holds already for network with square weight matrices.\n\nSuppose that there exists an efficient algorithm \\(\\mathcal{L}_{p}\\) that learns in the standard PAC framework depth-\\(3\\) neural networks where the minimal singular value of each weight matrix is lower bounded by \\(1/q(d)\\) for any polynomial \\(q(d)\\). We will use \\(\\mathcal{L}_{p}\\) to obtain an efficient algorithm \\(\\mathcal{L}_{s}\\) that learns depth-\\(3\\) networks with smoothed parameters as described above, and thus reach a contradiction.\n\nLet \\(\\boldsymbol{\\theta}\\in\\mathbb{R}^{p}\\) be \\(B\\)-bounded parameters of a depth-\\(3\\) network \\(N_{\\boldsymbol{\\theta}}:\\mathbb{R}^{d}\\to\\mathbb{R}\\), and let \\(\\tau,\\epsilon>0\\). Assume that \\(p,B,1/\\epsilon,1/\\tau\\leq\\operatorname{poly}(d)\\), and that the widths of the hidden layers in \\(\\mathcal{N}_{\\boldsymbol{\\theta}}\\) are \\(d\\). For random \\(\\boldsymbol{\\xi}\\sim\\mathcal{N}(\\boldsymbol{0},\\tau^{2}I_{p})\\) and \\(\\boldsymbol{\\hat{\\theta}}=\\boldsymbol{\\theta}+\\boldsymbol{\\xi}\\), the algorithm \\(\\mathcal{L}_{s}\\) has access to examples labeled by \\(N_{\\boldsymbol{\\hat{\\theta}}}\\). Using Lemma B.1 below with \\(t=\\frac{\\tau}{4}\\) and the union bound over the two weight matrices in \\(N_{\\boldsymbol{\\theta}}\\), we get that with probability at least \\(1-\\frac{2\\cdot 2.35}{\\sqrt{d}}\\geq 1-\\frac{1}{1000}\\) (for large enough \\(d\\)), the minimal singular values of all weight matrices in \\(\\boldsymbol{\\hat{\\theta}}\\) are at least \\(\\frac{\\tau}{d}\\geq\\frac{1}{q(d)}\\) for some sufficiently large polynomial \\(q(d)\\). Our algorithm \\(\\mathcal{L}_{s}\\) will simply run \\(\\mathcal{L}_{p}\\). Given that the minimal singular values of the weight matrices are at least \\(\\frac{1}{q(d)}\\), the algorithm \\(\\mathcal{L}_{p}\\) runs in time \\(\\operatorname{poly}(d)\\) and returns with probability at least \\(\\frac{3}{4}\\) a hypothesis \\(h\\) with \\(\\mathbb{E}_{\\mathbf{x}}\\left[(h(\\mathbf{x})-N_{\\boldsymbol{\\hat{\\theta}}}( \\mathbf{x}))^{2}\\right]\\leq\\epsilon\\). Overall, the algorithm \\(\\mathcal{L}_{s}\\) runs in \\(\\operatorname{poly}(d)\\) time, and with probability at least \\(\\frac{3}{4}-\\frac{1}{1000}\\) (over both \\(\\boldsymbol{\\xi}\\) and the internal randomness) returns a hypothesis \\(h\\) with loss at most \\(\\epsilon\\).\n\n**Lemma B.1** (Sankar et al. [37], Theorem 3.3).: _Let \\(W\\) be an arbitrary square matrix in \\(\\mathbb{R}^{d\\times d}\\), and let \\(P\\in\\mathbb{R}^{d\\times d}\\) be a random matrix, where each entry is drawn i.i.d. from \\(\\mathcal{N}(0,\\tau^{2})\\) for some \\(\\tau>0\\). Let \\(\\sigma_{d}\\) be the minimal singular value of the matrix \\(W+P\\). Then, for every \\(t>0\\) we have_\n\n\\[\\Pr_{P}\\left[\\sigma_{d}\\leq t\\right]\\leq 2.35\\cdot\\frac{t\\sqrt{d}}{\\tau}\\;.\\]\n\n## Appendix C Proof of Theorem 3.2\n\nThe proof follows similar ideas to the proof of Theorem 3.1. The main difference is that we need to handle here a smoothed discrete input distribution rather than the standard Gaussian distribution.\n\nFor a sufficiently large \\(n\\), let \\(\\mathcal{D}\\) be a distribution on \\(\\{0,1\\}^{n^{2}}\\), where each component is drawn i.i.d. from a Bernoulli distribution which takes the value \\(0\\) with probability \\(\\frac{1}{n}\\). Assume that there is a \\(\\operatorname{poly}(n)\\)-time algorithm \\(\\mathcal{L}\\) that learns depth-\\(3\\) neural networks with at most \\(n^{2}\\) hidden neurons and parameter magnitudes bounded by \\(n^{3}\\), with smoothed parameters and inputs, under the distribution \\(\\mathcal{D}\\), with \\(\\epsilon=\\frac{1}{n}\\) and \\(\\tau,\\omega=1/\\operatorname{poly}(n)\\) that we will specify later. Let \\(m(n)\\leq\\operatorname{poly}(n)\\) be the sample complexity of \\(\\mathcal{L}\\), namely, \\(\\mathcal{L}\\) uses a sample of size at most \\(m(n)\\) and returns with probability at least \\(\\frac{3}{4}\\) a hypothesis \\(h\\) with \\(\\mathbb{E}_{\\mathbf{z}\\sim\\widehat{\\mathcal{D}}}\\left[\\left(h(\\mathbf{z})-N_{ \\boldsymbol{\\hat{\\theta}}}(\\mathbf{z})\\right)^{2}\\right]\\leq\\epsilon=\\frac{1}{n}\\). Note that \\(\\widehat{\\mathcal{D}}\\) is the distribution \\(\\mathcal{D}\\) after smoothing with parameter \\(\\omega\\), and the vector \\(\\boldsymbol{\\hat{\\theta}}\\) is the parameters of the target network after smoothing with parameter \\(\\tau\\). Let \\(s>1\\) be a constant such that \\(n^{s}\\geq m(n)+n^{3}\\) for every sufficiently large \\(n\\). By Assumption 2.1, there exists a constant \\(k\\) and a predicate \\(P:\\{0,1\\}^{k}\\to\\{0,1\\}\\), such that is \\(\\frac{1}{3}\\)-PRG. We will show an efficient algorithm \\(\\mathcal{A}\\) with distinguishing advantage greater than \\(\\frac{1}{3}\\) and thus reach a contradiction.\n\nThroughout this proof, we will use some notations from the proof of Theorem 3.1. We repeat it here for convenience. For a hyperedge \\(S=(i_{1},\\ldots,i_{k})\\) we denote by \\(\\mathbf{z}^{S}\\in\\{0,1\\}^{kn}\\) the following encoding of \\(S\\): the vector \\(\\mathbf{z}^{S}\\) is a concatenation of \\(k\\) vectors in \\(\\{0,1\\}^{n}\\), such that the \\(j\\)-th vector has \\(0\\) in the \\(i_{j}\\)-th coordinate and \\(1\\) elsewhere. Thus, \\(\\mathbf{z}^{S}\\) consists of \\(k\\) size-\\(n\\) slices, each encoding a member of \\(S\\). For \\(\\mathbf{z}\\in\\{0,1\\}^{kn}\\), \\(i\\in[k]\\) and \\(j\\in[n]\\), we denote \\(z_{i,j}=z_{(i-1)n+j}\\). That is, \\(z_{i,j}\\) is the \\(j\\)-th component in the \\(i\\)-th slice in \\(\\mathbf{z}\\). For \\(\\mathbf{x}\\in\\{0,1\\}^{n}\\), let \\(P_{\\mathbf{x}}:\\{0,1\\}^{kn}\\to\\{0,1\\}\\) be such that for every hyperedge \\(S\\) we have \\(P_{\\mathbf{x}}(\\mathbf{z}^{S})=P(\\mathbf{x}_{S})\\). For \\(\\tilde{\\mathbf{z}}\\in\\mathbb{R}^{n^{2}}\\) we denote \\(\\tilde{\\mathbf{z}}_{[kn]}=(\\tilde{z}_{1},\\ldots,\\tilde{z}_{kn})\\), namely, the first \\(kn\\) components of \\(\\tilde{\\mathbf{z}}\\) (assuming \\(n^{2}\\geq kn\\)).\n\n### Defining the target network for \\(\\mathcal{L}\\)\n\nSince our goal is to use the algorithm \\(\\mathcal{L}\\) for breaking PRGs, in this subsection we define a neural network \\(\\tilde{N}:\\mathbb{R}^{n^{2}}\\to\\mathbb{R}\\) that we will later use as a target network for \\(\\mathcal{L}\\). The network \\(\\tilde{N}\\) contains the subnetworks \\(N_{1},N_{2}\\) that we define below.\n\nLet \\(N_{1}\\) be a depth-\\(1\\) neural network (i.e., one layer, with activations in the output neurons) with input dimension \\(kn\\), at most \\(\\log(n)\\) output neurons, and parameter magnitudes bounded by \\(n^{3}\\) (all bounds are for a sufficiently large \\(n\\)), which satisfies the following. We denote the set of output neurons of \\(N_{1}\\) by \\(\\mathcal{E}_{1}\\). Let \\(\\mathbf{z}^{\\prime}\\in\\{0,1\\}^{kn}\\) be an input to \\(N_{1}\\) such that \\(\\mathbf{z}^{\\prime}=\\mathbf{z}^{S}\\) for some hyperedge \\(S\\). Thus, even though \\(N_{1}\\) takes inputs in \\(\\mathbb{R}^{kn}\\), we consider now its behavior for an input \\(\\mathbf{z}^{\\prime}\\) with discrete components in \\(\\{0,1\\}\\). Fix some \\(\\mathbf{x}\\in\\{0,1\\}^{n}\\). Then, for \\(S\\) with \\(P_{\\mathbf{x}}(\\mathbf{z}^{S})=0\\) the inputs to all output neurons \\(\\mathcal{E}_{1}\\) are at most \\(-1\\), and for \\(S\\) with \\(P_{\\mathbf{x}}(\\mathbf{z}^{S})=1\\) there exists a neuron in \\(\\mathcal{E}_{1}\\) with input at least \\(2\\). Recall that our definition of a neuron's input includes the addition of the bias term. The construction of the network \\(N_{1}\\) is given in Lemma A.2. Note that the network \\(N_{1}\\) depends on \\(\\mathbf{x}\\). Let \\(N_{1}^{\\prime}:\\mathbb{R}^{kn}\\to\\mathbb{R}\\) be a depth-\\(2\\) neural network with no activation function in the output neuron, obtained from \\(N_{1}\\) by summing the outputs from all neurons \\(\\mathcal{E}_{1}\\).\n\nLet \\(N_{2}\\) be a depth-\\(1\\) neural network (i.e., one layer, with activations in the output neurons) with input dimension \\(kn\\), at most \\(2n\\) output neurons, and parameter magnitudes bounded by \\(n^{3}\\) (for a sufficiently large \\(n\\)), which satisfies the following. We denote the set of output neurons of \\(N_{2}\\) by \\(\\mathcal{E}_{2}\\). Let \\(\\mathbf{z}^{\\prime}\\in\\{0,1\\}^{kn}\\) be an input to \\(N_{2}\\) (note that it has components only in \\(\\{0,1\\}\\)). If \\(\\mathbf{z}^{\\prime}\\) is an encoding of a hyperedge then the inputs to all output neurons \\(\\mathcal{E}_{2}\\) are at most \\(-1\\), and otherwise there exists a neuron in \\(\\mathcal{E}_{2}\\) with input at least \\(2\\). The construction of the network \\(N_{2}\\) is given in Lemma A.4. Let \\(N_{2}^{\\prime}:\\mathbb{R}^{kn}\\to\\mathbb{R}\\) be a depth-\\(2\\) neural network with no activation function in the output neuron, obtained from \\(N_{2}\\) by summing the outputs from all neurons \\(\\mathcal{E}_{2}\\).\n\nLet \\(N^{\\prime}:\\mathbb{R}^{kn}\\to\\mathbb{R}\\) be a depth-\\(2\\) network obtained from \\(N_{1}^{\\prime},N_{2}^{\\prime}\\) as follows. For \\(\\mathbf{z}^{\\prime}\\in\\mathbb{R}^{kn}\\) we have \\(N^{\\prime}(\\mathbf{z}^{\\prime})=\\left[1-N_{1}^{\\prime}(\\mathbf{z}^{\\prime})-N _{2}^{\\prime}(\\mathbf{z}^{\\prime})\\right]_{+}\\). The network \\(N^{\\prime}\\) has at most \\(n^{2}\\) neurons, and parameter magnitudes bounded by \\(n^{3}\\) (all bounds are for a sufficiently large \\(n\\)). Finally, let \\(\\tilde{N}:\\mathbb{R}^{n^{2}}\\to\\mathbb{R}\\) be a depth-\\(2\\) neural network such that \\(\\tilde{N}(\\tilde{\\mathbf{z}})=N^{\\prime}\\left(\\tilde{\\mathbf{z}}_{[kn]}\\right)\\).\n\nDefining the noise magnitudes \\(\\tau,\\omega\\) and analyzing the perturbed network under perturbed inputs\n\nIn order to use the algorithm \\(\\mathcal{L}\\) w.r.t. some neural network with parameters \\(\\boldsymbol{\\theta}\\) and a certain input distribution, we need to implement an examples oracle, such that the examples are drawn from a smoothed input distribution, and labeled according to a neural network with parameters \\(\\boldsymbol{\\theta}+\\boldsymbol{\\xi}\\), where \\(\\boldsymbol{\\xi}\\) is a random perturbation. Specifically, we use \\(\\mathcal{L}\\) with an examples oracle where the input distribution \\(\\tilde{\\mathcal{D}}\\) is obtained from \\(\\mathcal{D}\\) by smoothing, and the labels correspond to a network \\(\\hat{N}:\\mathbb{R}^{n^{2}}\\to\\mathbb{R}\\) obtained from \\(\\tilde{N}\\) (w.r.t. an appropriate \\(\\mathbf{x}\\in\\{0,1\\}^{n}\\) in the construction of \\(N_{1}\\)) by adding a small perturbation to the parameters. The smoothing magnitudes \\(\\omega,\\tau\\) of the inputs and the network's parameters (respectively) are such that the following hold.\n\nWe first choose the parameter \\(\\tau=1/\\operatorname{poly}(n)\\) as follows. Let \\(f_{\\boldsymbol{\\theta}}:\\mathbb{R}^{n^{2}}\\to\\mathbb{R}\\) be any depth-\\(2\\) neural network parameterized by \\(\\boldsymbol{\\theta}\\in\\mathbb{R}^{r}\\) for some \\(r>0\\) with at most \\(n^{2}\\) neurons, and parameter magnitudes bounded by \\(n^{3}\\) (note that \\(r\\) is polynomial in \\(n\\)). Then, \\(\\tau\\) is such that with probability at least \\(1-\\frac{1}{n}\\) over \\(\\boldsymbol{\\xi}\\sim\\mathcal{N}(\\mathbf{0},\\tau^{2}I_{r})\\), we have \\(|\\xi_{i}|\\leq\\frac{1}{10}\\) for all \\(i\\in[r]\\), and the network \\(f_{\\boldsymbol{\\theta}+\\boldsymbol{\\xi}}\\) is such that for every input \\(\\tilde{\\mathbf{z}}\\in\\mathbb{R}^{n^{2}}\\) with \\(\\|\\tilde{\\mathbf{z}}\\|\\leq n\\) and every neuron we have: Let \\(a,b\\) be the inputs to the neuron in the computations \\(f_{\\mathbf{\\theta}}(\\tilde{\\mathbf{z}})\\) and \\(f_{\\mathbf{\\theta}+\\tilde{\\mathbf{\\xi}}}(\\tilde{\\mathbf{z}})\\) (respectively), then \\(|a-b|\\leq\\frac{1}{4}\\). Thus, \\(\\tau\\) is sufficiently small, such that w.h.p. adding i.i.d. noise \\(\\mathcal{N}(0,\\tau^{2})\\) to each parameter does not change the inputs to the neurons by more than \\(\\frac{1}{4}\\). Note that such an inverse-polynomial \\(\\tau\\) exists, since when the network size, parameter magnitudes, and input size are bounded by some \\(\\mathrm{poly}(n)\\), then the input to each neuron in \\(f_{\\mathbf{\\theta}}(\\tilde{\\mathbf{z}})\\) is \\(\\mathrm{poly}(n)\\)-Lipschitz as a function of \\(\\mathbf{\\theta}\\), and thus it suffices to choose \\(\\tau\\) that implies with probability at least \\(1-\\frac{1}{n}\\) that \\(\\|\\mathbf{\\xi}\\|\\leq\\frac{1}{q(n)}\\) for a sufficiently large polynomial \\(q(n)\\) (see Lemma A.7 for details).\n\nNext, we choose the parameter \\(\\omega=1/\\mathrm{poly}(n)\\) as follows. Let \\(f_{\\mathbf{\\theta}}:\\mathbb{R}^{n^{2}}\\to\\mathbb{R}\\) be any depth-\\(2\\) neural network parameterized by \\(\\mathbf{\\theta}\\) with at most \\(n^{2}\\) neurons, and parameter magnitudes bounded by \\(n^{3}+\\frac{1}{10}\\). Then, \\(\\omega\\) is such that for every \\(\\mathbf{z}\\in\\{0,1\\}^{n^{2}}\\), with probability at least \\(1-\\exp(-n/2)\\) over \\(\\mathbf{\\zeta}\\sim\\mathcal{N}(\\mathbf{0},\\omega^{2}I_{n^{2}})\\) the following holds for every neuron in the \\(f_{\\mathbf{\\theta}}\\): Let \\(a,b\\) be the inputs to the neuron in the computations \\(f_{\\mathbf{\\theta}}(\\mathbf{z})\\) and \\(f_{\\mathbf{\\theta}}(\\mathbf{z}+\\mathbf{\\zeta})\\) (respectively), then \\(|a-b|\\leq\\frac{1}{4}\\). Thus, \\(\\omega\\) is sufficiently small, such that w.h.p. adding noise \\(\\mathcal{N}(\\mathbf{0},\\omega^{2}I_{n^{2}})\\) to the input \\(\\mathbf{z}\\) does not change the inputs to the neurons by more than \\(\\frac{1}{4}\\). Note that such an inverse-polynomial \\(\\omega\\) exists, since when the network size and parameter magnitudes are bounded by some \\(\\mathrm{poly}(n)\\), then the input to each neuron in \\(f_{\\mathbf{\\theta}}(\\mathbf{z})\\) is \\(\\mathrm{poly}(n)\\)-Lipschitz as a function of \\(\\mathbf{z}\\), and thus it suffices to choose \\(\\omega\\) that implies with probability at least \\(1-\\exp(-n/2)\\) that \\(\\|\\mathbf{\\zeta}\\|\\leq\\frac{1}{q(n)}\\) for a sufficiently large polynomial \\(q(n)\\) (see Lemma A.7 for details).\n\nLet \\(\\tilde{\\mathbf{\\theta}}\\in\\mathbb{R}^{p}\\) be the parameters of the network \\(\\tilde{N}\\). Recall that the parameters vector \\(\\tilde{\\mathbf{\\theta}}\\) is the concatenation of all weight matrices and bias terms. Let \\(\\hat{\\mathbf{\\theta}}\\in\\mathbb{R}^{p}\\) be the parameters of \\(\\hat{N}\\), namely, \\(\\hat{\\mathbf{\\theta}}=\\tilde{\\mathbf{\\theta}}+\\mathbf{\\xi}\\) where \\(\\mathbf{\\xi}\\sim\\mathcal{N}(\\mathbf{0},\\tau^{2}I_{p})\\). By our choice of \\(\\tau\\) and the construction of the networks \\(N_{1},N_{2}\\), with probability at least \\(1-\\frac{1}{n}\\) over \\(\\mathbf{\\xi}\\), for every \\(\\mathbf{z}\\in\\{0,1\\}^{n^{2}}\\) the following holds: Let \\(\\mathbf{\\zeta}\\sim\\mathcal{N}(\\mathbf{0},\\omega^{2}I_{n^{2}})\\) and let \\(\\hat{\\mathbf{z}}=\\mathbf{z}+\\mathbf{\\zeta}\\). Then with probability at least \\(1-\\exp(-n/2)\\) over \\(\\mathbf{\\zeta}\\) the differences between inputs to all neurons in the computations \\(\\hat{N}(\\hat{\\mathbf{z}})\\) and \\(\\tilde{N}(\\mathbf{z})\\) are at most \\(\\frac{1}{2}\\). Indeed, w.h.p. for all \\(\\mathbf{z}\\in\\{0,1\\}^{n^{2}}\\) the computations \\(\\tilde{N}(\\mathbf{z})\\) and \\(\\hat{N}(\\mathbf{z})\\) are roughly similar (up to change of \\(1/4\\) in the input to each neuron), and w.h.p. the computations \\(\\hat{N}(\\mathbf{z})\\) and \\(\\hat{N}(\\hat{\\mathbf{z}})\\) are roughly similar (up to change of \\(1/4\\) in the input to each neuron). Thus, with probability at least \\(1-\\frac{1}{n}\\) over \\(\\mathbf{\\xi}\\), the network \\(\\hat{N}\\) is such that for every \\(\\mathbf{z}\\in\\{0,1\\}^{n^{2}}\\), we have with probability at least \\(1-\\exp(-n/2)\\) over \\(\\mathbf{\\zeta}\\) that the computation \\(\\hat{N}(\\hat{\\mathbf{z}})\\) satisfies the following properties, where \\(\\mathbf{z}^{\\prime}:=\\mathbf{z}_{[kn]}\\):\n\n1. If \\(\\mathbf{z}^{\\prime}=\\mathbf{z}^{S}\\) for some hyperedge \\(S\\), then the inputs to \\(\\mathcal{E}_{1}\\) satisfy: * If \\(P_{\\mathbf{x}}(\\mathbf{z}^{S})=0\\) the inputs to all neurons in \\(\\mathcal{E}_{1}\\) are at most \\(-\\frac{1}{2}\\). * If \\(P_{\\mathbf{x}}(\\mathbf{z}^{S})=1\\) there exists a neuron in \\(\\mathcal{E}_{1}\\) with input at least \\(\\frac{3}{2}\\).\n2. The inputs to \\(\\mathcal{E}_{2}\\) satisfy: * If \\(\\mathbf{z}^{\\prime}\\) is an encoding of a hyperedge then the inputs to all neurons \\(\\mathcal{E}_{2}\\) are at most \\(-\\frac{1}{2}\\). * Otherwise, there exists a neuron in \\(\\mathcal{E}_{2}\\) with input at least \\(\\frac{3}{2}\\).\n\n### Stating the algorithm \\(\\mathcal{A}\\)\n\nGiven a sequence \\((S_{1},y_{1}),\\ldots,(S_{n^{*}},y_{n^{*}})\\), where \\(S_{1},\\ldots,S_{n^{*}}\\) are i.i.d. random hyperedges, the algorithm \\(\\mathcal{A}\\) needs to distinguish whether \\(\\mathbf{y}=(y_{1},\\ldots,y_{n^{*}})\\) is random or that \\(\\mathbf{y}=(P(\\mathbf{x}_{S_{1}}),\\ldots,P(\\mathbf{x}_{S_{n^{*}}}))=(P_{ \\mathbf{x}}(\\mathbf{z}^{S_{1}}),\\ldots,P_{\\mathbf{x}}(\\mathbf{z}^{S_{n^{*}}}))\\) for a random \\(\\mathbf{x}\\in\\{0,1\\}^{n}\\). Let \\(\\mathcal{S}=((\\mathbf{z}^{S_{1}},y_{1}),\\ldots,(\\mathbf{z}^{S_{n^{*}}},y_{n^{*}}))\\).\n\nWe use the efficient algorithm \\(\\mathcal{L}\\) in order to obtain distinguishing advantage greater than \\(\\frac{1}{3}\\) as follows.\n\nLet \\(\\mathbf{\\xi}\\) be a random perturbation, and let \\(\\hat{N}\\) be the perturbed network as defined above, w.r.t. the unknown \\(\\mathbf{x}\\in\\{0,1\\}^{n}\\). Note that given a perturbation \\(\\mathbf{\\xi}\\), only the weights in the second layer of the subnetwork \\(N_{1}\\) in \\(\\hat{N}\\) are unknown, since all other parameters do not depend on \\(\\mathbf{x}\\). The algorithm \\(\\mathcal{A}\\) runs \\(\\mathcal{L}\\) with the following examples oracle. In the \\(i\\)-th call, the oracle first draws \\(\\mathbf{z}^{\\prime}\\in\\{0,1\\}^{kn}\\) such that each component is drawn i.i.d. from a Bernoulli distribution which takes the value \\(0\\) with probability \\(\\frac{1}{n}\\). If \\(\\mathbf{z}^{\\prime}\\) is an encoding of a hyperedge then the oracle replaces \\(\\mathbf{z}^{\\prime}\\) with \\(\\mathbf{z}^{S_{i}}\\). Let \\(\\mathbf{z}\\in\\{0,1\\}^{n^{2}}\\) be such that \\(\\mathbf{z}_{[kn]}=\\mathbf{z}^{\\prime}\\), and the other \\(n^{2}-kn\\) components of \\(\\mathbf{z}\\) are drawn i.i.d. from a Bernoulli distribution which takes the value \\(0\\) with probability \\(\\frac{1}{n}\\). Note that the vector \\(\\mathbf{z}\\) has the distribution \\(\\mathcal{D}\\), since replacing an encoding of a random hyperedge by an encoding of another random hyperedge does not change the distribution of \\(\\mathbf{z}^{\\prime}\\). Let \\(\\hat{\\mathbf{z}}=\\mathbf{z}+\\mathbf{\\zeta}\\), where \\(\\mathbf{\\zeta}\\sim\\mathcal{N}(\\mathbf{0},\\omega^{2}I_{n^{2}})\\). Note that \\(\\hat{\\mathbf{z}}\\) has the distribution \\(\\hat{\\mathcal{D}}\\). Let \\(\\hat{b}\\in\\mathbb{R}\\) be the bias term of the output neuron of \\(\\hat{N}\\). The oracle returns \\((\\hat{\\mathbf{z}},\\hat{y})\\), where the labels \\(\\hat{y}\\) are chosen as follows:\n\n* If \\(\\mathbf{z}^{\\prime}\\) is not an encoding of a hyperedge, then \\(\\hat{y}=0\\).\n* If \\(\\mathbf{z}^{\\prime}\\) is an encoding of a hyperedge:\n* If \\(y_{i}=0\\) we set \\(\\hat{y}=\\hat{b}\\).\n* If \\(y_{i}=1\\) we set \\(\\hat{y}=0\\).\n\nLet \\(h\\) be the hypothesis returned by \\(\\mathcal{L}\\). Recall that \\(\\mathcal{L}\\) uses at most \\(m(n)\\) examples, and hence \\(\\mathcal{S}\\) contains at least \\(n^{3}\\) examples that \\(\\mathcal{L}\\) cannot view. We denote the indices of these examples by \\(I=\\{m(n)+1,\\ldots,m(n)+n^{3}\\}\\), and the examples by \\(\\mathcal{S}_{I}=\\{(\\mathbf{z}^{\\mathcal{S}_{i}},y_{i})\\}_{i\\in I}\\). By \\(n^{3}\\) additional calls to the oracle, the algorithm \\(\\mathcal{A}\\) obtains the examples \\(\\hat{\\mathcal{S}_{I}}=\\{(\\hat{\\mathbf{z}}_{i},\\hat{y}_{i})\\}_{i\\in I}\\) that correspond to \\(\\mathcal{S}_{I}\\). Let \\(h^{\\prime}\\) be a hypothesis such that for all \\(\\tilde{\\mathbf{z}}\\in\\mathbb{R}^{n^{2}}\\) we have \\(h^{\\prime}(\\tilde{\\mathbf{z}})=\\max\\{0,\\min\\{\\hat{b},h(\\tilde{\\mathbf{z}})\\}\\}\\), thus, for \\(\\hat{b}\\geq 0\\) the hypothesis \\(h^{\\prime}\\) is obtained from \\(h\\) by clipping the output to the interval \\([0,\\hat{b}]\\). Let \\(\\ell_{I}(h^{\\prime})=\\frac{1}{|I|}\\sum_{i\\in I}(h^{\\prime}(\\hat{\\mathbf{z}}_{i })-\\hat{y}_{i})^{2}\\). Now, if \\(\\ell_{I}(h^{\\prime})\\leq\\frac{2}{n}\\), then \\(\\mathcal{A}\\) returns \\(1\\), and otherwise it returns \\(0\\). We remark that the decision of our algorithm is based on \\(h^{\\prime}\\) (rather than \\(h\\)) since we need the outputs to be bounded, in order to allow using Hoeffding's inequality in our analysis, which we discuss in the next subsection.\n\n### Analyzing the algorithm \\(\\mathcal{A}\\)\n\nNote that the algorithm \\(\\mathcal{A}\\) runs in \\(\\operatorname{poly}(n)\\) time. We now show that if \\(\\mathcal{S}\\) is pseudorandom then \\(\\mathcal{A}\\) returns \\(1\\) with probability greater than \\(\\frac{2}{3}\\), and if \\(\\mathcal{S}\\) is random then \\(\\mathcal{A}\\) returns \\(1\\) with probability less than \\(\\frac{1}{3}\\). To that end, we use similar arguments to the proof of Theorem 3.1.\n\nIn Lemma C.1, we show that if \\(\\mathcal{S}\\) is pseudorandom then with probability at least \\(\\frac{39}{40}\\) (over \\(\\mathbf{\\xi}\\sim\\mathcal{N}(\\mathbf{0},\\tau^{2}I_{p})\\) and \\(\\mathbf{\\zeta}_{i}\\sim\\mathcal{N}(\\mathbf{0},\\omega^{2}I_{n^{2}})\\) for all \\(i\\in[m(n)]\\)) the examples \\((\\hat{\\mathbf{z}}_{1},\\hat{y}_{1}),\\ldots,(\\hat{\\mathbf{z}}_{m(n)},\\hat{y}_{m (n)})\\) returned by the oracle are realized by \\(\\hat{N}\\). Recall that the algorithm \\(\\mathcal{L}\\) is such that with probability at least \\(\\frac{3}{4}\\) (over \\(\\mathbf{\\xi}\\sim\\mathcal{N}(\\mathbf{0},\\tau^{2}I_{p})\\), the i.i.d. inputs \\(\\hat{\\mathbf{z}}_{i}\\sim\\hat{\\mathcal{D}}\\), and possibly its internal randomness), given a size-\\(m(n)\\) dataset labeled by \\(\\hat{N}\\), it returns a hypothesis \\(h\\) such that \\(\\mathbb{E}_{\\hat{\\mathbf{z}}\\sim\\hat{\\mathcal{D}}}\\left[(h(\\hat{\\mathbf{z}})- \\hat{N}(\\hat{\\mathbf{z}}))^{2}\\right]\\leq\\frac{1}{n}\\). Hence, with probability at least \\(\\frac{3}{4}-\\frac{1}{40}\\) the algorithm \\(\\mathcal{L}\\) returns such a good hypothesis \\(h\\), given \\(m(n)\\) examples labeled by our examples oracle. Indeed, note that \\(\\mathcal{L}\\) can return a bad hypothesis only if the random choices are either bad for \\(\\mathcal{L}\\) (when used with realizable examples) or bad for the realizability of the examples returned by our oracle. By the definition of \\(h^{\\prime}\\) and the construction of \\(\\hat{N}\\), if \\(h\\) has small error then \\(h^{\\prime}\\) also has small error, namely,\n\n\\[\\mathbb{E}_{\\hat{\\mathbf{z}}\\sim\\hat{\\mathcal{D}}}\\left[(h^{\\prime}(\\hat{ \\mathbf{z}})-\\hat{N}(\\hat{\\mathbf{z}}))^{2}\\right]\\leq\\mathbb{E}_{\\hat{\\mathbf{ z}}\\sim\\hat{\\mathcal{D}}}\\left[(h(\\hat{\\mathbf{z}})-\\hat{N}(\\hat{\\mathbf{z}}))^{2} \\right]\\leq\\frac{1}{n}\\;.\\]\n\nLet \\(\\hat{\\ell}_{I}(h^{\\prime})=\\frac{1}{|I|}\\sum_{i\\in I}(h^{\\prime}(\\hat{ \\mathbf{z}}_{i})-\\hat{N}(\\hat{\\mathbf{z}}_{i}))^{2}\\). Recall that by our choice of \\(\\tau\\) we have \\(\\Pr[\\hat{b}>\\frac{11}{10}]\\leq\\frac{1}{n}\\). Since, \\((h^{\\prime}(\\hat{\\mathbf{z}})-\\hat{N}(\\hat{\\mathbf{z}}))^{2}\\in[0,\\hat{b}^{2}]\\) for all \\(\\hat{\\mathbf{z}}\\in\\mathbb{R}^{n^{2}}\\), by Hoeffding's inequality, we have for a sufficiently large \\(n\\) that\n\n\\[\\Pr\\left[\\left|\\hat{\\ell}_{I}(h^{\\prime})-\\mathbb{E}_{\\hat{\\mathcal{ S}}_{I}}\\,\\hat{\\ell}_{I}(h^{\\prime})\\right|\\geq\\frac{1}{n}\\right] =\\Pr\\left[\\left|\\hat{\\ell}_{I}(h^{\\prime})-\\mathbb{E}_{\\hat{ \\mathcal{S}}_{I}}\\,\\hat{\\ell}_{I}(h^{\\prime})\\right|\\geq\\frac{1}{n}\\right] \\hat{b}\\leq\\frac{11}{10}\\right]\\cdot\\Pr\\left[\\hat{b}\\leq\\frac{11}{10}\\right]\\] \\[\\leq 2\\exp\\left(-\\frac{2n^{3}}{n^{2}(11/10)^{4}}\\right)\\cdot 1+1 \\cdot\\frac{1}{n}\\] \\[\\leq\\frac{1}{40}\\;.\\]Moreover, by Lemma C.1,\n\n\\[\\Pr\\left[\\ell_{I}(h^{\\prime})\\neq\\hat{\\ell}_{I}(h^{\\prime})\\right]\\leq\\Pr\\left[ \\exists i\\in I\\text{ s.t. }\\hat{y}_{i}\\neq\\hat{N}(\\hat{\\mathbf{z}}_{i})\\right]\\leq\\frac{1}{40}\\;.\\]\n\nOverall, by the union bound we have with probability at least \\(1-\\left(\\frac{1}{4}+\\frac{1}{40}+\\frac{1}{40}+\\frac{1}{40}\\right)>\\frac{2}{3}\\) for sufficiently large \\(n\\) that:\n\n* \\(\\mathbb{E}_{\\hat{\\mathcal{S}}_{I}}\\,\\hat{\\ell}_{I}(h^{\\prime})=\\mathbb{E}_{ \\hat{\\mathbf{z}}\\sim\\hat{\\mathcal{D}}}\\left[(h^{\\prime}(\\hat{\\mathbf{z}})-\\hat{ N}(\\hat{\\mathbf{z}}))^{2}\\right]\\leq\\frac{1}{n}\\).\n* \\(\\left|\\hat{\\ell}_{I}(h^{\\prime})-\\mathbb{E}_{\\hat{\\mathcal{S}}_{I}}\\,\\hat{\\ell }_{I}(h^{\\prime})\\right|\\leq\\frac{1}{n}\\).\n* \\(\\ell_{I}(h^{\\prime})-\\hat{\\ell}_{I}(h^{\\prime})=0\\).\n\nCombining the above, we get that if \\(\\mathcal{S}\\) is pseudorandom, then with probability greater than \\(\\frac{2}{3}\\) we have\n\n\\[\\ell_{I}(h^{\\prime})=\\left(\\ell_{I}(h^{\\prime})-\\hat{\\ell}_{I}(h^{\\prime}) \\right)+\\left(\\hat{\\ell}_{I}(h^{\\prime})-\\underset{\\hat{\\mathcal{S}}_{I}}{ \\mathbb{E}}\\,\\hat{\\ell}_{I}(h^{\\prime})\\right)+\\underset{\\hat{\\mathcal{S}}_{I}} {\\mathbb{E}}\\,\\hat{\\ell}_{I}(h^{\\prime})\\leq 0+\\frac{1}{n}+\\frac{1}{n}=\\frac{2}{n}\\;.\\]\n\nWe now consider the case where \\(\\mathcal{S}\\) is random. For an example \\(\\hat{\\mathbf{z}}_{i}=\\mathbf{z}_{i}+\\boldsymbol{\\zeta}_{i}\\) returned by the oracle, we denote \\(\\mathbf{z}_{i}^{\\prime}=(\\mathbf{z}_{i})_{[kn]}\\in\\{0,1\\}^{kn}\\). Thus, \\(\\mathbf{z}_{i}^{\\prime}\\) is the input that the oracle used before adding the \\(n^{2}-kn\\) additional components and adding noise \\(\\boldsymbol{\\zeta}_{i}\\). Let \\(\\mathcal{Z}^{\\prime}\\subseteq\\{0,1\\}^{kn}\\) be such that \\(\\mathbf{z}^{\\prime}\\in\\mathcal{Z}^{\\prime}\\) iff \\(\\mathbf{z}^{\\prime}=\\mathbf{z}^{\\mathcal{S}}\\) for some hyperedge \\(S\\). If \\(\\mathcal{S}\\) is random, then by the definition of our examples oracle, for every \\(i\\in[m(n)+n^{3}]\\) such that \\(\\mathbf{z}_{i}^{\\prime}\\in\\mathcal{Z}^{\\prime}\\), we have \\(\\hat{y}_{i}=\\hat{b}\\) with probability \\(\\frac{1}{2}\\) and \\(\\hat{y}_{i}=0\\) otherwise. Also, by the definition of the oracle, \\(\\hat{y}_{i}\\) is independent of \\(S_{i}\\), independent of the \\(n^{2}-kn\\) additional components that where added, and independent of the noise \\(\\boldsymbol{\\zeta}_{i}\\sim\\mathcal{N}(\\mathbf{0},\\omega^{2}I_{n^{2}})\\) that corresponds to \\(\\hat{\\mathbf{z}}_{i}\\).\n\nIf \\(\\hat{b}\\geq\\frac{9}{10}\\) then for a sufficiently large \\(n\\) the hypothesis \\(h^{\\prime}\\) satisfies for each random example \\((\\hat{\\mathbf{z}}_{i},\\hat{y}_{i})\\in\\hat{\\mathcal{S}}_{I}\\) the following:\n\n\\[\\Pr_{(\\hat{\\mathbf{z}}_{i},\\hat{y}_{i})} \\left[(h^{\\prime}(\\hat{\\mathbf{z}}_{i})-\\hat{y}_{i})^{2}\\geq \\frac{1}{5}\\right]\\] \\[\\geq\\Pr_{(\\hat{\\mathbf{z}}_{i},\\hat{y}_{i})}\\left[(h^{\\prime}( \\hat{\\mathbf{z}}_{i})-\\hat{y}_{i})^{2}\\geq\\frac{1}{5}\\left|\\,\\,\\mathbf{z}_{i}^ {\\prime}\\in\\mathcal{Z}^{\\prime}\\right|\\cdot\\Pr\\left[\\mathbf{z}_{i}^{\\prime}\\in \\mathcal{Z}^{\\prime}\\right]\\right.\\] \\[\\geq\\frac{1}{2}\\cdot\\Pr\\left[\\mathbf{z}_{i}^{\\prime}\\in\\mathcal{ Z}^{\\prime}\\right]\\;.\\]\n\nIn Lemma A.10, we show that for a sufficiently large \\(n\\) we have \\(\\Pr\\left[\\mathbf{z}_{i}^{\\prime}\\in\\mathcal{Z}^{\\prime}\\right]\\geq\\frac{1}{\\log(n)}\\). Hence,\n\n\\[\\Pr_{(\\hat{\\mathbf{z}}_{i},\\hat{y}_{i})}\\left[(h^{\\prime}(\\hat{\\mathbf{z}}_{i })-\\hat{y}_{i})^{2}\\geq\\frac{1}{5}\\right]\\geq\\frac{1}{2}\\cdot\\frac{1}{\\log(n) }\\geq\\frac{1}{2\\log(n)}\\;.\\]\n\nThus, if \\(\\hat{b}\\geq\\frac{9}{10}\\) then we have\n\n\\[\\underset{\\hat{\\mathcal{S}}_{I}}{\\mathbb{E}}\\left[\\ell_{I}(h^{\\prime})\\right] \\geq\\frac{1}{5}\\cdot\\frac{1}{2\\log(n)}=\\frac{1}{10\\log(n)}\\;.\\]\n\nTherefore, for large \\(n\\) we have\n\n\\[\\Pr\\left[\\underset{\\hat{\\mathcal{S}}_{I}}{\\mathbb{E}}\\left[\\ell_{I}(h^{\\prime}) \\right]\\geq\\frac{1}{10\\log(n)}\\right]\\geq 1-\\frac{1}{n}\\geq\\frac{7}{8}\\;.\\]Since, \\((h^{\\prime}(\\hat{\\mathbf{z}})-\\hat{y})^{2}\\in[0,\\hat{b}^{2}]\\) for all \\(\\hat{\\mathbf{z}},\\hat{y}\\) returned by the examples oracle, and the examples \\(\\hat{\\mathbf{z}}_{i}\\) for \\(i\\in I\\) are i.i.d., then by Hoeffding's inequality, we have for a sufficiently large \\(n\\) that\n\n\\[\\Pr\\left[\\left|\\ell_{I}(h^{\\prime})-\\operatorname*{\\mathbb{E}}_{ \\hat{\\mathcal{S}}_{t}}\\ell_{I}(h^{\\prime})\\right|\\geq\\frac{1}{n}\\right] =\\Pr\\left[\\left|\\ell_{I}(h^{\\prime})-\\operatorname*{\\mathbb{E}}_ {\\hat{\\mathcal{S}}_{t}}\\ell_{I}(h^{\\prime})\\right|\\geq\\frac{1}{n}\\right]\\hat{b }\\leq\\frac{11}{10}\\right]\\cdot\\Pr\\left[\\hat{b}\\leq\\frac{11}{10}\\right]\\] \\[\\leq\\frac{1}{8}\\;.\\]\n\nHence, for large enough \\(n\\), with probability at least \\(1-\\frac{1}{8}-\\frac{1}{8}=\\frac{3}{4}>\\frac{2}{3}\\) we have both \\(\\operatorname*{\\mathbb{E}}_{\\hat{\\mathcal{S}}_{I}}\\left[\\ell_{I}(h^{\\prime}) \\right]\\geq\\frac{1}{10\\log(n)}\\) and \\(\\left|\\ell_{I}(h^{\\prime})-\\operatorname*{\\mathbb{E}}_{\\hat{\\mathcal{S}}_{I} }\\ell_{I}(h^{\\prime})\\right|\\leq\\frac{1}{n}\\), and thus\n\n\\[\\ell_{I}(h^{\\prime})\\geq\\frac{1}{10\\log(n)}-\\frac{1}{n}>\\frac{2}{n}\\;.\\]\n\nOverall, if \\(\\mathcal{S}\\) is pseudorandom then with probability greater than \\(\\frac{2}{3}\\) the algorithm \\(\\mathcal{A}\\) returns \\(1\\), and if \\(\\mathcal{S}\\) is random then with probability greater than \\(\\frac{2}{3}\\) the algorithm \\(\\mathcal{A}\\) returns \\(0\\). Thus, the distinguishing advantage is greater than \\(\\frac{1}{3}\\). This concludes the proof of the theorem. It remains to prove the deffered lemma on the realizability of the examples returned by the examples oracle:\n\n**Lemma C.1**.: _If \\(\\mathcal{S}\\) is pseudorandom then with probability at least \\(\\frac{39}{40}\\) over \\(\\boldsymbol{\\xi}\\sim\\mathcal{N}(\\mathbf{0},\\tau^{2}I_{p})\\) and \\(\\boldsymbol{\\zeta}_{i}\\sim\\mathcal{N}(\\mathbf{0},\\omega^{2}I_{n^{2}})\\) for \\(i\\in[m(n)+n^{3}]\\), the examples \\((\\hat{\\mathbf{z}}_{1},\\hat{y}_{1}),\\ldots,(\\hat{\\mathbf{z}}_{m(n)+n^{3}},\\hat {y}_{m(n)+n^{3}})\\) returned by the oracle are realized by \\(\\hat{N}\\)._\n\nProof.: By our choice of \\(\\tau\\) and \\(\\omega\\) and the construction of \\(N_{1},N_{2}\\), with probability at least \\(1-\\frac{1}{n}\\) over \\(\\boldsymbol{\\xi}\\sim\\mathcal{N}(\\mathbf{0},\\tau^{2}I_{p})\\), we have \\(|\\xi_{j}|\\leq\\frac{1}{10}\\) for all \\(j\\in[p]\\), and for every \\(\\mathbf{z}\\in\\{0,1\\}^{n^{2}}\\) the following holds: Let \\(\\boldsymbol{\\zeta}\\sim\\mathcal{N}(\\mathbf{0},\\omega^{2}I_{n^{2}})\\) and let \\(\\hat{\\mathbf{z}}=\\mathbf{z}+\\boldsymbol{\\zeta}\\). Then with probability at least \\(1-\\exp(-n/2)\\) over \\(\\boldsymbol{\\zeta}\\) the inputs to the neurons \\(\\mathcal{E}_{1},\\mathcal{E}_{2}\\) in the computation \\(\\hat{N}(\\hat{\\mathbf{z}})\\) satisfy Properties (Q1) and (Q2). Hence, with probability at least \\(1-\\frac{1}{n}-(m(n)+n^{3})\\exp(-n/2)\\geq 1-\\frac{2}{n}\\) (for a sufficiently large \\(n\\)), \\(|\\xi_{j}|\\leq\\frac{1}{10}\\) for all \\(j\\in[p]\\), and Properties (Q1) and (Q2) hold for the computations \\(\\hat{N}(\\hat{\\mathbf{z}}_{i})\\) for all \\(i\\in[m(n)+n^{3}]\\). It remains to show that if \\(|\\xi_{j}|\\leq\\frac{1}{10}\\) for all \\(j\\in[p]\\) and Properties (Q1) and (Q2) hold, then the examples \\((\\hat{\\mathbf{z}}_{1},\\hat{y}_{1}),\\ldots,(\\hat{\\mathbf{z}}_{m(n)+n^{3}},\\hat {y}_{m(n)+n^{3}})\\) are realized by \\(\\hat{N}\\).\n\nLet \\(i\\in[m(n)+n^{3}]\\). We denote \\(\\hat{\\mathbf{z}}_{i}=\\mathbf{z}_{i}+\\boldsymbol{\\zeta}_{i}\\), namely, the \\(i\\)-th example returned by the oracle was obtained by adding noise \\(\\boldsymbol{\\zeta}_{i}\\) to \\(\\mathbf{z}_{i}\\in\\{0,1\\}^{n^{2}}\\). We also denote \\(\\mathbf{z}_{i}^{\\prime}=(\\mathbf{z}_{i})_{[kn]}\\in\\{0,1\\}^{kn}\\). Since \\(|\\xi_{j}|\\leq\\frac{1}{10}\\) for all \\(j\\in[p]\\), and all incoming weights to the output neuron in \\(\\tilde{N}\\) are \\(-1\\), then in \\(\\hat{N}\\) all incoming weights to the output neuron are in \\(\\left[-\\frac{11}{10},-\\frac{9}{10}\\right]\\), and the bias term in the output neuron, denoted by \\(\\hat{b}\\), is in \\(\\left[\\frac{9}{10},\\frac{11}{10}\\right]\\). Consider the following cases:\n\n* If \\(\\mathbf{z}_{i}^{\\prime}\\) is not an encoding of a hyperedge then \\(\\hat{y}_{i}=0\\). Moreover, in the computation \\(\\hat{N}(\\hat{\\mathbf{z}}_{i})\\), there exists a neuron in \\(\\mathcal{E}_{2}\\) with output at least \\(\\frac{3}{2}\\) (by Property (Q2)). Since all incoming weights to the output neuron in \\(\\hat{N}\\) are in \\(\\left[-\\frac{11}{10},-\\frac{9}{10}\\right]\\), and \\(\\hat{b}\\in\\left[\\frac{9}{10},\\frac{11}{10}\\right]\\), then the input to the output neuron (including the bias term) is at most \\(\\frac{11}{10}-\\frac{3}{2}\\cdot\\frac{9}{10}<0\\), and thus its output is \\(0\\).\n* If \\(\\mathbf{z}^{\\prime}\\) is an encoding of a hyperedge \\(S\\), then by the definition of the examples oracle we have \\(S=S_{i}\\). Hence:\n* If \\(y_{i}=0\\) then the oracle sets \\(\\hat{y}_{i}=\\hat{b}\\). Since \\(\\mathcal{S}\\) is pseudorandom, we have \\(P_{\\mathbf{x}}(\\mathbf{z}^{S})=P_{\\mathbf{x}}(\\mathbf{z}^{S_{i}})=y_{i}=0\\). Hence, in the computation \\(\\hat{N}(\\hat{\\mathbf{z}}_{i})\\) the inputs to all neurons in \\(\\mathcal{E}_{1},\\mathcal{E}_{2}\\) are at most \\(-\\frac{1}{2}\\) (by Properties (Q1) and (Q2)), and thus their outputs are \\(0\\). Therefore, \\(\\hat{N}(\\hat{\\mathbf{z}}_{i})=\\hat{b}\\).\n* If \\(y_{i}=1\\) then the oracle sets \\(\\hat{y}_{i}=0\\). Since \\(\\mathcal{S}\\) is pseudorandom, we have \\(P_{\\mathbf{x}}(\\mathbf{z}^{S})=P_{\\mathbf{x}}(\\mathbf{z}^{S_{i}})=y_{i}=1\\). Hence, in the computation \\(\\hat{N}(\\hat{\\mathbf{z}}_{i})\\) there exists a neuron in \\(\\mathcal{E}_{1}\\) with output at least \\(\\frac{3}{2}\\) (by Property (Q1)). Since all incoming weights to the output neuron in \\(\\hat{N}\\) are in \\(\\left[-\\frac{11}{10},-\\frac{9}{10}\\right]\\), and \\(\\hat{b}\\in\\left[\\frac{9}{10},\\frac{11}{10}\\right]\\), then the input to output neuron (including the bias term) is at most \\(\\frac{11}{10}-\\frac{3}{2}\\cdot\\frac{9}{10}<0\\), and thus its output is \\(0\\).", "response": "## Review\n\n### Summary\n\nThis paper addresses the computational complexity of learning neural networks, specifically focusing on depth-3 ReLU networks under smoothed analysis. It builds on prior work, establishing that learning depth-3 networks is computationally hard, even when considering assumptions that facilitate learning for depth-2 networks. The authors provide new lower bounds under a smoothed analysis framework, arguing that their results extend previous findings which were limited to degenerate cases. While the paper is theoretically rigorous and well-structured, some reviewers noted concerns regarding the incremental nature of the contributions compared to existing literature.\n\n### Strengths\n\n- Understanding the boundary between hard and easy cases of learning neural networks is an important problem in learning theory.\n- Provides a thorough summary of previous work, aiding comprehension of its contributions.\n- Presents technical results in a clear and logical manner.\n- The results are significant in showing the hardness of learning 3-layer networks under smoothed distributions, expanding on prior findings.\n- The paper is rigorous in its analysis, presenting results in a logical order.\n\n### Weaknesses\n\n- The novelty of the findings may be limited, as they seem to build on previous work without substantial advancement.\n- Lack of clearer comparison to prior results, especially regarding the work of Daniely and Vardi.\n- The motivation for studying computational complexity in this specific setting could be better articulated.\n- Some sections, particularly Section 4, are difficult to understand and could benefit from better organization and clarity.\n- The connection of the findings to practical applications in neural network training is not sufficiently addressed.\n\n### Questions\n\n- Can the proofs for depth-3 neural networks be extended to depth-k networks for k >= 3?\n- What is the relevance of the learning problem studied in relation to training neural networks?\n- Are we discussing computational complexity or sample complexity in the context of this paper?\n- Could the authors clarify the significance of the assumptions made in their proofs?\n- Will the authors consider investigating the efficient learning of 3-depth networks without activation functions at the output?\n\n### Soundness\n\n**Score:** 3\n\n**Description:** Good overall, with strong technical results but some concerns about novelty and clarity.\n\n### Presentation\n\n**Score:** 3\n\n**Description:** Good, but several sections could be better organized for clarity and ease of understanding.\n\n### Contribution\n\n**Score:** 3\n\n**Description:** Good, though the contributions may be seen as somewhat incremental compared to existing literature.\n\n### Rating\n\n**Score:** 6\n\n**Description:** Weak Accept: The paper is technically solid with moderate-to-high impact, but it requires some improvements in clarity and motivation.\n\n### Paper Decision\n\n**Decision:** Accept (poster)\n\n**Reasons:** The paper presents relevant and technically sound results in the field of neural network learning. Despite some concerns regarding its novelty and clarity, the contributions are significant enough to warrant acceptance. The authors should clarify the motivation and implications of their findings, especially in relation to prior work.\n"}
{"query": "You are a highly experienced, conscientious, and fair academic reviewer, please help me review this paper. The review should be organized into nine sections: \n1. Summary: A summary of the paper in 100-150 words.\n2. Strengths/Weaknesses/Questions: The Strengths/Weaknesses/Questions of paper, which should be listed in bullet points, with each point supported by specific examples from the article where possible.\n3. Soundness/Contribution/Presentation: Rate the paper's Soundness/Contribution/Presentation, and match this score to the corresponding description from the list below and provide the result. The possible scores and their descriptions are: \n    1 poor\n    2 fair\n    3 good\n    4 excellent\n4. Rating: Give this paper an appropriate rating, match this rating to the corresponding description from the list below and provide the result. The possible Ratings and their descriptions are: \n    1 strong reject\n    2 reject, significant issues present\n    3 reject, not good enough\n    4 possibly reject, but has redeeming facets\n    5 marginally below the acceptance threshold\n    6 marginally above the acceptance threshold\n    7 accept, but needs minor improvements \n    8 accept, good paper\n    9 strong accept, excellent work\n    10 strong accept, should be highlighted at the conference   \n5. Paper Decision: It must include the Decision itself(Accept or Reject) and the reasons for this decision, based on the criteria of originality, methodological soundness, significance of results, and clarity and logic of presentation.\n\nHere is the template for a review format, you must follow this format to output your review result:\n**Summary:**\nSummary content\n\n**Strengths:**\n- Strength 1\n- Strength 2\n- ...\n\n**Weaknesses:**\n- Weakness 1\n- Weakness 2\n- ...\n\n**Questions:**\n- Question 1\n- Question 2\n- ...\n\n**Soundness:**\nSoundness result\n\n**Presentation:**\nPresentation result\n\n**Contribution:**\nContribution result\n\n**Rating:**\nRating result\n\n**Paper Decision:**\n- Decision: Accept/Reject\n- Reasons: reasons content\n\n\nPlease ensure your feedback is objective and constructive. The paper is as follows:\n\n# Offline Imitation Learning with Variational Counterfactual Reasoning\n\n Zexu Sun\\({}^{\\lx@sectionsign}\\), Bowei He\\({}^{\\dagger}\\), Jinxin Liu\\({}^{\\ddagger}\\), Xu Chen\\({}^{\\lx@sectionsign}\\), Chen Ma\\({}^{\\dagger}\\), Shuai Zhang\\({}^{\\ddagger}\\)\n\n\\({}^{\\lx@sectionsign}\\)Gaoling School of Artificial Intelligence, Renmin University of China\n\n\\({}^{\\dagger}\\)Department of Computer Science, City University of Hong Kong\n\n\\({}^{\\ddagger}\\)School of Engineering, Westlake University \\({}^{\\ddagger}\\)DiDi Chuxing\n\n{sunzexu21, xu.chen}@ruc.edu.cn, boweihe2-c@my.cityu.edu.hk\n\nliujinxin@westlake.edu.cn, chema@cityu.edu.hk, shuai.zhang@tju.edu.cn\n\nCorresponding author\n\n###### Abstract\n\nIn offline imitation learning (IL), an agent aims to learn an optimal expert behavior policy without additional online environment interactions. However, in many real-world scenarios, such as robotics manipulation, the offline dataset is collected from suboptimal behaviors without rewards. Due to the scarce expert data, the agents usually suffer from simply memorizing poor trajectories and are vulnerable to the variations in the environments, lacking the capability of generalizing to new environments. To automatically generate high-quality expert data and improve the generalization ability of the agent, we propose a framework named Offline Imitation Learning with Counterfactual data Augmentation (OILCA) by doing counterfactual inference. In particular, we leverage identifiable variational autoencoder to generate _counterfactual_ samples for expert data augmentation. We theoretically analyze the influence of the generated expert data and the improvement of generalization. Moreover, we conduct extensive experiments to demonstrate that our approach significantly outperforms various baselines on both DeepMind Control Suite benchmark for in-distribution performance and CausalWorld benchmark for out-of-distribution generalization.\n\n## 1 Introduction\n\nBy utilizing the pre-collected expert data, imitation learning (IL) allows us to circumvent the difficulty in designing proper rewards for decision-making tasks and learning an expert policy. Theoretically, as long as adequate expert data are accessible, we can easily learn an imitator policy that maintains a sufficient capacity to approximate the expert behaviors [25, 30, 29, 31]. However, in practice, several challenging issues hinder its applicability to practical tasks. In particular, expert data are often limited, and due to the typical requisite for online interaction with the environment, performing such online IL may be costly or unsafe in real-world scenarios such as self-driving or industrial robotics [11, 34, 37]. Alternatively, in such settings, we might instead have access to large amounts of pre-collected unlabeled data, which are of unknown quality and may consist of both good-performing and poor-performing trajectories. For example, in self-driving tasks, a number of human driving behaviors may be available; in industrial robotics domains, one may have access to large amounts of robot data. The question then arises: can we perform offline IL with only limited expert data and the previously collected unlabeled data, thus relaxing the costly online IL requirements?\n\nTraditional behavior cloning (BC) [5] directly mimics historical behaviors logged in offline data (both expert data and unlabeled data) via supervised learning. However, in our above setting, BCsuffers from unstable training, as it relies on sufficient high-quality offline data, which is unrealistic. Besides, utilizing unlabeled data indiscriminately will lead to severe catastrophes: Bad trajectories mislead policy learning, and good trajectories fail to provide strong enough guidance signals for policy learning. In order to better distinguish the effect of different quality trajectories on policy learning, various solutions are proposed correspondingly. For example, ORIL [39] learns a reward model to relabel previously collected data by contrasting expert and unlabeled trajectories from a fixed dataset; DWBC [36] introduces a discriminator-weighted task to assist the policy learning. However, such discriminator-based methods are prone to overfitting and suffer from poor generalization ability, especially when only very limited expert data are provided.\n\nIndeed, the expert data play an important role in offline IL and indicate the well-performing agent's intention. Thus, how to fully understand the expert's behaviors or preferences via limited available expert data becomes pretty crucial. In this paper, we propose to investigate counterfactual techniques for interpreting such expert behaviors. Specifically, we leverage the _variational counterfactual reasoning_[23, 4] to augment the expert data, as typical imitation learning data augmentation methods [3, 7] easily generate noisy data and inevitably bias the learning agent. Here, we conduct the counterfactual reasoning of the expert by answering the following question:\n\n_\"What action might the expert take if a different state is observed?\"_\n\nThroughout this paper, we consider the structure causal model (SCM) underlying the offline training data and introduce an exogenous variable that influences the states and yet is unobserved. This variable is utilized as the minimal edits to existing expert training data so that we can generate counterfactual states that cause an agent to change its behavior. Intuitively, this exogenous variable captures variations of features in the environment. By introducing additional variations in the states during training, we encourage the model to rely less on the idiosyncrasies of a given environment. In detail, we leverage an identifiable generative model to generate the counterfactual expert data, thus enhancing the agent's capabilities of generalization in test environments (Figure 1).\n\nThe main contributions of this paper are summarized as follows:\n\n* We propose a novel learning framework OILCA for offline IL. Using both training data and the augmentation model, we can generate counterfactual expert data and improve the generalization of the learned agent.\n* We theoretically analyze the disentanglement identifiability of the constructed exogenous variable and the influence of augmented counterfactual expert data via a sampler policy. We also guarantee the improvement of generalization ability from the perspective of error bound.\n* We conduct extensive experiments and provide related analysis. The empirical results about in-distribution performance on DeepMind Control Suite benchmark and out-of-distribution generalization on CausalWorld benchmark both demonstrate the effectiveness of our method.\n\n## 2 Related Works\n\nOffline IL.A significant inspiration for this work grows from the offline imitation learning technique on how to learn policies from the demonstrations. Most of these methods take the idea of behavior cloning (BC) [5] that utilizes supervised learning to learn to act. However, due to the presence of suboptimal demonstrations, the performance of BC is limited to mediocre levels on many datasets. To address this issue, ORIL [39] learns a reward function and uses it to relabel offline trajectories.\n\nFigure 1: Agent is trained with the collected dataset containing limited expert data and large amounts of unlabeled data, and tested on both in-distribution and out-of-distribution environments.\n\nHowever, it suffers from high computational costs and the difficulty of performing offline RL under distributional shifts. Trained on all data, BCND [26] reuses another policy learned by BC as the weight of the original BC objective, but its performance can even be worse if the suboptimal data occupies the major part of the offline dataset. LobsDICE [12] learns to imitate the expert policy via optimization in the space of stationary distributions. It solves a single convex minimization problem, which minimizes the divergence between the two-state transition distributions induced by the expert and the agent policy. CEIL [16] explicitly learns a hindsight embedding function together with a contextual policy. To achieve the expert matching objective for IL, CEIL advocates for optimizing a contextual variable such that it biases the contextual policy towards mimicking expert behaviors. DWBC [36] introduces an additional discriminator to distinguish expert and unlabeled demonstrations, and the outputs of the discriminator serve as the weights of the BC loss. CLUE [17] proposes to learn an intrinsic reward that is consistent with the expert intention via enforcing the embeddings of expert data to a calibrated contextual representation. OILA aims to augment the the scarce expert data to improve the performance of the learned policy.\n\nCausal Dynamics RLAdopting this formalism allows one to cast several important problems within RL as questions of causal inference, such as off-policy evaluation [6; 22], learning baselines for model-free RL [20], and policy transfer [10; 15]. CTRL [19] applies SCM dynamics to the data augmentation in continuous sample spaces and discusses the conditions under which the generated transitions are uniquely identifiable counterfactual samples. This approach models state and action variables as unstructured vectors, emphasizing benefits in modeling action interventions for scenarios such as clinical healthcare where exploratory policies cannot be directly deployed. MOCODA [24] applies a learned locally factored dynamics model to an augmented distribution of states and actions to generate counterfactual transitions for RL. FOCUS [38] can reconstruct the causal structure accurately and illustrate the feasibility of learning causal structure in offline RL. OILCA uses the identifiable generative model to infer the distribution of the exogenous variable in the causal MDP, then performs the counterfactual data augmentation to augment the scarce expert data in offline IL.\n\n## 3 Preliminaries\n\n### Problem Definition\n\nWe consider the causal Markov Decision Process (MDP) [19] with an additive noise. In our problem setting, we have an offline static dataset consisting of _i.i.d_ tuples \\(\\mathcal{D}_{\\text{all}}=\\left\\{s_{t}^{i},a_{t}^{i},s_{t+1}^{i}\\right\\}_{i=1} ^{n_{\\text{all}}}\\) s.t. \\((s_{t},a_{t})\\sim\\rho(s_{t},a_{t}),s_{t+1}\\sim f_{e}(s_{t},a_{t},u_{t+1})\\), where \\(\\rho(s_{t},a_{t})\\) is an offline state-action distribution resulting from some behavior policies, \\(f_{e}(s_{t},a_{t},u_{t+1})\\) represents the causal transition mechanism, \\(u_{t+1}\\) is the sample of the exogenous variable \\(u\\), which is unobserved, and \\(\\varepsilon\\) is a small permutation. Let \\(\\mathcal{D}_{E}\\) and \\(\\mathcal{D}_{U}\\) be the sets of expert and unlabeled demonstrations respectively, our goal is to only leverage the offline batch data \\(\\mathcal{D}_{\\text{all}}\\coloneqq\\mathcal{D}_{E}\\cup\\mathcal{D}_{U}\\) to learn an optimal policy \\(\\pi\\) without any online interaction.\n\n### Counterfactual Reasoning\n\nWe provide a brief background on counterfactual reasoning. Further details can be found in [23].\n\n**Definition 1** (Structural Causal Model (SCM)).: _A structural causal model \\(\\mathcal{M}\\) over variables \\(\\mathbf{X}=\\{X_{1},\\dots,X_{n}\\}\\) consists of a set of independent exogenous variables \\(\\mathbf{U}=\\{\\mathbf{u}_{1},\\dots,\\mathbf{u}_{n}\\}\\) with prior distributions \\(P\\left(\\mathbf{u}_{i}\\right)\\) and a set of functions \\(f_{1},\\dots,f_{n}\\) such that \\(X_{i}=f_{i}\\left(\\mathbf{P}\\dot{\\mathbf{A}}_{i},\\mathbf{u}_{i}\\right)\\), where \\(\\mathbf{P}\\mathbf{A}_{i}\\subset\\mathbf{X}\\) are parents of \\(X_{i}\\). Therefore, the distribution of the SCM, which is denoted \\(P^{\\mathcal{M}}\\), is determined by the functions and the prior distributions of exogenous variables._\n\nInferring the exogenous random variables based on the observations, we can intervene in the observations and inspect the consequences.\n\n**Definition 2** (\\(do\\)-intervention in SCM).: _An intervention \\(I=do\\left(X_{i}:=f_{i}\\left(\\tilde{\\mathbf{P}\\mathbf{A}}_{i},\\mathbf{u}_{i} \\right)\\right)\\) is defined as replacing some functions \\(f_{i}\\left(\\mathbf{P}\\mathbf{A}_{i},\\mathbf{u}_{i}\\right)\\) with \\(f_{i}\\left(\\mathbf{P}\\dot{\\mathbf{A}}_{i},\\mathbf{u}_{i}\\right)\\), where \\(\\dot{\\mathbf{P}\\mathbf{A}}_{i}\\) is the intervened parents of \\(X_{i}\\). The intervened SCM is indicated as \\(\\mathcal{M}^{I}\\), and, consequently, its distribution is denoted as \\(P^{\\mathcal{M};I}\\)._\n\nThe counterfactual inference with which we can answer the _what if_ questions will be obtained in the following process:1. Infer the posterior distribution of exogenous variable \\(P\\left(\\mathbf{u}_{i}\\mid\\mathbf{X}=\\mathbf{x}\\right)\\), where \\(\\mathbf{x}\\) is a set of observations. Replace the prior distribution \\(P\\left(\\mathbf{u}_{i}\\right)\\) with the posterior distribution \\(P\\left(\\mathbf{u}_{i}\\mid\\mathbf{X}=\\mathbf{x}\\right)\\) in the SCM.\n2. We denote the resulted SCM as \\(\\mathcal{M}_{\\mathbf{x}}\\) and its distribution as \\(P^{\\mathcal{M}_{\\mathbf{x}}}\\), perform an intervention \\(I\\) on \\(\\mathcal{M}_{\\mathbf{x}}\\) to reach \\(P^{\\mathcal{M}_{\\mathbf{x}};I}\\).\n3. Return the output of \\(P^{\\mathcal{M}_{\\mathbf{x}};I}\\) as the counterfactual inference.\n\nSCM representation of causal MDPWe encode causal MDP (under a policy \\(\\pi\\)) into an SCM \\(\\mathcal{M}\\). The SCM shown in Figure 2 consists of an exogenous variable \\(u\\) and a set of functions that transmit the state \\(s_{t}\\), action \\(a_{t}\\) and latent representation \\(u_{t+1}\\) of \\(u\\) to the next state \\(s_{t+1}\\)_e.g._\\(s_{t+1}\\sim\\boldsymbol{f}_{\\boldsymbol{\\varepsilon}}\\left(s_{t},a_{t},u_{t+1}\\right)\\), where \\(\\varepsilon\\) is a small perturbation, and subsequently, to the next action \\(a_{t+1}\\), _e.g._\\(a_{t+1}\\sim\\pi(\\cdot\\mid s_{t+1})\\). To perform counterfactual inference on the SCM \\(\\mathcal{M}\\), we intervene in the original parents pair \\(\\left(s_{t},a_{t}\\right)\\) to \\(\\left(\\tilde{s}_{t},\\tilde{a}_{t}\\right)\\). Specifically, we sample the latent representation \\(\\tilde{u}_{t+1}\\) from the distribution of exogenous variable \\(u\\), and then generate the counterfactual next state \\(\\tilde{s}_{t+1}\\), _i.e._\\(\\tilde{s}_{t+1}\\sim\\boldsymbol{f}_{\\boldsymbol{\\varepsilon}}\\left(\\tilde{s}_{t},\\tilde{a}_{t},\\tilde{u}_{t+1}\\right)\\), and also the counterfactual next action \\(\\tilde{a}_{t+1}\\), _i.e._\\(\\tilde{a}_{t+1}\\sim\\tilde{\\pi}(\\cdot\\mid\\tilde{s}_{t+1})\\), where \\(\\tilde{\\pi}\\) is the sampler policy.\n\n### Variational Autoencoder and Identifiability\n\nWe briefly introduce the Variational Autoencoder (VAE) and present its identifiability results. The VAE can be conceptualized as the amalgamation of a generative latent variable model and an associated inference model, both of which are parameterized by neural networks [13]. Specifically, the VAE learns the joint distribution \\(p_{\\boldsymbol{\\theta}}(\\boldsymbol{x},\\boldsymbol{z})=p_{\\boldsymbol{\\theta }}(\\boldsymbol{x}\\mid\\boldsymbol{z})p_{\\boldsymbol{\\theta}}(\\boldsymbol{z})\\), where \\(p_{\\boldsymbol{\\theta}}(\\boldsymbol{x}\\mid\\boldsymbol{z})\\) represents the conditional distribution of observing \\(\\boldsymbol{x}\\) given \\(\\boldsymbol{z}\\). Here, \\(\\boldsymbol{\\theta}\\) denotes the set of generative parameters, and \\(p_{\\boldsymbol{\\theta}}(\\boldsymbol{z})=\\Pi_{i=1}^{I}p_{\\boldsymbol{\\theta}} \\left(z_{i}\\right)\\) represents the factorized prior distribution of the latent variables. By incorporating an inference model \\(q_{\\phi}(\\boldsymbol{z}\\mid\\boldsymbol{x})\\), the parameters \\(\\boldsymbol{\\phi}\\) and \\(\\boldsymbol{\\theta}\\) can be jointly optimized by maximizing the evidence lower bound (ELBO) on the marginal likelihood \\(p_{\\theta}(\\boldsymbol{x})\\).\n\n\\[\\mathcal{L} =\\mathbb{E}_{q_{\\boldsymbol{\\phi}}(\\boldsymbol{z}\\mid\\boldsymbol {x})}\\left[\\log p_{\\boldsymbol{\\theta}}(\\boldsymbol{x}\\mid\\boldsymbol{z}) \\right]-D_{\\mathrm{KL}}\\left(q_{\\boldsymbol{\\phi}}(\\boldsymbol{z}\\mid \\boldsymbol{x})\\|p(\\boldsymbol{z})\\right) \\tag{1}\\] \\[=\\log p_{\\boldsymbol{\\theta}}(\\boldsymbol{x})-D_{\\mathrm{KL}} \\left(q_{\\boldsymbol{\\phi}}(\\boldsymbol{z}\\mid\\boldsymbol{x})\\|p_{ \\boldsymbol{\\theta}}(\\boldsymbol{z}\\mid\\boldsymbol{x})\\right)\\leq\\log p_{ \\boldsymbol{\\theta}}(\\boldsymbol{x}),\\]\n\nwhere \\(D_{\\mathrm{KL}}\\) denotes the KL-divergence between the approximation and the true posterior, and \\(\\mathcal{L}\\) is a lower bound of the marginal likelihood \\(p_{\\boldsymbol{\\theta}}(\\boldsymbol{x})\\) because of the non-negativity of the KL-divergence. Recently, it has been shown that VAEs with unconditional prior distributions \\(p_{\\boldsymbol{\\theta}}(\\boldsymbol{z})\\) are not identifiable [18], but the latent factors \\(\\boldsymbol{z}\\) can be identified with a conditionally factorized prior distribution \\(p_{\\boldsymbol{\\theta}}(\\boldsymbol{x}\\mid\\boldsymbol{z})\\) over the latent variables to break the symmetry [9].\n\n## 4 Offline Imitation Learning with Counterfactual Data Augmentation\n\nAt a high level, OILCA consists of following steps: (1) data augmentation via variational counterfactual reasoning, and (2) cooperatively learning a discriminator and a policy by using \\(\\mathcal{D}_{U}\\) and augmented \\(\\mathcal{D}_{E}\\). In this section, we detail these two steps and present our method's theoretical guarantees.\n\nFigure 2: SCM of causal Markov Decision Process (MDP). We incorporate an exogenous variable in the SCM that is learned and utilized for counterfactual reasoning about _do_-intervention.\n\n### Counterfactual Data Augmentation\n\nOur intuition is that the counterfactual expert data can model deeper relations between states and actions, which can help the learned policy be generalized better. Hence, in this section, we aim to generate counterfactual expert data by answering the _what if_ question in Section 1. Consequently, counterfactual expert data augmentation is especially suitable for some real-world settings, where executing policies during learning can be too costly or slow.\n\nAs presented in Section 3.2, the counterfactual data can be generated by using the posterior of the exogenous variable obtained from the observations. Thus, we can briefly introduce the conditional VAE [13; 28] to build the latent representation of the exogenous variable \\(u\\). Moreover, Considering the identifiability of unsupervised disentangled representation learning [18], an additionally observed variable \\(c\\) is needed, where \\(c\\) could be, for example, the time index or previous data points in a time series, some kind of (possibly noisy) class label, or another concurrently observed variable [9]. Formally, let \\(\\mathbf{\\theta}=(\\mathbf{f},\\mathbf{T},\\mathbf{\\lambda})\\) be the parameters of the following conditional generative model:\n\n\\[p_{\\mathbf{\\theta}}(s_{t+1},u\\mid s_{t},a_{t},c)=p_{\\mathbf{f}}(s_{t+1}\\mid s_{t},a_{t},u)p_{\\mathbf{T},\\mathbf{\\lambda}}(u\\mid c), \\tag{2}\\]\n\nwhere we first define:\n\n\\[p_{\\mathbf{f}}(s_{t+1}\\mid s_{t},a_{t},u)=p_{\\mathbf{e}}(s_{t+1}-\\mathbf{f}_{s_{t},a_{t}}( u)). \\tag{3}\\]\n\nEquation (2) describes the generative mechanism of \\(s_{t+1}\\) given the underlying exogenous variable \\(u\\), along with \\((s_{t},a_{t})\\). Equation (3) implies that the observed representation \\(s_{t+1}\\) is an additive noise function, _i.e._, \\(s_{t+1}=\\mathbf{f}_{s_{t},a_{t}}(u)+\\mathbf{\\varepsilon}\\) where \\(\\mathbf{\\varepsilon}\\) is independent of \\(\\mathbf{f}_{s_{t},a_{t}}\\) or \\(u\\). Moreover, this formulation also can be found in the SCM representation of causal MDP in Section 3.2, where we treat the function \\(f\\) as the parameter \\(\\mathbf{f}\\) of the model.\n\nFollowing standard conditional VAE [28] derivation, the evidence lower bound (ELBO) for each sample for \\(s_{t+1}\\) of the above generative model can be written as:\n\n\\[\\log p(s_{t+1}\\mid s_{t},a_{t},c)\\geq \\mathcal{L}(\\mathbf{\\theta},\\mathbf{\\phi})\\coloneqq\\log p_{\\mathbf{f}}(s_{t+1} \\mid s_{t},a_{t},u)+\\log p(c)- \\tag{4}\\] \\[D_{\\text{KL}}(q_{\\mathbf{\\phi}}(u\\mid s_{t},a_{t},s_{t+1},c)\\|_{\\mathbf{ P},\\mathbf{\\lambda}}(u\\mid c)).\\]\n\nNote that the ELBO in Equation (4) contains an additional term of the \\(\\log p(c)\\) that does not affect identifiability but improves the estimation for the conditional prior [21], we present the theoretical guarantees of disentanglement identifiability in Section 4.3. Moreover, the encoder \\(q_{\\mathbf{\\phi}}\\) contains all the observable variables. The reason is that, as shown in Figure 2(a), from a causal perspective, \\(s_{t},a_{t}\\) and \\(u\\) form a collider at \\(s_{t+1}\\), which means that when given \\(s_{t+1}\\), \\(s_{t}\\) and \\(a_{t}\\) are related to \\(u\\).\n\nWe seek to augment the expert data in \\(\\mathcal{D}_{E}\\). As shown in Figure 2(b), once the posterior of exogenous variable \\(q_{\\mathbf{\\phi}}\\) (\\(u\\mid s_{t},a_{t},s_{t+1},c\\)) is obtained, we can perform _do_-intervention. In particular, we intervene in the parents \\((s_{t},a_{t})\\) of \\(s_{t+1}\\) in \\(\\mathcal{D}_{E}\\) by re-sampling a different pair \\((\\tilde{s}_{t},\\tilde{a}_{t})\\) from the collected data in \\(\\mathcal{D}_{U}\\). Moreover, we sample the latent representation \\(\\tilde{u}_{t+1}\\) from the learned posterior distribution of exogenous variable \\(u\\). Then utilizing \\(\\tilde{s}_{t}\\), \\(\\tilde{a}_{t}\\), and \\(\\tilde{u}_{t+1}\\), we can generate the counterfactual next state according to Equation (3), which is denoted as \\(\\tilde{s}_{t+1}\\). Subsequently, we pre-train a sampler policy \\(\\tilde{\\pi}_{E}\\) with the original \\(\\mathcal{D}_{E}\\) to sample the counterfactual next action \\(\\tilde{a}_{t+1}=\\tilde{\\pi}_{E}(\\cdot\\mid\\tilde{s}_{t+1})\\). Thus, for all the generated tuples \\((\\tilde{s}_{t+1},\\tilde{a}_{t+1})\\), we constitute them in \\(\\mathcal{D}_{E}\\).\n\n### Offline Agent Imitation Learning\n\nIn general, the counterfactual data augmentation of our method can enhance many offline IL methods. It is worth noting that as described in [40], discriminator-based methods are easy to be over-fitted, which can more directly show the importance and effectiveness of the counterfactual augmented data. Thus, in this section, also to best leverage the unlabeled data, we use Discriminator-Weighted Behavioral Cloning (DWBC) [36], a state-of-the-art discriminator-based offline IL method. This method introduces a unified framework to learn the policy and discriminator cooperatively. The discriminator training gets information from the policy \\(\\pi_{\\mathbf{\\omega}}\\) as additional input, yielding a new discriminating task whose learning objective is as follows:\n\n\\[\\mathcal{L}_{\\mathbf{\\psi}}(\\mathcal{D}_{E},\\mathcal{D}_{U})= \\eta\\mathop{\\mathbb{E}}_{(s_{t},a_{t})\\sim\\mathcal{D}_{E}}[-\\log D _{\\mathbf{\\psi}}(s_{t},a_{t},\\log\\pi_{\\mathbf{\\omega}})]+\\mathop{\\mathbb{E}}_{(s_{t},a _{t}^{\\prime})\\sim\\mathcal{D}_{U}}[-\\log(1-D_{\\mathbf{\\psi}}(s_{t}^{\\prime},a_{t} ^{\\prime},\\log\\pi_{\\mathbf{\\omega}})] \\tag{5}\\] \\[-\\eta\\mathop{\\mathbb{E}}_{(s_{t},a_{t})\\sim\\mathcal{D}_{E}}[-\\log (1-D_{\\mathbf{\\psi}}(s_{t},a_{t},\\log\\pi_{\\mathbf{\\omega}}))],\\]where \\(D_{\\mathbf{\\psi}}\\) is the discriminator, \\(\\eta\\) is called the class prior. In the previous works [35; 39], \\(\\eta\\) is a fixed hyperparameter often assigned as 0.5.\n\nNotice that now \\(\\pi_{\\mathbf{\\omega}}\\) appears in the input of \\(D_{\\mathbf{\\psi}}\\), which means that imitation information from \\(\\log\\pi_{\\mathbf{\\omega}}\\) will affect \\(\\mathcal{L}_{\\mathbf{\\psi}}\\), and further impact the learning of \\(D_{\\mathbf{\\psi}}\\). Thus, inspired by the idea of adversarial training, DWBC [36] introduces a new learning objective for BC Task:\n\n\\[\\begin{split}\\mathcal{L}_{\\pi}=&\\alpha\\mathop{ \\mathbb{E}}_{(s_{t},a_{t})\\sim\\mathcal{D}_{E}}[-\\log\\pi_{\\mathbf{\\omega}}(a_{t}\\mid s _{t})]-\\mathop{\\mathbb{E}}_{(s_{t},a_{t})\\sim\\mathcal{D}_{E}}\\left[-\\log\\pi_{ \\mathbf{\\omega}}(a_{t}\\mid s_{t})\\cdot\\frac{\\eta}{d(1-d)}\\right]\\\\ &+\\mathop{\\mathbb{E}}_{(s^{\\prime}_{t},a^{\\prime}_{t})\\sim\\mathcal{ D}_{U}}\\left[-\\log\\pi_{\\mathbf{\\omega}}(a^{\\prime}_{t}\\mid s^{\\prime}_{t})\\cdot\\frac{1}{1 -d}\\right],\\quad\\alpha>1.\\end{split} \\tag{6}\\]\n\nwhere \\(d\\) represents \\(D_{\\mathbf{\\psi}}(s_{t},a_{t},\\log\\pi_{\\mathbf{\\omega}})\\) for simplicity, \\(\\alpha\\) is the weight factor (\\(\\alpha>1\\)). The detailed training procedure of OILCA is shown in Algorithm 1. Moreover, we also present more possible combinations with other offline IL methods and the related results in Appendix F.\n\n```\n0: Dataset \\(\\mathcal{D}_{E}\\), \\(\\mathcal{D}_{U}\\), \\(\\mathcal{D}_{\\text{all}}\\), pre-trained sampler policy \\(\\hat{\\pi}_{E}\\), hyperparameters \\(\\eta\\), \\(\\alpha\\), initial variational counterfactual parameters \\(\\mathbf{\\theta},\\mathbf{\\phi}\\), discriminator parameters \\(\\mathbf{\\psi}\\), policy parameters \\(\\mathbf{\\omega}\\), data augmentation batch number \\(B\\).\n0: Learned policy parameters \\(\\mathbf{\\omega}\\).\n1:while counterfactual training do\\(\\triangleright\\) Variational counterfactual reasoning\n2: Sample \\((s_{t},a_{t},s_{t+1},c)\\sim\\mathcal{D}_{\\text{all}}\\) to form a training batch\n3: Update \\(\\mathbf{\\theta}\\) and \\(\\mathbf{\\phi}\\) according to Equation (4)\n4:endwhile\n5:for\\(b=1\\) to \\(B\\)do\\(\\triangleright\\) Expert data augmentation\n6: Sample \\((s_{t},a_{t},s_{t+1},c)\\sim\\mathcal{D}_{E}\\), \\((\\tilde{s}_{t},\\tilde{a}_{t})\\sim\\mathcal{D}_{U}\\) to form an augmentation batch\n7: Generate the counterfactual \\(\\tilde{s}_{t+1}\\) according to Equation (2), then predict \\(\\tilde{a}_{t+1}=\\hat{\\pi}_{E}(\\cdot\\mid\\tilde{s}_{t+1})\\)\n8:\\(\\mathcal{D}_{E}\\cup(\\tilde{s}_{t+1},\\tilde{a}_{t+1})\\)\n9:endfor\n10:while agent training do\\(\\triangleright\\) Learning the discriminator and policy cooperatively\n11: Sample \\((s_{t},a_{t})\\sim\\mathcal{D}_{E}\\) and \\((s^{\\prime}_{t},a^{\\prime}_{t})\\sim\\mathcal{D}_{U}\\) to form a training batch\n12: Update \\(\\mathbf{\\psi}\\) according Equation (5) every 100 training steps\\(\\triangleright\\) Discriminator learning\n13: Update \\(\\mathbf{\\omega}\\) according to Equation (6) every 1 training step\\(\\triangleright\\) Policy learning\n14:endwhile\n```\n\n**Algorithm 1** Training procedure of OILCA.\n\n### Theoretical Analysis\n\nIn this section, we theoretically analyze our method, which mainly contains three aspects: (1) disentanglement identifiability, (2) the influence of the augmented data, and (3) the generalization ability of the learned policy.\n\nOur disentanglement identifiability extends the theory of iVAE [9]. To begin with, some assumptions are needed. Considering the SCM in Figure 2(a), we assume the data generation process in Assumption 1.\n\n**Assumption 1**.: _(a) The distribution of the exogenous variable \\(u\\) is independent of time but dependent on the auxiliary variable c. (b) The prior distributions of exogenous variable \\(u\\) are different across auxiliary variable c. (c) The transition mechanism \\(p(s_{t+1}\\mid s_{t},a_{t},u)\\) are invariant across different auxiliary variable c. (d) Given the exogenous variable \\(u\\), the next state \\(s_{t+1}\\) is independent of the auxiliary variable c. i.e. \\(s_{t+1}\\perp c\\mid u\\)._\n\nPart of the above assumption is also used in [19]; considering the identifiability of iVAE, we also add some necessary assumptions, which are also practical. In the following discussion, we will show that when the underlying data-generating mechanism satisfies Assumption 1, the exogenous variable \\(u\\) can be identified up to permutation and affine transformations if the conditional prior distribution \\(p(u|c)\\) belongs to a general exponential family distribution.\n\n**Assumption 2**.: _The prior distribution of the exogenous variable \\(p(u|c)\\) follows a general exponential family with its parameter specified by an arbitrary function \\(\\mathbf{\\lambda}(c)\\) and sufficient statistics \\(\\mathbf{T}(u)=[\\mathbf{T}_{\\mathbf{d}}(u),\\mathbf{T}_{NN}(u)]\\), here \\(\\mathbf{T}_{\\mathbf{d}}(u)\\) is defined by the concatenation of \\([\\mathbf{T}_{1}(u^{1})^{T},\\cdot,\\cdot,\\mathbf{T}_{d}(u^{d})^{T}]^{T}\\) from a factorized exponential family and the outputs of a neural network \\(\\mathbf{T}_{NN}(u)\\) with universal approximation power. The probability density can be written as:_\n\n\\[p_{\\mathbf{T},\\mathbf{\\lambda}}(u\\mid c)=\\frac{\\mathbf{Q}(u)}{\\mathbf{Z}(u)}\\exp \\bigl{[}\\mathbf{T}(u)^{T}\\mathbf{\\lambda}(c)\\bigr{]}. \\tag{7}\\]\n\nUnder the Assumption 2, and leveraging the ELBO in Equation (4), we can obtain the following identifiability of the parameters in the model. For convenience, we omit the subscript of \\(\\mathbf{f}_{s_{t},a_{t}}\\) as \\(\\tilde{\\mathbf{f}}\\).\n\n**Theorem 1**.: _Assume that we observe data sampled from a generative model defined according to Equation (2)-(3) and Equation (7) with parameters \\(\\left(\\mathbf{f},\\mathbf{T},\\mathbf{\\lambda}\\right)\\), the following holds:_\n\n1. _The set_ \\(\\{s_{t+1}\\in\\mathcal{S}:\\varphi_{\\mathbf{e}}(s_{t+1}=0)\\}\\) _has measure zero, where_ \\(\\varphi_{\\mathbf{e}}\\) _is the characteristic function of the density_ \\(p_{\\mathbf{e}}\\) _defined in Equation (_3_)._\n2. _The function_ \\(\\mathbf{f}\\) _is injective and all of its second-order cross partial derivatives exist._\n3. _The sufficient statistics_ \\(\\mathbf{T}_{\\mathbf{d}}\\) _are twice differentiable._\n4. _There exist_ \\(k+1\\) _distinct points_ \\(c^{0},\\ldots,c^{k}\\) _such that the matrix_ \\[L=\\left(\\mathbf{\\lambda}\\left(c^{1}\\right)-\\mathbf{\\lambda}\\left(c^{0}\\right),\\ldots, \\mathbf{\\lambda}\\left(c^{k}\\right)-\\mathbf{\\lambda}\\left(c^{0}\\right)\\right)\\] (8) _of size_ \\(k\\times k\\) _is invertible._\n\n_Then, the parameters \\(\\mathbf{\\theta}=(\\mathbf{f},\\mathbf{T},\\mathbf{\\lambda})\\) are identifiable up to an equivalence class induced by permutation and component-wise transformations._\n\nTheorem 1 guarantees the identifiability of Equation (2). We present its proof in Appendix A.\n\nMoreover, Theorem 1 further implies a consistent result on the conditional VAE. If the variational distribution of encoder \\(q_{\\mathbf{\\phi}}\\) is a broad parametric family that includes the true posterior, we have the following results.\n\n**Theorem 2**.: _Assume the following holds:_\n\n1. _There exists the_ \\((\\mathbf{\\theta},\\mathbf{\\phi})\\) _such that the family of distributions_ \\(q_{\\mathbf{\\phi}}\\left(u\\mid s_{t},a_{t},s_{t+1},c\\right)\\) _contains_ \\(p_{\\mathbf{\\theta}}\\left(u\\mid s_{t},a_{t},s_{t+1},c\\right)\\)_._\n2. _We maximize_ \\(\\mathcal{L}(\\mathbf{\\theta},\\mathbf{\\phi})\\) _with respect to both_ \\(\\mathbf{\\theta}\\) _and_ \\(\\mathbf{\\phi}\\)_._\n\n_Then, given infinite data, OILCA can learn the true parameters \\(\\mathbf{\\theta}^{*}:=(\\mathbf{f}^{*},\\mathbf{T}^{*},\\mathbf{\\lambda}^{*})\\)._\n\nWe present the corresponding proof in Appendix B. Theorem 2 is proved by assuming our conditional VAE is flexible enough to ensure the ELBO is tight for some parameters and the optimization algorithm can achieve the global maximum of ELBO.\n\nIn our framework, the current generated expert sample pairs \\((\\tilde{s}_{t+1},\\tilde{a}_{t+1})\\) are estimated based on the sampler policy \\(\\hat{\\pi}_{E}\\). However, \\(\\hat{\\pi}_{E}\\) may be not perfect, and its predicted results may contain noise. Thus, we would like to answer: \"given the noise level of the sampler policy, how many samples one need to achieve sufficiently well performance?\". Using \\(\\kappa\\in(0,0.5)\\) indicates the noise level of \\(\\hat{\\pi}_{E}\\). If \\(\\hat{\\pi}_{E}\\) can exactly recover the true action \\(a_{t+1}\\) (i.e., \\(\\kappa=0\\) ), then the generated sequences are perfect without any noise. On the contrary, \\(\\kappa=0.5\\) means that \\(\\hat{\\pi}_{E}\\) can only produce random results, and the generated sequences are fully noisy. Then we have the following theorem:\n\n**Theorem 3**.: _Given a hypothesis class \\(\\mathcal{H}\\), for any \\(\\epsilon,\\delta\\in(0,1)\\) and \\(\\kappa\\in(0,0.5)\\), if \\(\\hat{\\pi}_{E}\\in\\mathcal{H}\\) is the pretrained policy model learned based on the empirical risk minimization (ERM), and the sample complexity (i.e., number of samples) is larger than \\(\\frac{2\\log\\left(\\frac{2k}{2}\\right)}{e^{2}(1-2\\kappa)^{2}}\\), then the error between the model estimated and true results is smaller than \\(\\epsilon\\) with probability larger than \\(1-\\delta\\)._\n\nThe related proof details are presented in Appendix C. From Theorem 3, we can see: in order to guarantee the same performance with a given probability (i.e., \\(\\epsilon\\) and \\(\\delta\\) are fixed), one needs to generate more than \\(\\frac{2\\log\\left(\\frac{2k}{2}\\right)}{e^{2}(1-2\\kappa)^{2}}\\) sequences. If the noise level \\(\\kappa\\) is larger, more samples have to be generated. Extremely, if the pre-trained policy can only produce fully noisy information (i.e., \\(\\kappa=0.5\\)), then infinity number of samples are required, which is impossible in reality.\n\nFor the generalization ability, [8] explains the efficacy of counterfactual augmented data by the empirical evidence. In the context of offline IL, a straightforward yet very relevant conclusion from the analysis of generalization ability is the strong dependence on the number of expert data [25]. We work with finite state and action spaces (\\(|\\mathcal{S}|,|\\mathcal{A}|<\\infty\\)), and for the learned policy \\(\\pi_{\\omega}\\), we can analyze the generalization ability from the perspective of error upper bound with the optimal expert policy \\(\\pi^{\\star}\\).\n\n**Theorem 4**.: _Let \\(|\\mathcal{D}_{E}|\\) be the number of empirical expert data used to train the policy and \\(\\epsilon\\) be the expected upper bound of generalization error. There exists constant \\(h\\) such that, if_\n\n\\[|\\mathcal{D}_{E}|\\geq\\frac{h|\\mathcal{S}||\\mathcal{A}|\\log(|\\mathcal{S}|/ \\delta)}{\\epsilon^{2}}, \\tag{9}\\]\n\n_and each state \\(s_{t}\\) is sampled uniformly, then, with probability at least \\(1-\\delta\\), we have:_\n\n\\[\\max_{s_{t}}\\left\\|\\pi^{\\star}(\\cdot\\mid s_{t})-\\pi_{\\omega}(\\cdot\\mid s_{t}) \\right\\|_{1}\\leq\\epsilon. \\tag{10}\\]\n\n_which shows that increasing \\(|\\mathcal{D}_{E}|\\) drastically improves the generalization guarantee._\n\nNote that we provide the proof details for Theorem 4 in Appendix D.\n\n## 5 Experiments\n\nIn this section, we evaluate the performance of OILCA, aiming to answer the following questions.\n\n**Q1:** With the synthetic toy environment of the causal MDP, can OILCA disentangle the latent representations of the exogenous variable? **Q2:** For an in-distribution test environment, can OILCA improve the performance? **Q3:** For an out-of-distribution test environment, can OILCA improve the generalization? In addition, we use five baseline methods: BC-exp (BC on the expert data \\(\\mathcal{D}_{E}\\)), BC-all (BC on all demonstrations \\(\\mathcal{D}_{\\text{all}}\\)), ORIL [39], BCND [26], LobsDICE [12], DWBC [36]. More details about these methods are presented in Appendix E. Furthermore, the hyper-parameters of our method and baselines are all detail-tuned for better performance.\n\n### Simulations on Toy Environment (Q1)\n\nIn real-world situations, we can hardly get the actual distributions of the exogenous variable. To evaluate the disentanglement identifiability in variational counterfactual reasoning, we build a toy environment to show that our method can indeed get the disentangled distributions of the exogenous variable.\n\nToy environmentFor the toy environment, we consider a simple 2D navigation task. The agent can move a fixed distance in each of the four directions. States are continuous and are considered as the 2D position of the agent. The goal is to navigate to a specific target state within a bounding box. The reward is the negative distance between the agent's state and the target state. For the initialization of the environment, we consider \\(C\\) kinds of Gaussian distributions (Equation (7)) for the exogenous variable, where \\(C\\) = \\(3\\), and the conditional variable \\(c\\) is the class label. We design the transition function by using a multi-layer perceptron (MLP) with invertible activations. We present the details about data collection and model learning in Appendix E.\n\nResultsWe visualize the identifiability in such a 2D case in Figure 3, where we plot the sources, the posterior distributions learned by OILCA, and the posterior distributions learned by a vanilla conditional VAE, respectively. Our method recovers the original sources to trivial indeterminacies (rotation and sign flip), whereas the vanilla conditional VAE fails to separate the latent variables well. To show the effectiveness of our method in the constructed toy environment, we also conduct repeated experiments for 1K episodes (500 steps per episode) to compare OILCA with all baseline methods. The results are presented in Figure 4, showing that OILCA achieves the highest average return. To analyze the influence of the number of augmented samples (Theorem 4), we also conduct the experiments with varying numbers of counterfactual expert data; the result is shown in Figure 5. The X-axis represents the percentage of \\(|\\mathcal{D}_{E}|/|\\mathcal{D}_{U}|\\), where \\(\\mathcal{D}_{E}\\) is the augmented expert data. We can observe that within a certain interval, the generalization ability of the learned policy does improve with the increase of \\(|\\mathcal{D}_{E}|\\).\n\n### In-distribution Performance (Q2)\n\nWe use DeepMind Control Suite[32] to evaluate the performance of in-distribution performance. Similar to [39], we also define an episode as positive if its episodic reward is among the top 20% episodes for the task. For the auxiliary variable \\(c\\), we add three kinds of different Gaussian noise distributions into the environment (encoded as \\(c=\\{0,1,2\\}\\)). Moreover, we present the detailed data collection and statistics in Appendix E.1). We report the results in Table 1. We can observe that OILCA outperforms baselines on all tasks; this shows the effectiveness of the augmented counterfactual expert data. We also find that the performance of ORIL and DWBC tends to decrease in testing for some tasks (ORIL: Cheetah Run, Walker Walk; DWBC: Cheetah Run, Manipulator Insert Ball); this \"overfitting\" phenomenon also occurs in experiments of previous offline RL works [14, 33]. This is perhaps due to limited data size and model generalization bottleneck. DWBC gets better results on most tasks, but for some tasks, such as Manipulator Insert Ball and the Walker Walk, OILCA achieves more than twice the average return than DWBC.\n\n### Out-of-distribution Generalization (Q3)\n\nTo evaluate the out-of-distribution generalization of OILCA, we use a benchmark named CausalWorld[2], which provides a combinatorial family of tasks with a common causal structure and\n\n\\begin{table}\n\\begin{tabular}{l|c c c c c c c} \\hline \\hline Task Name & BC-exp & BC-all & ORIL & BCND & LobsDICE & DWBC & **OILCA** \\\\ \\hline \\multirow{3}{*}{Cartpole Swingup} & 195.44 & 269.03 & 221.24 & 243.52 & 292.96 & 382.55 & **608.38** \\\\  & \\(\\pm\\) 7.39 & \\(\\pm\\) 7.06 & \\(\\pm\\) 14.49 & \\(\\pm\\) 11.33 & \\(\\pm\\) 11.05 & \\(\\pm\\) 8.95 & \\(\\pm\\) 35.54 \\\\ Cheetah Run & 66.59 & 90.01 & 45.08 & 96.06 & 74.53 & 66.87 & **116.05** \\\\  & \\(\\pm\\) 11.09 & \\(\\pm\\) 31.74 & \\(\\pm\\) 9.88 & \\(\\pm\\) 16.15 & \\(\\pm\\) 7.75 & \\(\\pm\\) 4.60 & \\(\\pm\\) 14.65 \\\\ Finger Turn Hard & 129.20 & 104.56 & 185.57 & 204.67 & 190.93 & 243.47 & 298.73 \\\\  & \\(\\pm\\) 4.51 & \\(\\pm\\) 8.32 & \\(\\pm\\) 26.75 & \\(\\pm\\) 13.18 & \\(\\pm\\) 12.19 & \\(\\pm\\) 17.12 & \\(\\pm\\) 5.11 \\\\ Fish Swim & 74.59 & 68.87 & 84.90 & 153.28 & 188.84 & 212.39 & **290.28** \\\\  & \\(\\pm\\)11.73 & \\(\\pm\\) 11.93 & \\(\\pm\\) 19.96 & \\(\\pm\\) 19.29 & \\(\\pm\\) 11.28 & \\(\\pm\\)7.62 & \\(\\pm\\) 10.07 \\\\ Humanoid Run & 77.59 & 138.93 & 96.88 & 257.01 & 120.87 & 302.33 & **460.96** \\\\  & \\(\\pm\\)8.63 & \\(\\pm\\) 9.14 & \\(\\pm\\)10.76 & \\(\\pm\\) 11.21 & 410.66 & \\(\\pm\\) 14.53 & \\(\\pm\\) 17.55 \\\\ Manipulator Insert Ball & 91.61 & 98.59 & 98.25 & 141.71 & 197.79 & 107.86 & **296.83** \\\\  & \\(\\pm\\)7.49 & \\(\\pm\\)1.62 & \\(\\pm\\) 26.88 & \\(\\pm\\) 15.30 & \\(\\pm\\) 1.98 & \\(\\pm\\) 15.01 & \\(\\pm\\) 3.43 \\\\ Manipulator Insert Peg & 92.02 & 119.63 & 105.86 & 220.66 & 299.19 & 238.39 & **305.66** \\\\  & \\(\\pm\\)15.04 & \\(\\pm\\)6.37 & \\(\\pm\\) 5.10 & \\(\\pm\\) 15.14 & \\(\\pm\\) 3.40 & \\(\\pm\\) 19.76 & \\(\\pm\\) 4.91 \\\\ Walker Stand & 169.14 & 192.14 & 181.23 & 279.66 & 252.34 & 280.07 & **395.51** \\\\  & \\(\\pm\\) 8.00 & \\(\\pm\\) 37.91 & \\(\\pm\\) 10.31 & \\(\\pm\\) 12.69 & \\(\\pm\\) 7.73 & \\(\\pm\\) 5.79 & \\(\\pm\\) 8.05 \\\\ Walker Walk & 29.44 & 75.79 & 41.43 & 157.44 & 102.14 & 166.95 & **377.19** \\\\  & \\(\\pm\\) 2.18 & \\(\\pm\\) 6.34 & \\(\\pm\\) 4.05 & \\(\\pm\\) 9.31 & \\(\\pm\\) 5.94 & \\(\\pm\\) 10.68 & \\(\\pm\\) 15.87 \\\\ \\hline \\hline \\end{tabular}\n\\end{table}\nTable 1: Results for in-distribution performance on DeepMind Control Suite. We report the average return of episodes (with the length of 1K steps) over five random seeds. The best results and second best results are **bold** and underlined, respectively.\n\nFigure 4: Performance of OILCA with the growing percentage of \\(|\\mathcal{D}_{E}|/|\\mathcal{D}_{U}|\\). We plot the average return’s mean and standard errors over five random seeds.\n\nFigure 5: Performance of OILCA with the growing percentage of \\(|\\mathcal{D}_{E}|/|\\mathcal{D}_{U}|\\). We plot the mean and the standard errors over five random seeds.\n\nunderlying factors. Furthermore, the auxiliary variable \\(c\\) represents the different _do_-interventions on the task environments [2]. We collect the offline data by using three different _do_-interventions on environment features (stage_color, stage_friction. floor_friction) to generate offline datasets, while other features are set as the defaults. More data collection and statistics details are presented in Appendix E.1. We show the comparative results in Table 2. It is evident that OILCA also achieves the best performance on all the tasks; ORIL and DWBC perform the second-best results on most tasks. BCND performs poorly compared to other methods. The reason may be that the collected data in space \\(\\mathbf{A}\\) can be regarded as low-quality data when evaluating on space \\(\\mathbf{B}\\), which may lead to the poor performance of a single BC and cause cumulative errors for the BC ensembles of BCND. LobsDICE even performs poorer than BC-exp on most tasks. This is because the KL regularization, which regularizes the learned policy to stay close to \\(\\mathcal{D}_{U}\\) is too conservative, resulting in a suboptimal policy, especially when \\(\\mathcal{D}_{U}\\) contains a large collection of noisy data. This indeed hurts the performance of LobsDICE for out-of-distribution generalization.\n\n## 6 Conclusion\n\nIn this paper, we propose an effective and generalizable offline imitation learning framework OILCA, which can learn a policy from the expert and unlabeled demonstrations. We apply a novel identifiable counterfactual expert data augmentation approach to facilitate the offline imitation learning. We also analyze the influence of generated data and the generalization ability theoretically. The experimental results demonstrate that our method achieves better performance in simulated and public tasks.\n\n## Acknowledgement\n\nThis work is supported in part by National Key R&D Program of China (2022ZD0120103), National Natural Science Foundation of China (No. 62102420), Beijing Outstanding Young Scientist Program NO. BJWZYJH012019100020098, Intelligent Social Governance Platform, Major Innovation & Planning Interdisciplinary Platform for the \"Double-First Class\" Initiative, Renmin University of China, Public Computing Cloud, Renmin University of China, fund for building world-class universities (disciplines) of Renmin University of China, Intelligent Social Governance Platform.\n\n## References\n\n* [1] Alekh Agarwal, Nan Jiang, Sham M. Kakade, and Wen Sun. Reinforcement learning: Theory and algorithms, 2021.\n\n\\begin{table}\n\\begin{tabular}{l|c c c c c c c} \\hline \\hline Task Name & BC-exp & BC-all & ORIL & BCND & LobsDICE & DWBC & **OILCA** \\\\ \\hline \\multirow{2}{*}{Reaching} & 281.18 & 176.54 & 339.40 & 228.33 & 243.29 & 479.92 & **976.60** \\\\  & \\(\\pm\\) 16.64 & \\(\\pm\\) 9.75 & \\(\\pm\\) 12.98 & \\(\\pm\\) 7.14 & \\(\\pm\\) 9.84 & \\(\\pm\\) 18.75 & \\(\\pm\\) 20.13 \\\\  & 256.64 & 235.58 & 283.91 & 191.23 & 206.44 & 298.09 & **405.08** \\\\  & \\(\\pm\\) 12.70 & \\(\\pm\\) 10.23 & \\(\\pm\\) 19.72 & \\(\\pm\\) 12.64 & \\(\\pm\\) 15.35 & \\(\\pm\\) 14.94 & \\(\\pm\\) 24.03 \\\\  & 270.01 & 258.54 & 388.15 & 221.89 & 377.78 & 366.26 & **491.09** \\\\ Picking & \\(\\pm\\) 13.13 & \\(\\pm\\) 16.53 & \\(\\pm\\) 19.21 & \\(\\pm\\) 7.68 & \\(\\pm\\) 12.09 & \\(\\pm\\) 8.77 & \\(\\pm\\) 6.44 \\\\  & 294.06 & 225.42 & 270.75 & 259.12 & 266.09 & 349.66 & **490.24** \\\\  & \\(\\pm\\) 7.34 & \\(\\pm\\) 12.44 & \\(\\pm\\) 14.87 & \\(\\pm\\) 8.01 & \\(\\pm\\) 10.31 & \\(\\pm\\) 7.39 & \\(\\pm\\) 11.69 \\\\ Stacking2 & 496.63 & 394.91 & 388.55 & 339.18 & 362.47 & 481.07 & **831.82** \\\\  & \\(\\pm\\) 7.68 & \\(\\pm\\) 16.98 & \\(\\pm\\) 10.93 & \\(\\pm\\) 9.46 & \\(\\pm\\) 17.05 & \\(\\pm\\) 10.11 & \\(\\pm\\) 11.78 \\\\ Towers & 667.81 & 784.88 & 655.96 & 139.30 & 535.74 & 768.68 & **994.82** \\\\  & \\(\\pm\\) 9.27 & \\(\\pm\\) 17.17 & \\(\\pm\\) 15.14 & \\(\\pm\\) 18.22 & \\(\\pm\\) 13.76 & \\(\\pm\\) 24.27 & \\(\\pm\\) 5.76 \\\\ Stacked Blocks & 581.91 & 452.88 & 702.15 & 341.97 & 250.44 & 1596.96 & **2617.71** \\\\  & \\(\\pm\\) 26.92 & \\(\\pm\\) 18.78 & \\(\\pm\\) 15.30 & \\(\\pm\\) 33.69 & \\(\\pm\\) 14.08 & \\(\\pm\\) 81.84 & \\(\\pm\\) 88.07 \\\\  & 496.32 & 529.82 & 882.27 & 288.55 & 317.95 & 700.23 & **1348.49** \\\\  & \\(\\pm\\) 26.92 & \\(\\pm\\) 31.01 & \\(\\pm\\) 46.79 & \\(\\pm\\) 19.63 & \\(\\pm\\) 32.03 & \\(\\pm\\) 13.71 & \\(\\pm\\) 55.05 \\\\ General & 492.78 & 547.32 & 647.95 & 195.06 & 458.27 & 585.98 & **891.14** \\\\  & \\(\\pm\\)17.64 & \\(\\pm\\) 8.49 & \\(\\pm\\) 24.39 & \\(\\pm\\)9.80 & \\(\\pm\\) 18.69 & \\(\\pm\\) 19.25 & \\(\\pm\\) 23.12 \\\\ \\hline \\hline \\end{tabular}\n\\end{table}\nTable 2: Results for out-of-distribution generalization on CausalWorld. We report the average return of episodes (length varies for different tasks) over five random seeds. All the models are trained on _space_\\(\\mathbf{A}\\) and tested on _space_\\(\\mathbf{B}\\) to show the out-of-distribution performance [2].\n\n* [2] Ossama Ahmed, Frederik Trauble, Anirudh Goyal, Alexander Neitz, Yoshua Bengio, Bernhard Scholkopf, Manuel Wuthrich, and Stefan Bauer. Causalworld: A robotic manipulation benchmark for causal structure and transfer learning. _arXiv preprint arXiv:2010.04296_, 2020.\n* [3] Dafni Antotsiou, Carlo Ciliberto, and Tae-Kyun Kim. Adversarial imitation learning with trajectorial augmentation and correction. In _2021 IEEE International Conference on Robotics and Automation (ICRA)_, pages 4724-4730. IEEE, 2021.\n* [4] Leon Bottou, Jonas Peters, Joaquin Quinonero-Candela, Denis X Charles, D Max Chickering, Elon Portugaly, Dipankar Ray, Patrice Simard, and Ed Snelson. Counterfactual reasoning and learning systems: The example of computational advertising. _Journal of Machine Learning Research_, 14(11), 2013.\n* [5] Ivan Bratko, Tanja Urbancic, and Claude Sammut. Behavioural cloning: phenomena, results and problems. _IFAC Proceedings Volumes_, 28(21):143-149, 1995.\n* [6] Lars Buesing, Theophane Weber, Yori Zwols, Sebastien Racaniere, Arthur Guez, Jean-Baptiste Lespiau, and Nicolas Heess. Woulda, coulda, shoulda: Counterfactually-guided policy search. _arXiv preprint arXiv:1811.06272_, 2018.\n* [7] Satoshi Hoshino, Tomoki Hisada, and Ryota Oikawa. Imitation learning based on data augmentation for robotic reaching. In _2021 60th Annual Conference of the Society of Instrument and Control Engineers of Japan (SICE)_, pages 417-424. IEEE, 2021.\n* [8] Divyansh Kaushik, Amrith Setlur, Eduard Hovy, and Zachary C Lipton. Explaining the efficacy of counterfactually augmented data. _arXiv preprint arXiv:2010.02114_, 2020.\n* [9] Ilyes Khemakhem, Diederik Kingma, Ricardo Monti, and Aapo Hyvarinen. Variational autoencoders and nonlinear ica: A unifying framework. In _International Conference on Artificial Intelligence and Statistics_, pages 2207-2217. PMLR, 2020.\n* [10] Taylor W Killian, Marzyeh Ghassemi, and Shalmali Joshi. Counterfactually guided policy transfer in clinical settings. In _Conference on Health, Inference, and Learning_, pages 5-31. PMLR, 2022.\n* [11] Beomjoon Kim and Joelle Pineau. Socially adaptive path planning in human environments using inverse reinforcement learning. _International Journal of Social Robotics_, 8:51-66, 2016.\n* [12] Geon-Hyeong Kim, Jongmin Lee, Youngsoo Jang, Hongseok Yang, and Kee-Eung Kim. Lobsdice: offline imitation learning from observation via stationary distribution correction estimation. _arXiv preprint arXiv:2202.13536_, 2022.\n* [13] Diederik P Kingma and Max Welling. Auto-encoding variational bayes. _arXiv preprint arXiv:1312.6114_, 2013.\n* [14] Aviral Kumar, Justin Fu, Matthew Soh, George Tucker, and Sergey Levine. Stabilizing off-policy q-learning via bootstrapping error reduction. _Advances in Neural Information Processing Systems_, 32, 2019.\n* [15] Yao Lai, Jinxin Liu, Zhentao Tang, Bin Wang, Jianye Hao, and Ping Luo. Chipformer: Transferable chip placement via offline decision transformer. _arXiv preprint arXiv:2306.14744_, 2023.\n* [16] Jinxin Liu, Li He, Yachen Kang, Zifeng Zhuang, Donglin Wang, and Huazhe Xu. Ceil: Generalized contextual imitation learning. _arXiv preprint arXiv:2306.14534_, 2023.\n* [17] Jinxin Liu, Lipeng Zu, Li He, and Donglin Wang. Clue: Calibrated latent guidance for offline reinforcement learning. _arXiv preprint arXiv:2306.13412_, 2023.\n* [18] Francesco Locatello, Stefan Bauer, Mario Lucic, Gunnar Raetsch, Sylvain Gelly, Bernhard Scholkopf, and Olivier Bachem. Challenging common assumptions in the unsupervised learning of disentangled representations. In _international conference on machine learning_, pages 4114-4124. PMLR, 2019.\n\n* [19] Chaochao Lu, Biwei Huang, Ke Wang, Jose Miguel Hernandez-Lobato, Kun Zhang, and Bernhard Scholkopf. Sample-efficient reinforcement learning via counterfactual-based data augmentation. _arXiv preprint arXiv:2012.09092_, 2020.\n* [20] Thomas Mesnard, Theophane Weber, Fabio Viola, Shantanu Thakoor, Alaa Saade, Anna Harutyunyan, Will Dabney, Tom Stepleton, Nicolas Heess, Arthur Guez, et al. Counterfactual credit assignment in model-free reinforcement learning. _arXiv preprint arXiv:2011.09464_, 2020.\n* [21] Graziano Mita, Maurizio Filippone, and Pietro Michiardi. An identifiable double vae for disentangled representations. In _International Conference on Machine Learning_, pages 7769-7779. PMLR, 2021.\n* [22] Michael Oberst and David Sontag. Counterfactual off-policy evaluation with gumbel-max structural causal models. In _International Conference on Machine Learning_, pages 4881-4890. PMLR, 2019.\n* [23] Judea Pearl. Models, reasoning and inference. _Cambridge, UK: CambridgeUniversityPress_, 19 (2), 2000.\n* [24] Silviu Pitis, Elliot Creager, Ajay Mandlekar, and Animesh Garg. Mocoda: Model-based counterfactual data augmentation. _arXiv preprint arXiv:2210.11287_, 2022.\n* [25] Stephane Ross, Geoffrey Gordon, and Drew Bagnell. A reduction of imitation learning and structured prediction to no-regret online learning. In _Proceedings of the fourteenth international conference on artificial intelligence and statistics_, pages 627-635. JMLR Workshop and Conference Proceedings, 2011.\n* [26] Fumihiro Sasaki and Ryota Yamashina. Behavioral cloning from noisy demonstrations. In _International Conference on Learning Representations_, 2021.\n* [27] Shai Shalev-Shwartz and Shai Ben-David. _Understanding machine learning: From theory to algorithms_. Cambridge university press, 2014.\n* [28] Kihyuk Sohn, Honglak Lee, and Xinchen Yan. Learning structured output representation using deep conditional generative models. _Advances in neural information processing systems_, 28, 2015.\n* [29] Jonathan Spencer, Sanijban Choudhury, Arun Venkatraman, Brian Ziebart, and J Andrew Bagnell. Feedback in imitation learning: The three regimes of covariate shift. _arXiv preprint arXiv:2102.02872_, 2021.\n* [30] Wen Sun, Anirudh Vemula, Byron Boots, and Drew Bagnell. Provably efficient imitation learning from observation alone. In _International conference on machine learning_, pages 6036-6045. PMLR, 2019.\n* [31] Gokul Swamy, Sanjiban Choudhury, J Andrew Bagnell, and Steven Wu. Of moments and matching: A game-theoretic framework for closing the imitation gap. In _International Conference on Machine Learning_, pages 10022-10032. PMLR, 2021.\n* [32] Yuval Tassa, Yotam Doron, Alistair Muldal, Tom Erez, Yazhe Li, Diego de Las Casas, David Budden, Abbas Abdolmaleki, Josh Merel, Andrew Lefrancq, et al. Deepmind control suite. _arXiv preprint arXiv:1801.00690_, 2018.\n* [33] Yifan Wu, George Tucker, and Ofir Nachum. Behavior regularized offline reinforcement learning. _arXiv preprint arXiv:1911.11361_, 2019.\n* [34] Zheng Wu, Liting Sun, Wei Zhan, Chenyu Yang, and Masayoshi Tomizuka. Efficient sampling-based maximum entropy inverse reinforcement learning with application to autonomous driving. _IEEE Robotics and Automation Letters_, 5(4):5355-5362, 2020.\n* [35] Danfei Xu and Misha Denil. Positive-unlabeled reward learning. In _Conference on Robot Learning_, pages 205-219. PMLR, 2021.\n\n* [36] Haoran Xu, Xianyuan Zhan, Honglei Yin, and Huiling Qin. Discriminator-weighted offline imitation learning from suboptimal demonstrations. In _International Conference on Machine Learning_, pages 24725-24742. PMLR, 2022.\n* [37] Zhenting Zhao, Bowei He, Wenhao Luo, and Rui Liu. Collective conditioned reflex: A bio-inspired fast emergency reaction mechanism for designing safe multi-robot systems. _IEEE Robotics and Automation Letters_, 7(4):10985-10990, 2022.\n* [38] Zheng-Mao Zhu, Xiong-Hui Chen, Hong-Long Tian, Kun Zhang, and Yang Yu. Offline reinforcement learning with causal structured world models. _arXiv preprint arXiv:2206.01474_, 2022.\n* [39] Konrad Zolna, Alexander Novikov, Ksenia Konyushkova, Caglar Gulcehre, Ziyu Wang, Yusuf Aytar, Misha Denil, Nando de Freitas, and Scott Reed. Offline learning from demonstrations and unlabeled experience. _arXiv preprint arXiv:2011.13885_, 2020.\n* [40] Konrad Zolna, Scott Reed, Alexander Novikov, Sergio Gomez Colmenarejo, David Budden, Serkan Cabi, Misha Denil, Nando de Freitas, and Ziyu Wang. Task-relevant adversarial imitation learning. In _Conference on Robot Learning_, pages 247-263. PMLR, 2021.", "response": "## Review\n\n### Summary\n\nThis paper presents OILCA, a novel framework for offline imitation learning that utilizes causality-regularized data augmentation to generate counterfactual samples, thereby addressing the scarcity of expert data. It provides a theoretical analysis of the identifiability of the constructed exogenous variables and demonstrates the effectiveness of the method through extensive experiments, particularly in the CausalWorld benchmark. The empirical results are promising, showcasing significant performance improvements over multiple baselines across various environments. However, more thorough evaluations and clarifications regarding certain methodological aspects are suggested to bolster the claims made.\n\n### Strengths\n\n- The idea of using counterfactual data augmentation is novel and relevant to the RL community.\n- The paper is well-written, making the core ideas easy to follow.\n- The method shows solid experimental results across multiple tasks.\n- The theoretical framework provided strengthens the contributions of the paper.\n- The proposed method is compatible with various offline imitation learning algorithms.\n\n### Weaknesses\n\n- Insufficient discussion of related work, particularly recent advancements in the field.\n- The clarity of the introduction and definitions, particularly regarding spurious relations and the do-intervention process, needs improvement.\n- The paper relies heavily on a pretrained expert policy, raising questions about the reliability of the generated data.\n- The experimental evaluation lacks sufficient ablation studies and could benefit from a more detailed description of the baselines used.\n- Concerns about the potential instability in training due to the many components involved in OILCA.\n\n### Questions\n\n- How does the algorithm handle hyper-parameter sensitivity, especially given the context of offline reinforcement learning?\n- What specific settings were changed compared to previous algorithms, and how do they affect performance?\n- Is there a risk of compounding errors when generating counterfactual states based on already generated data?\n- What are the criteria for selecting the variable 'c' used in the method?\n\n### Soundness\n\n**Score:** 3\n\n**Description:** The soundness of the theoretical framework and empirical results is good, but some methodological clarifications are needed to fully support the claims.\n\n### Presentation\n\n**Score:** 3\n\n**Description:** The presentation is generally good, but there are areas where clarity could be enhanced, particularly in definitions and the flow of the introduction.\n\n### Contribution\n\n**Score:** 3\n\n**Description:** The contributions are significant, particularly in introducing a novel approach to data augmentation in offline imitation learning, but could be strengthened with additional context and related work.\n\n### Rating\n\n**Score:** 7\n\n**Description:** Accept: The paper is technically solid and shows promise in its contributions, but minor improvements in clarity and depth of evaluation are needed.\n\n### Paper Decision\n\n**Decision:** Accept (poster)\n\n**Reasons:** The paper presents an innovative approach to offline imitation learning that combines theoretical rigor with empirical validation. While it has some weaknesses regarding presentation and related work, the overall contributions are significant and impactful, warranting acceptance.\n"}
{"query": "You are a highly experienced, conscientious, and fair academic reviewer, please help me review this paper. The review should be organized into nine sections: \n1. Summary: A summary of the paper in 100-150 words.\n2. Strengths/Weaknesses/Questions: The Strengths/Weaknesses/Questions of paper, which should be listed in bullet points, with each point supported by specific examples from the article where possible.\n3. Soundness/Contribution/Presentation: Rate the paper's Soundness/Contribution/Presentation, and match this score to the corresponding description from the list below and provide the result. The possible scores and their descriptions are: \n    1 poor\n    2 fair\n    3 good\n    4 excellent\n4. Rating: Give this paper an appropriate rating, match this rating to the corresponding description from the list below and provide the result. The possible Ratings and their descriptions are: \n    1 strong reject\n    2 reject, significant issues present\n    3 reject, not good enough\n    4 possibly reject, but has redeeming facets\n    5 marginally below the acceptance threshold\n    6 marginally above the acceptance threshold\n    7 accept, but needs minor improvements \n    8 accept, good paper\n    9 strong accept, excellent work\n    10 strong accept, should be highlighted at the conference   \n5. Paper Decision: It must include the Decision itself(Accept or Reject) and the reasons for this decision, based on the criteria of originality, methodological soundness, significance of results, and clarity and logic of presentation.\n\nHere is the template for a review format, you must follow this format to output your review result:\n**Summary:**\nSummary content\n\n**Strengths:**\n- Strength 1\n- Strength 2\n- ...\n\n**Weaknesses:**\n- Weakness 1\n- Weakness 2\n- ...\n\n**Questions:**\n- Question 1\n- Question 2\n- ...\n\n**Soundness:**\nSoundness result\n\n**Presentation:**\nPresentation result\n\n**Contribution:**\nContribution result\n\n**Rating:**\nRating result\n\n**Paper Decision:**\n- Decision: Accept/Reject\n- Reasons: reasons content\n\n\nPlease ensure your feedback is objective and constructive. The paper is as follows:\n\n# Balance Risk and Reward: A Batched-Bandit Strategy for Automated Phased Release\n\nYufan Li\\({}^{1}\\),  Jialiang Mao\\({}^{2}\\),  Iavor Bojinov\\({}^{3}\\)\n\n\\({}^{1}\\)Harvard University \\({}^{2}\\)LinkedIn Corporation \\({}^{3}\\)Harvard Business School\n\nyufan_li@g.harvard.edu, jimao@linkedin.com, ibojinov@hbs.com\n\n###### Abstract\n\nPhased releases are a common strategy in the technology industry for gradually releasing new products or updates through a sequence of A/B tests in which the number of treated units gradually grows until full deployment or deprecation. Performing phased releases in a principled way requires selecting the proportion of units assigned to the new release in a way that balances the risk of an adverse effect with the need to iterate and learn from the experiment rapidly. In this paper, we formalize this problem and propose an algorithm that automatically determines the release percentage at each stage in the schedule, balancing the need to control risk while maximizing ramp-up speed. Our framework models the challenge as a constrained batched bandit problem that ensures that our pre-specified experimental budget is not depleted with high probability. Our proposed algorithm leverages an adaptive Bayesian approach in which the maximal number of units assigned to the treatment is determined by the posterior distribution, ensuring that the probability of depleting the remaining budget is low. Notably, our approach analytically solves the ramp sizes by inverting probability bounds, eliminating the need for challenging rare-event Monte Carlo simulation. It only requires computing means and variances of outcome subsets, making it highly efficient and parallelizable.\n\n## 1 Introduction\n\nPhased release, also known as staged rollout, is a widely used strategy in the technology industry that involves gradually releasing a new product or update to larger audiences over time [17, 30]. For example, Apple's App Store offers a phased release option where application updates are released over a 7-day period on a fixed schedule [1]. Google Play Console provides a similar feature with more flexibility in the release schedule [16]. Typically, the audiences are randomly selected at each stage from the set of all customers, and so phased releases can be thought of as a sequence of A/B tests (or randomized experiments) in which the proportion of units assigned to the treatment group changes until either the product or update is fully launched or deprecated [26, 18, 3, 33, 6]. The process of combining phased releases with A/B tests is often called controlled rollout or iterative experiments and provides companies with an important mechanism to gather feedback on early product versions [30, 20, 5].\n\nThe key advantage of phased release is its ability to mitigate risks associated with launching a new product or update directly to all users. The potential impact of faulty features is limited by releasing the update first to a small percentage of the users (i.e., the treatment group). However, this risk-averse approach introduces an opportunity cost for slowly launching beneficial features, which quickly adds up for companies that release thousands of features yearly [34]. Therefore, when designing a phased release schedule, it is important to determine the release percentage (known as ramp schedule) at each stage that balances the need to control risk while maximizing the speed of ramp-up. \\({}^{*}\\) This paper proposes an algorithm to address this challenge by automatically determining the releasepercentage for the next phase based on observations from previous stages. Specifically, we frame the challenge as a budget-constrained batched bandit problem. For each batch, we aim to determine the assignment probabilities of newly arrived users while keeping the probability of depleting a pre-specified experimental budget, where the experiment's cost is the cumulative treatment effect that are not directly observed. Formally, we derive recursive relations that decompose the risk of ruin (depleting the budget) of a phased release to the individual stages in the sense that the risk of ruin of the entire experiment is controlled if stage-wise ruin probabilities are controlled. Our algorithm is Bayesian in the sense that it learns from past observations by computing the posteriors of a conjugate Gaussian model and uses these parameters to infer the remaining budget and other cost-related quantities. However, the algorithm is robust to misspecifications and works well even when underlying outcomes are far from Gaussian by law of large number and central limit theorem; nevertheless, in Appendix E, we provide an extension to non-Gaussian outcomes. Finally, the next stage's assignment probabilities are derived from the posterior distribution and the stage-wise risk tolerances. Notably, our approach solves ramp sizes analytically from inverting the ruin probability upper bounds, avoiding challenging rare-event Monte-Carlo simulation for budget depletion events and data imputation procedures for unobserved counterfactual outcomes.\n\n### Literature review\n\nWhile many firms have guidelines on how to conduct a phased release process, these guidelines are often ad-hoc and qualitative, making it difficult to create executable ramp schedules. The SQR framework in [34] is the first attempt to address this problem by providing quantitative guidance. Our work differs significantly from SQR. Our algorithm adopts a fully Bayesian approach, enabling us to incorporate prior information on the risk of a feature in a probabilistic manner when initiating a ramp. Additionally, unlike SQR, our approach introduces a \"shared budget\" over the entire phased release, allowing the budget to be sequentially adjusted based on the observations from prior iterations. Finally, our algorithm is robust to modifications made to the treatment during experiments and different outcome models.\n\nOur work is notably distinct from the risk-averse multiarmed bandit approaches considered in previous research [19; 14; 35; 28; 23; 10; 8]. In these approaches, the agent considers the expected variability in expected rewards to identify and avoid less predictable (and therefore risky) actions, without considering a budget constraint. A related literature focuses on batched and Bayesian variants of these methods in multi-stage clinical trials [4; 27; 21; 24; 2]. While this literature also aims to determine treatment assignment for each stage of the experiment, it differs from our setting in two key aspects: (i) the objective is to maximize treatment effect while balancing exploration of treatment arms, rather than rapidly ramping up experiments, and (ii) to the best of our knowledge, no clinical trials paper has addressed the imposition of a budget for potential adverse treatment effects. Hence, bandit approaches developed for clinical trials cannot be directly applied to our setting. To illustrate the difference, we present a numerical simulation of a Thompson sampling-based Bayesian bandit from [27] and highlight that budget spent and the aggressiveness of the ramp-up schedule depends on model tuning in a very unpredictable way, making the ramp-up schedule far from ideal. Another related literature is budgeted multiarmed bandits [32; 31; 9; 29; 12]. However, most budgeted bandit algorithms are developed for settings very different from ours and do not consider risk-of-ruin control or handle unobserved costs. Therefore, these algorithms cannot be directly applied to our specific scenario.\n\nNotation.Let \\(\\mathbb{N}\\) be the set of non-negative integers and \\(\\mathbb{N}_{+}:=\\mathbb{N}\\setminus\\{0\\}\\). Let \\(\\mathbb{R}\\), \\(\\mathbb{R}_{+}\\) denote the set of real numbers and positive real numbers respectively. \\([N]:=1,\\ldots,N\\) for \\(N\\in\\mathbb{N}_{+}\\). \\(\\sigma(\\cdot)\\) is the generated \\(\\sigma\\)-algebra. \\(X\\in\\mathcal{F}\\) if random variable \\(X\\) is measurable to \\(\\mathcal{F}\\). \\([X\\mid\\mathcal{F}]\\) denotes a random variable with distribution \\(\\mathbb{P}(X\\in\\cdot\\mid\\mathcal{F})\\) for a random variable \\(X\\) and \\(\\sigma\\)-algebra \\(\\mathcal{F}\\).\n\n## 2 Risk-of-ruin-constrained experiment and strategy overview\n\n### Risk-of-ruin-constrained experiment\n\nConsider a scenario in which a single feature is released to a sequence of subpopulations \\(\\mathcal{N}_{t}\\) consisting of \\(N_{t}\\) units at stages \\(t=1,...,T\\), where \\(T\\) is not necessarily fixed. At each stage, we randomly assign treatment to a group of units denoting the indexing set \\(\\mathcal{T}_{t}\\), while the control group with indexing set \\(\\mathcal{C}_{t}\\). The size of the treatment group at stage \\(t\\) is \\(|\\mathcal{T}_{t}|=m_{t}\\) and the size of the control group is \\(|\\mathcal{C}_{t}|=N_{T}-m_{t}\\).\n\nOur paper adopts the Neyman-Rubin framework for causal inference [22; 25; 7], where the potential outcome of each unit \\(i\\in\\mathcal{N}_{t}\\) during experiment stage \\(t\\) under control and treatment are denoted by \\(Y_{i,t}(0)\\in\\mathbb{R}\\) and \\(Y_{i,t}(1)\\in\\mathbb{R}\\), respectively1. Appendix E provides the extension to multivariate outcomes. The treatment assignment of unit \\(i\\in\\mathcal{N}_{t}\\) at stage \\(t\\) is denoted by \\(W_{i,t}\\). Since each unit only receives a single treatment at each stage, we only observe \\(Y_{i,t}(W_{i,t})\\) during the experiment, not the counterfactual \\(Y_{i,t}(1-W_{i,t})\\) (we are explicitly assuming that there is full compliance).\n\nFootnote 1: We are implicitly assuming that there is no interference between experimental units, that is, each unit’s outcomes do not depend on any other unit’s assignments [11].\n\nLet \\(\\mathcal{F}_{t}=\\sigma\\Big{(}\\big{(}Y_{\\mathcal{T}_{k},k}(1)\\big{)}_{k\\in[t]}, \\left(Y_{\\mathcal{C}_{k},k}(0)\\right)_{k\\in[t]},\\left(W_{i,k}\\right)_{i\\in \\mathcal{N},k\\in[t]}\\Big{)}\\) be the \\(\\sigma\\)-algebra generated by the treatment assignment and the observed experiment outcome in the first \\(t\\) stages, with \\(\\mathcal{F}_{0}\\) representing the trivial \\(\\sigma\\)-algebra. In our setting, the experimenter aims to ramp up the experiment to the \"max-power stage\" (50% of the population placed in treatment) as quickly as possible while avoiding the risk of a large negative business impact (or cost). We define the cost of the experiment as the treatment effect on the treated. See generalized cost in Appendix E.\n\n**Definition 2.1** (Experiment cost).: The cost of the experiment from stage \\(t\\in[T]\\) is \\(r_{t}:=\\sum_{i\\in\\mathcal{T}_{t}}Y_{i,t}(1)-Y_{i,t}(0)\\), where \\(r_{t}=0\\) if \\(\\mathcal{T}_{t}=\\emptyset\\). The cumulative cost is \\(R_{t}:=\\sum_{k\\in[t]}r_{k}\\).\n\nThroughout, \\(r_{t}<0\\) corresponds to a negative business impact; our goal is to control the experiment cost by setting a budget \\(B<0\\) and imposing the cost constraint \\(R_{T}>B\\). Since the outcomes are stochastic, we require this cost constraint to be satisfied with probability at least \\(1-\\delta\\) for some \\(\\delta\\in[0,1)\\) set before the experiment. Our goal is then to adaptively determine the size of the treatment group based on the observed data while satisfying our risk constraint. We refer to such an experiment as a risk-of-ruin-constrained (RRC) experiment.\n\n**Definition 2.2** (RRC experiment).: Fix any \\(B<0,\\delta\\in[0,1)\\). A \\((\\delta,B)\\)-RRC experiment running for \\(T\\) stages selects the size of \\(\\mathcal{T}_{t}\\) before \\(t\\)-th stage of the experiment such that \\(\\mathbb{P}(R_{T}>B)\\geq 1-\\delta\\).\n\n### Strategy overview\n\nOur goal is to determine the number of users to assign to the treatment \\(m_{t}\\in[0,N_{t}/2]\\), such that \\(\\sum_{i=1}^{T}m_{t}\\) is maximized while ensuring that the experiment is \\((B,\\delta)\\)-RRC.\n\nOur strategy is to decompose the overall constraint (_i.e._, that the experiment is \\((B,\\delta)\\)-RRC) into a sequence of stage-wise adaptive constraints using Theorem 3.1, below. We then sequentially maximize \\(m_{t}\\) under the stage-wise constraint for the \\(t\\)-th stage. Under the Gaussian model, the stage-wise constraints can be directly solved using a simple quadratic equation to obtain the maximum \\(m_{t}\\).\n\nAlthough our algorithm solves a relaxed version of the original optimization problem, our solution has crucial practical implications. Theorem 3.1 decomposes the \\((B,\\delta)\\)-RRC constraints in an adaptive fashion, _i.e._, if the feature turns out to be safe, the stage-wise constraints will relax in response, and the experiment will ramp up quickly.\n\n## 3 Model and the algorithm\n\n### Decompose the risk of ruin\n\nOur experimental design is based on the following theorem, which identifies a sequence of sufficient conditions for a sequential experiment to be \\((\\delta,B)\\)-RRC. We defer its proof to Appendix A.\n\n**Theorem 3.1**.: _Fix \\(B<0\\) and \\(\\delta\\in[0,1)\\). For any stopping time \\(T\\geq 1\\), let \\((b_{t})_{t\\in[T]}\\) be a budget sequence, such that \\(b_{t}\\stackrel{{(i)}}{{\\geq}}B,\\forall t\\in[T]\\), and \\((\\Delta_{t})_{t\\in[T]}\\) be a risk tolerance sequence, such that \\(\\Delta_{t}\\in[0,1),\\forall t\\in[T]\\) and \\(1-\\prod_{t=1}^{T}\\left(1-\\Delta_{t}\\right)\\overset{(ii)}{\\leq}\\delta\\). Then, if \\((\\mathcal{T}_{t})_{t\\in[T]}\\) is chosen such that for \\(t=1\\),_\n\n\\[\\mathbb{P}\\bigg{(}r_{1}\\leq b_{1}\\bigg{)}\\overset{(iii)}{\\leq}\\Delta_{1} \\tag{1}\\]\n\n_and for any \\(t=2,...,T\\), almost surely,_\n\n\\[\\begin{cases}\\mathcal{T}_{t}=\\emptyset,&\\text{if }\\mathbb{P}\\left(R_{t-1}>B \\mid\\mathcal{F}_{t-1}\\right)=0\\\\ \\mathbb{P}\\left(R_{t}\\leq b_{t}\\mid R_{t-1}>B,\\mathcal{F}_{t-1}\\right)\\overset {(iv)}{\\leq}\\Delta_{t},&\\text{otherwise}\\end{cases} \\tag{2}\\]\n\n_then \\(\\mathbb{P}\\left(R_{T}>B\\right)\\geq 1-\\delta\\). This inequality is tight when \\((i)\\)-\\((iv)\\) are all equalities and \\(r_{t}\\leq 0,\\forall t\\in[T]\\) almost surely. Furthermore, if we set \\(\\mathcal{T}_{t}\\leftarrow\\emptyset\\), (1), (2) always hold._\n\nRecall, \\(B<0\\) denotes the budget and \\(\\delta\\in[0,1)\\) is our risk tolerance that controls the risk of ruin (_i.e.,_ the probability of exceeding the budget); both need to be fixed a _priori_. In general, smaller \\(\\delta\\) leads to more conservative experimentation and slower releases. The sequence \\((b_{t})_{t\\geq 1}\\) \"rations\" the budget: setting \\(b_{t}<B\\) at stage \\(t\\) reserves \\(B-b_{t}\\) budget for later stages, which may be beneficial when the released feature is expected to undergo modifications during the experiment [20]. To quickly scale the experiment, we can set \\(b_{t}=B,\\forall t\\). The sequence \\((\\Delta_{t})_{t\\geq 1}\\) distributes the overall tolerance \\(\\delta\\) to individual stages by \\(\\delta=1-\\prod_{i-t}^{T}\\left(1-\\Delta_{t}\\right)\\) allowing us to customize the tolerance for individual stages. If \\(T\\) is fixed a priori, we can uniformly distribute the tolerance by setting \\(\\Delta_{t}=1-(1-\\delta)^{1/T},\\forall t\\in[T]\\).\n\nTheorem 3.1 breaks the risk constraint \\(\\mathbb{P}\\left(R_{T}\\leq B\\right)\\leq\\delta\\) into stage-wise constraints in the form of (1),(2). The idea is to control current-stage cumulative experiment cost \\(R_{t}\\) given past observations \\(\\mathcal{F}_{t}\\) and determine the treatment assignment based on posterior inference of the remaining budget. In our setting, the goal is to maximize \\(m_{t}=|\\mathcal{T}_{t}|\\) subject to (1),(2). Note that the first line in (2) stops the experiment when the model estimates the budget is exhausted, while the second line sets the stage cost \\(r_{t}\\) below the remaining budget \\(b_{t}-R_{t-1}\\) with high probability. We require assumptions on the data-generating model to derive an explicit algorithm, which we present in the next sections.\n\nGenerally, the total number of stages \\(T\\), stage-wise budget and tolerance \\((b_{t})_{t\\geq 1}\\), \\((\\Delta_{t})_{t\\geq 1}\\) can be determined dynamically during the process. That is, we can define \\(\\Delta_{t}\\) and \\(b_{t}\\) just before stage \\(t\\) as long as \\(\\prod_{r=1}^{t}\\left(1-\\Delta_{t}\\right)\\leq 1-\\delta\\) and \\(b_{t}\\geq B\\). If we plan to terminate the experiment after stage \\(t\\), we can define \\(\\Delta_{t}\\) such that \\(\\prod_{r=1}^{t}\\left(1-\\Delta_{r}\\right)=1-\\delta\\) and \\(b_{t}\\geq B\\) is attained in which case \\(T=t\\). If is also possible to have \\(T=+\\infty\\) and \\(\\prod_{t=1}^{\\infty}\\left(1-\\Delta_{t}\\right)=1-\\delta\\). For example, choosing \\(\\Delta_{t}=\\left(\\gamma_{\\star}/t\\right)^{2}\\) where \\(\\gamma_{\\star}\\) is the unique solution of \\(\\text{sinc}(\\gamma_{\\star})=1-\\delta\\) on \\([0,1]\\) (cf. [13, Eq. (1)]) satisfy our condition.\n\nNote that the decomposition scheme in Theorem 3.1 is formulated such that \\(\\mathbb{P}\\left(R_{T}\\leq B\\right)\\leq\\delta\\) is tight if inequalities (i)--(iv) are tight. Practically, this means that the ramp-up schedules obtained through this approach are typically not overly conservative, unlike approaches that leverage a union-bound for risk decomposition. Finally, Theorem 3.1 holds for a general definition of the cost \\(r_{t}:=r_{t}\\left(\\left(Y_{i,t}\\right)_{i\\in\\mathcal{N}_{t}},\\mathcal{T}_{t}\\right)\\) such that \\(r_{t}=0\\) if \\(\\mathcal{T}_{t}=\\emptyset\\), making it useful in other budgeted online problems beyond our setting.\n\n### Gaussian outcome model\n\nFor this subsection, make the following model assumptions on the outcomes distribution; appendix E provides the extension to general outcome models.\n\n**Definition 3.1** (Conjugate Gaussian outcomes).: Let the unknown model parameters \\(\\mu_{\\text{true}}\\left(0\\right),\\mu_{\\text{true}}\\left(1\\right)\\in\\mathbb{R}\\) satisfy the prior \\(\\mu_{\\text{true}}\\left(w\\right)\\sim N\\Big{(}\\mu_{0}(w),\\sigma_{0}{\\left(w \\right)}^{2}\\Big{)}\\) for \\(w=0,1\\) independently, where \\(\\mu_{0}(0),\\mu_{0}(1)\\in\\mathbb{R},\\sigma_{0}{\\left(0\\right)}^{2},\\sigma_{0}{ \\left(1\\right)}^{2}\\in\\mathbb{R}_{+}\\) are hyperparameters. The experiment outcome of unit \\(i\\) at stage \\(t\\) are distributed independently and identically as\n\n\\[\\left(\\begin{array}{c}Y_{i,t}(0)\\\\ Y_{i,t}(1)\\end{array}\\right)\\overset{\\text{iid}}{\\sim}N\\left(\\left(\\begin{array}[] {c}\\mu_{\\text{true}}\\left(0\\right)\\\\ \\mu_{\\text{true}}\\left(1\\right)\\end{array}\\right),\\left(\\begin{array}{cc} \\sigma{\\left(0\\right)}^{2}&0\\\\ 0&\\sigma{\\left(1\\right)}^{2}\\end{array}\\right)\\right) \\tag{3}\\]\n\nwhere \\(\\sigma{\\left(0\\right)}^{2},\\sigma{\\left(1\\right)}^{2}\\in\\mathbb{R}_{+}\\) are hyperparameters.\n\nThe unknown parameters \\(\\mu_{\\text{true}}(0)\\) and \\(\\mu_{\\text{true}}(1)\\) represent the intrinsic quality of the feature before and after the update, as measured by a specific metric. If \\(\\mu_{\\text{true}}(1)-\\mu_{\\text{true}}(0)<0\\), the feature update is likely to have a negative business impact, _i.e._, \\(Y_{i,t}(1)-Y_{i,t}(0)<0\\).\n\nTo derive the posterior distribution we need the following statistics. For \\(t=1,w=0,1\\)\n\n\\[\\begin{split}& S_{0}^{\\mathcal{C}}(w)=s_{0}^{\\mathcal{C}}(w)=S_{0}^{ \\mathcal{T}}(w)=s_{0}^{\\mathcal{T}}(w)=M_{0}(w)=0\\\\ &\\mu_{p,t=1}(w)=\\mu_{0}(w),\\quad\\sigma_{p,t=1}(w)^{2}=\\sigma_{0}( w)^{2}.\\end{split} \\tag{4}\\]\n\nand for \\(t\\geq 2,w=0,1\\)\n\n\\[\\begin{split}& s_{t}^{\\mathcal{T}}(w):=\\sum\\limits_{i\\in\\mathcal{T} _{t}}Y_{i,t}(w),\\;S_{t}^{\\mathcal{T}}(w):=\\sum\\limits_{r\\in[t]}s_{r}^{\\mathcal{ T}}(w),\\;s_{t}^{\\mathcal{C}}(w):=\\sum\\limits_{i\\in\\mathcal{C}_{t}}Y_{i,t}(w),\\;S_{t}^{ \\mathcal{C}}(w):=\\sum\\limits_{r\\in[t]}s_{r}^{\\mathcal{C}}(w)\\end{split} \\tag{5a}\\] \\[\\mu_{p,t}(w):=\\frac{1}{\\frac{1}{\\sigma_{0}(w)^{2}}+\\frac{M_{t-1}( w)}{\\sigma(w)^{2}}}\\Bigg{(}\\frac{\\mu_{0}(w)}{\\sigma_{0}(w)^{2}}+\\frac{ \\mathbb{I}(w=0)S_{t-1}^{\\mathcal{C}}(w)+\\mathbb{I}(w=1)S_{t-1}^{\\mathcal{T}}(w )}{\\sigma(w)^{2}}\\Bigg{)}\\] (5b) \\[\\sigma_{p,t}(w)^{2}=\\Big{(}\\sigma_{0}(w)^{-2}+M_{t-1}^{(w)}\\sigma(w)^{-2} \\Big{)}^{-1}\\,, \\tag{5c}\\]\n\nwhere \\(M_{t}^{(1)}:=\\sum_{r\\in[t]}m_{r}\\) and \\(M_{t}^{(0)}:=\\sum_{r\\in[t]}N_{r}-m_{r}\\) are the cumulative number of users in the treatment and control groups up to stage \\(t\\), respectively. In (5a), \\(S_{t}^{\\mathcal{T}}(w)\\) and \\(S_{t}^{\\mathcal{C}}(w)\\) represent the cumulative sum of outcomes for \\(w=0,1\\) in the treatment and control groups up to stage \\(t\\), while \\(s_{t}^{\\mathcal{T}}(w)\\) and \\(s_{t}^{\\mathcal{C}}(w)\\) represent the sum of outcomes at stage \\(t\\). In equation (5b), \\(\\mu_{p,t}(w)\\) represents the posterior mean of \\(\\mu_{\\text{true}}\\left(w\\right)\\), while in equation (5c), \\(\\sigma_{p,t}(w)^{2}\\) represents the posterior variance of \\(\\mu_{\\text{true}}\\left(w\\right)\\), for \\(w=0,1\\). When lacking prior information, we suggest using a non-informative priors by setting \\(\\mu_{0}(0)=\\mu_{0}(1)=0\\) and \\(\\sigma_{0}(0)^{2},\\sigma_{0}(1)^{2}\\) sufficiently large.\n\nThe model parameters \\(\\sigma(0)^{2}\\) and \\(\\sigma(1)^{2}\\) at stage \\(t\\geq 2\\) can be estimated using unbiased and consistent estimators. For \\(w=0,1\\) let\n\n\\[\\sigma(w)^{2}\\leftarrow\\frac{\\sum\\limits_{r\\in[t-1]}\\sum\\limits_{i\\in \\mathcal{C}_{r}}\\bigg{(}Y_{i,r}(w)-\\frac{1}{M_{t-1}(w)}\\bigg{(}\\mathbb{I}(w=0) S_{t-1}^{\\mathcal{C}}(w)+\\mathbb{I}(w=1)S_{t-1}^{\\mathcal{T}}(w)\\bigg{)}\\bigg{)}^{2}}{M _{t-1}(w)-1}. \\tag{6}\\]\n\nFor \\(t=1\\), some prior estimate can be used, either from a similar experiment or from a small-scale pretrial run.\n\n### An algorithm for the sample size in an RRC experiment\n\nWe now derive an explicit algorithm from Theorem 3.1 that to outputs \\((m_{t})_{t\\geq 1}\\), the treatment group size at stage \\(t\\), such that, the experiment is \\((\\delta,B)\\)-RRC. Recall that for an experiment to be \\((\\delta,B)\\)-RRC, it suffices that (1), (2) holds for each \\(t\\geq 1\\). Under Definition 3.1, we have that (i) \\((Y_{i,t})_{i,t}\\) are exchangeable random variables (ii) for any \\(t\\geq 2\\), \\(\\mathbb{P}(S_{t-1}^{\\mathcal{T}}(0)<S_{t-1}^{\\mathcal{T}}(1)-B|\\mathcal{F}_{t- 1})>0\\) almost surely for any choice of \\(m_{[t-1]}\\). Combining these observations, we get that (1), (2) hold if for each \\(t\\geq 1\\),\n\n\\[\\mathbb{P}\\bigg{(}s_{t}^{\\mathcal{T}}(1)-S_{t}^{\\mathcal{T}}(0)\\leq b_{t}-S_{t- 1}^{\\mathcal{T}}(1)\\bigg{|}S_{t-1}^{\\mathcal{T}}(0)<S_{t-1}^{\\mathcal{T}}(1)-B,\\mathcal{F}_{t-1}\\bigg{)}\\leq\\Delta_{t}. \\tag{7}\\]\n\nLemma 3.2 provides an upper bound of the left hand size of (7); the proof is in Appendix B.\n\n**Lemma 3.2** (Stochastic domination).: _Assume the outcomes \\((Y_{i,t}(0),Y_{i,t}(1))_{i,t}\\) are generated as in Definition 3.1. For any \\(\\geq 1\\), almost surely,_\n\n\\[\\begin{split}&\\mathbb{P}\\bigg{(}s_{t}^{\\mathcal{T}}(1)-S_{t}^{ \\mathcal{T}}(0)\\leq b_{t}-S_{t-1}^{\\mathcal{T}}(1)\\bigg{|}S_{t-1}^{\\mathcal{T} }(0)<S_{t-1}^{\\mathcal{T}}(1)-B,\\mathcal{F}_{t-1}\\bigg{)}\\\\ &\\qquad\\qquad\\leq\\mathbb{P}\\bigg{(}s_{t}^{\\mathcal{T}}(1)-S_{t}^{ \\mathcal{T}}(0)\\leq b_{t}-S_{t-1}^{\\mathcal{T}}(1)\\bigg{|}\\mathcal{F}_{t-1} \\bigg{)}.\\end{split} \\tag{8}\\]Using Lemma 3.2, for (1) and (2) to hold, it suffices to choose any \\(m_{t}\\in\\mathbb{N}\\) such that\n\n\\[\\mathbb{P}\\bigg{(}s_{t}^{\\mathcal{T}}(1)-S_{t}^{\\mathcal{T}}(0)\\leq b_{t}-S_{t-1 }^{\\mathcal{T}}(1)\\bigg{|}\\mathcal{F}_{t-1}\\bigg{)}\\leq\\Delta_{t} \\tag{9}\\]\n\nand set \\(m_{t}=0\\) if such \\(m_{t}\\) does not exist. From posterior-predictive formulas for the conjugate Gaussian model in Definition 3.1 (see (15e), (15f) in Appendix C), we have \\(\\bigg{[}s_{t}^{\\mathcal{T}}(1)-S_{t}^{\\mathcal{T}}(0)\\bigg{|}\\mathcal{F}_{t-1 }\\bigg{]}\\sim N(\\tilde{\\mu}_{t}(m_{t}),\\tilde{\\sigma}_{t}^{2}\\left(m_{t}\\right))\\), where\n\n\\[\\tilde{\\mu}_{t}\\left(m\\right) :=\\mu_{p,t}(1)\\cdot m-\\mu_{p,t}(0)\\cdot\\Big{(}m+M_{t-1}^{(1)} \\Big{)} \\tag{10a}\\] \\[\\tilde{\\sigma}_{t}^{2}\\left(m\\right) :=m^{2}\\cdot\\sigma_{p,t}(1)^{2}+m\\cdot\\sigma(1)^{2}+\\Big{(}m+M_{t -1}^{(1)}\\Big{)}^{2}\\cdot\\sigma_{p,t}(0)^{2}+\\Big{(}m+M_{t-1}^{(1)}\\Big{)} \\cdot\\sigma(0)^{2}. \\tag{10b}\\]\n\nCombining the above with (9) yields the following Lemma.\n\n**Lemma 3.3**.: _Assume the outcomes \\(\\left(Y_{i,t}(0),Y_{i,t}(1)\\right)_{i,t}\\) are generated as in Definition 3.1. For each \\(t\\geq 1\\), the inequality (9) holds if and only if_\n\n\\[\\frac{b_{t}-S_{t-1}^{\\mathcal{T}}(1)-\\tilde{\\mu}_{t}\\left(m_{t}\\right)}{\\tilde {\\sigma}_{t}\\left(m_{t}\\right)}\\leq q_{t}:=\\Phi^{-1}(\\Delta_{t}) \\tag{11}\\]\n\n_where \\(\\Phi^{-1}\\) denotes inverse CDF of the standard normal distribution._\n\nReplace the inequality in (11) with equality and square both sides gives us the quadratic equation \\(A_{t}\\cdot m_{t}^{2}+B_{t}\\cdot m_{t}+C_{t}=0\\) where\n\n\\[A_{t} :=q_{t}^{2}\\big{(}\\sigma_{p,t}(1)^{2}+\\sigma_{p,t}(0)^{2}\\big{)}- \\left(\\mu_{p,t}(1)-\\mu_{p,t}(0)\\right)^{2} \\tag{12}\\] \\[B_{t} :=q_{t}^{2}\\Big{(}\\sigma(1)^{2}+\\sigma(0)^{2}+2\\sigma_{p,t}(0)^{2 }M_{t-1}^{(1)}\\Big{)}\\] \\[\\qquad\\quad+2\\left(b_{t}-S_{t-1}^{\\mathcal{T}}(1)+\\mu_{p,t}(0)M_ {t-1}^{(1)}\\right)\\left(\\mu_{p,t}(1)-\\mu_{p,t}(0)\\right)\\] \\[C_{t} :=q_{t}^{2}\\sigma_{p,t}(0)^{2}\\left(M_{t-1}^{(1)}\\right)^{2}+q_{t} ^{2}\\sigma(0)^{2}M_{t-1}^{(1)}-\\Big{(}b_{t}-S_{t-1}^{\\mathcal{T}}(1)+\\mu_{p,t }(0)M_{t-1}^{(1)}\\Big{)}^{2}\\,.\\]\n\nAlgorithm 1 finds the floor transform of the solutions of this equation and chooses \\(m_{t}\\) as the largest, positive integer that satisfies (11). If such a solution cannot be found, then either we do not have enough budget or the cost of the experiment is negligible (this accrues when \\(\\mu_{\\mathrm{true}}(1)-\\mu_{\\mathrm{true}}(0)\\gg 0\\)), and the inequality in (11) will be strict for any choice of \\(m_{t}\\). In the former case, Algorithm 1 sets \\(m_{t}=0\\); in the latter case, it sets \\(m_{t}=\\lfloor N_{t}/2\\rfloor\\). Therefore, by construction, the sequence \\((m_{t})_{t\\geq 1}\\) output by Algorithm 1 guarantees that (1) and (2) hold, thereby defining a \\((\\delta,B)\\)-RRC experiment. Note that this approach directly solves for \\(m_{t}\\) from the quadratic equation \\(A_{t}m_{t}^{2}+B_{t}m_{t}+C_{t}=0\\), bypassing the challenging task of estimating tail probabilities for potential choices of \\(m_{t}\\) through Monte-Carlo methods. By Algorithm 1, we can also conduct posterior inference on treatment effect after stage \\(t\\) using \\(\\mu_{p,t+1}(w),\\sigma_{p,t+1}(w),w=0,1\\) and estimate the remaining budget by \\(B-\\sum_{r=1}^{t}m_{r}(\\mu_{p,r+1}(1)-\\mu_{p,r+1}(0))\\).\n\n**Theorem 3.2**.: _Assume the outcomes \\(\\left(Y_{i,t}(0),Y_{i,t}(1)\\right)_{i,t}\\) are generated as in Definition 3.1. The experiment by Algorithm 1 is \\((\\delta,B)\\)-RRC._\n\nEven though our algorithms is derived from the conjugate Gaussian model we have found that it remains effective for broader outcome models. This is because the learning occurs essentially through computation of the first and second moments of past outcomes as in (5), (6), and Algorithm 1 tends to be successful so long as they are predictive of the outcome moments in future stages. The risk of ruin control remains approximately valid due to the law of large numbers and standard central limit theorem under specific conditions; see next section.\n\nFinally, in Algorithm 1, the assumption is made that the population size \\(N_{t}\\) for the next stage is known to ensure that \\(m_{t}\\) does not exceed \\(\\lfloor N_{t}/2\\rfloor\\). In practice, we recommend estimating \\(N_{t}\\) and using the model output \\(m_{t}\\) to calculate the assignment probability \\(p_{t}=m_{t}/N_{t}\\), the allows the experimenter to assign each incoming user to the treatment group with a probability of \\(p_{t}\\).\n\n```\n0:\\(B<0\\), \\(\\delta\\in[0,1)\\)\n1:Initialize\\(t\\gets 1\\), \\(\\prod_{r=1}^{0}\\left(1-\\Delta_{r}\\right)\\gets 1\\)\n2:while\\(\\prod_{r=1}^{t-1}\\left(1-\\Delta_{r}\\right)>1-\\delta\\)do\n3: choose\\(\\Delta_{t}\\in\\left[0,\\frac{1-\\delta}{\\prod_{r=1}^{t-1}\\left(1-\\Delta_{r}\\right)}-1 \\right],b_{t}\\geq B\\)\n4: estimate data variance \\(\\sigma(w)^{2},w=0,1\\) (if unknown) using (6)\n5: compute\\(\\mu_{p,t}(w),\\sigma_{p,t}^{2}(w),w=0,1\\) by (5b), (5c), (4) and \\(q_{t},A_{t},B_{t},C_{t}\\) by (11), (12)\n6:if\\(\\frac{b_{t}-S_{t-1}^{T}(1)-\\tilde{\\mu}_{t}\\left(\\left\\lfloor N_{t}/2\\right\\rfloor \\right)}{\\sqrt{\\sigma_{t}^{2}\\left(\\left\\lfloor N_{t}/2\\right\\rfloor\\right)}} \\leq q_{t}\\)then\\(m_{t}\\leftarrow\\left\\lfloor N_{t}/2\\right\\rfloor\\)\n7:elseif\\(B_{t}^{2}-4A_{t}C_{t}<0\\)then\\(m_{t}\\gets 0\\)\n8:else\n9:\\(\\mathcal{M}_{t}\\leftarrow\\left\\{\\left\\lfloor\\frac{-B_{t}+\\sqrt{B_{t}^{2}-4A_{t }C_{t}}}{2A_{t}}\\right\\rfloor,\\left\\lfloor\\frac{-B_{t}-\\sqrt{B_{t}^{2}-4A_{t}C _{t}}}{2A_{t}}\\right\\rfloor\\right\\}\\)\n10:\\(\\mathcal{V}_{t}\\leftarrow\\left\\{m\\in\\mathcal{M}_{t}\\cap\\left[0,\\frac{N_{t}}{2} \\right]:\\frac{b_{t}-S_{t-1}^{T}(1)-\\tilde{\\mu}_{t}(m)}{\\tilde{\\sigma}_{t}(m)} \\leq q_{t}\\right\\}\\)\n11:if\\(\\mathcal{V}_{t}\\neq\\emptyset\\)then\n12:\\(m_{t}\\leftarrow\\max\\mathcal{V}_{t}\\)\n13:else\n14:\\(m_{t}\\gets 0\\)\n15:endif\n16:endif\n17:Output\\(m_{t}\\), conduct stage \\(t\\)-experiment and observe outcomes \\(s_{t}^{\\mathcal{T}}\\left(1\\right),s_{t}^{\\mathcal{C}}\\left(0\\right)\\)\n18:compute\\(M_{t}(0),M_{t}(1),S_{t}^{\\mathcal{T}}\\left(1\\right),S_{t}^{\\mathcal{C}}\\left(0\\right)\\) by (5)\n19:update\\(t\\gets t+1\\)\n20:endwhile\n```\n\n**Algorithm 1** Output ramp size adaptively\n\n### Robustness to non-identically distributed and non-Gaussian outcomes\n\nWe now derive conditions for the validity of Algorithm 1 under the assumption that experiment outcomes are independent.\n\n**Definition 3.4**.: The experiment outcomes \\(\\left(Y_{i,t}(0),Y_{i,t}(1)\\right)\\) are independent across different units \\(i\\) and experiment stage \\(t\\).\n\nDefinition 3.4 allows \\(Y_{i,t}(0)\\) and \\(Y_{i,t}(1)\\) to be dependent and/or discrete-valued (_e.g._, binary outcomes). In addition, the outcome distribution \\(\\left(Y_{i,t}(0),Y_{i,t}(1)\\right)\\) can differ across \\(i,t\\); for instance, treatment effect may be non-stationary. The validity of Algorithm 1 under Definition 3.4 is now given in Theorem 3.3; we defer the proof to Appendix D.\n\n**Theorem 3.3**.: _Assume the outcomes \\(\\left(Y_{i,t}(0),Y_{i,t}(1)\\right)_{i,t}\\) satisfy Definition 3.4. The experiment by Algorithm 1 is \\(\\left(\\delta,B\\right)\\)-RRC if, for each stage \\(t\\geq 1\\) where \\(m_{t}\\neq 0\\), the following conditions hold_\n\n\\[\\mathbb{P}\\left(\\frac{s_{t}^{\\mathcal{T}}\\left(1\\right)-S_{t}^{\\mathcal{T}} \\left(0\\right)-\\tilde{\\mu}_{t}}{\\tilde{\\sigma}_{t}}\\leq z_{t}\\mid\\mathcal{F}_ {t-1}\\right)\\leq\\Phi\\left(z_{t}\\right),\\quad z_{t}\\leq\\frac{\\tilde{\\mu}_{t}- \\tilde{\\mu}_{t}}{\\tilde{\\sigma}_{t}-\\tilde{\\sigma}_{t}}, \\tag{13}\\]\n\n_where_\n\n\\[\\tilde{\\mu}_{t}:=\\mathbb{E}\\left[s_{t}^{\\mathcal{T}}\\left(1\\right)-S_{t}^{ \\mathcal{T}}\\left(0\\right)\\mid\\mathcal{F}_{t-1}\\right],\\;\\tilde{\\sigma}_{t}^{2 }:=\\mathbb{V}\\left[s_{t}^{\\mathcal{T}}\\left(1\\right)-S_{t}^{\\mathcal{T}}\\left( 0\\right)\\mid\\mathcal{F}_{t-1}\\right],\\;z_{t}:=\\frac{b_{t}-S_{t-1}^{\\mathcal{T} }\\left(1\\right)-\\tilde{\\mu}_{t}}{\\tilde{\\sigma}_{t}},\\]\n\n_and \\(\\tilde{\\mu}_{t}:=\\tilde{\\mu}_{t}(m_{t}),\\tilde{\\sigma}_{t}:=\\tilde{\\sigma}_{t} (m_{t})\\) are defined by (10a), (10b), and \\(s_{t}^{\\mathcal{T}}\\left(1\\right),S_{t}^{\\mathcal{T}}\\left(0\\right)\\) are defined in (5)._\n\nWe expect the first condition in (13) to hold as a consequence of central limit theorem for independent but non-identical random variables. Suppose \\(\\Delta_{t}\\leq 0.5,\\forall t\\) (_i.e., \\(z_{t}\\leq 0\\)_). By law of large number for independent but non-identical random variables, the second condition in (_13_) holds if (i) we have chosen prior and model parameters conservatively such that\n\n\\[\\mu_{0}(1)-\\mu_{0}(0)\\leq\\frac{1}{m_{t}}\\sum_{i\\in\\mathcal{T}_{1} }\\mathbb{E}\\left(Y_{i,1}(1)-Y_{i,1}(0)\\right)\\] \\[\\sigma(0)^{2}+\\sigma(1)^{2}+m_{t}\\cdot\\left(\\sigma_{0}(1)^{2}+ \\sigma_{0}(0)^{2}\\right)\\geq\\frac{1}{m_{t}}\\sum_{i\\in\\mathcal{T}_{t}}\\mathbb{ V}\\left(Y_{i,t}(1)-Y_{i,t}(0)\\right)\\]and (ii) if the treatment effects increase or stay roughly constant throughout the experiments\n\n\\[\\frac{1}{m_{t}}\\sum_{i\\in\\mathcal{T}_{t}}\\mathbb{E}\\left(Y_{i,t}(1)-Y_{i,t}(0) \\right)\\geq\\frac{1}{M_{t-1}^{(1)}}\\sum_{r\\in[t-1]}\\sum_{i\\in\\mathcal{T}_{r}} \\mathbb{E}\\left[Y_{i,t}(1)-Y_{i,t}(0)\\right]\\]\n\nand our variance estimates \\(\\sigma{(0)}^{2},\\sigma{(1)}^{2}\\) are accurate or conservative in the sense that\n\n\\[\\sigma{(0)}^{2}\\geq\\frac{1}{M_{t-1}^{(1)}}\\sum_{r\\in[t-1]}\\sum_{i\\in\\mathcal{T }_{r}}\\mathbb{V}\\left[Y_{i,t}(0)\\right],\\ \\ \\sigma{(0)}^{2}+\\sigma{(1)}^{2}\\geq \\frac{1}{m_{t}}\\sum_{i\\in\\mathcal{T}_{t}}\\mathbb{V}\\left(Y_{i,t}(1)-Y_{i,t}(0 )\\right).\\]\n\nIn summary, under Definition 3.4, the validity of Algorithm 1 depends on the accuracy and conservatism of the model's estimates based on past stages for the true treatment effect and volatility in the next stage. The algorithm's effectiveness may be compromised when there is a sudden decrease in treatment effect or a surge in outcome volatility in the next stage; see discussion in Appendix D.\n\n## 4 Numerical and empirical experiments\n\nSimulated ramp scheduleWe now examine the following three experimental scenarios, for each \\(\\left(Y_{i,t}(1),Y_{i,t}(0)\\right)_{i,t}\\) are iid sampled from (3) with variance \\(\\sigma{(1)}^{2}=\\sigma{(0)}^{2}=10\\) and means given below:\n\n* PTE: Positive treatment effect, with \\(\\mu_{\\text{true}}(0)=0\\), \\(\\mu_{\\text{true}}(1)=1\\);\n* NTE: Negative treatment effect, with \\(\\mu_{\\text{true}}(0)=1\\) and \\(\\mu_{\\text{true}}(1)=0\\);\n* PNTE: Negative to positive treatment effect, with \\(\\mu_{\\text{true}}(1)(t)=\\min\\left(-2+0.5(t-1),2\\right)\\).\n\nFor each scenario, we set \\(T=10\\) with \\(N_{t}=500,\\forall t\\) and we choose non-informative prior \\(\\mu_{0}(w)=0,\\sigma_{0}(w)^{2}=100,w=0,1\\). We assume model variance is known; however, using (6) to estimate the variance gives similar results. We repeat each scenario 500 times. Figure 1 (a)--(c) show median, 25% and 75% quantile of the simulated ramp schedules \\((m_{t})_{t\\in[T]}\\) and (g),(h) show the budget surpluses \\(\\sum_{r\\in[t]}\\sum_{i\\in\\mathcal{T}_{r}}\\left(Y_{i,t}(1)-Y_{i,t}(0)\\right)-B\\) produced given different choices of \\(B,\\delta,(b_{t})_{t\\in[T]},(\\Delta)_{t\\in[T]}\\).\n\nAcross the various scenarios, our model gives a reasonable ramp schedule. Large \\(B,\\delta\\) typically leads to more treated units and faster ramp-up. For the NPTE scenario, inadequate budget and low ruin tolerance can result in a failure to ramp up to 50% (\\(m_{t}=250\\)). We also found that reserving the budget for later stages by decreasing \\(b_{t}\\) or \\(\\Delta_{t}\\) in the initial stages leads to a faster ramp-up because more budget is available to support a swift increase when the treatment effect turns positive. This suggests that the experimenter may want to consider reserving some budget for later stages if the treatment effect \\(\\mu_{p,t}(1)-\\mu_{p,t}(0)\\) has not stabilized.\n\nFor PNTE scenario, we compare our method to a Thompson-sampling bandit with tuning parameters \\(c\\) and prior \\(\\mu_{0}(1)=-2,\\mu_{0}(0)=0,\\sigma_{0}{(0)}^{2}=\\sigma_{0}{(1)}^{2}=0.05\\) (see [27] and Appendix G for details). The prior is chosen so that the bandit can initialize conservatively depending on \\(c\\). It can be seen in Figure 1 (e),(i) that the ramp schedule generated is rather sub-optimal and does not respect the budget. It also follows a rigid pattern where with small \\(c\\), the ramp-up initializes too aggressively, and for large \\(c\\), the ramp-up proceeds too conservatively. These results demonstrate that our approach significantly outperforms the main existing alternative.\n\nSemi-real LinkedIn ramp schedule comparisonAppendix F gives group-level statistics from a 6-stage phased release run at LinkedIn. Due to privacy constraints, the individual-level data is not available and is simulated from (4) using stage-wise \\(\\mu_{\\text{true}}\\left(w\\right),\\sigma{(w)}^{2},w=0,1\\) (both unobserved). The ramp-up schedules for different tuning parameters are shown in Figure 1,(d). It is noteworthy that the ramp-up schedule employed by LinkedIn's data scientists, which was chosen without considering a specific budget, is roughly consistent with the budget-rationing schedule denoted as \"ration-budget LinkedIn\" in the caption. Our results suggest that deducing the budget and risk tolerance associated with an experiment retroactively using our method is possible. We also run the experiment using Thompson sampling Bayesian bandit with the same prior as for NPTE above. In Figure 1(f), we again observe the rigidity issue: with small \\(c\\), the ramp-up initializes too aggressively, and for large \\(c\\), the ramp-up proceeds too conservatively.\n\nFigure 1: Line plots (a)—(i) show the median, 25%, 75% quantiles of either the treatment group sizes or the budget surplus for the 500 simulations of the different experiment setups (PTE, NTE, NPTE, Linkedln) using our model and Thompson-sampling Bayesian bandit. Under the legends “\\((B,\\delta)\\)”, we set \\(b_{t}=B,\\Delta_{t}=1-(1-\\delta)^{1/T},\\forall t\\). We also use (i) “ration budget” to denote \\((B,\\delta)=(-500,0.01),b_{t}=-400,\\forall t\\leq 5,b_{t}=-500,\\forall t>5\\) and \\(\\Delta_{t}=1-(1-\\delta)^{1/T},\\forall t\\); (ii) “ration tolerance” to denote \\((B,\\delta)=(-500,0.01),b_{t}=-500,\\forall t\\) and \\(\\Delta_{t}=0.0001,\\forall t\\leq 5,\\Delta_{t}=0.0019,\\forall t>5\\) (iii) “actual Linkedln” to denote the actual ramp up schedule used by Linkedln data scientists (iv) “ration-budget Linkedln” to denote \\((B,\\delta)=(-1500,0.01),b_{t}=-400,t\\geq 4,b_{t}=-1500,t>4,\\Delta_{t}=1-(1- \\delta)^{1/T}\\). Particularly, (e), (f), (i) are results using Thompson-sampling Bayesian bandit with different values of tuning parameter \\(c\\), denoted by “TOM” in sub-caption, for experiment NPTE, Linkedln. We set \\(B=-500\\) to produce (h), although the model is not budget-aware. Histograms (j)—(n) show distribution of the budget used over 5000 simulations. The red dashed line marks the budget available \\(B=-500\\). “\\(x\\)% \\(/\\)\\(5\\%\\)” in the sub-captions denotes that the actual risk of ruin is \\(x\\%\\) and the ruin tolerance is \\(\\delta=5\\%\\).\n\nBudget-spent distributionTo explore how our algorithms controls the risk of ruin and budget spending, we simulate following experiments for 5,000 times and plot distribution of the budget spent \\(\\sum_{r\\in[t]}\\sum_{i\\in\\mathcal{T}_{r}}\\left(Y_{i,t}(1)-Y_{i,t}(0)\\right)\\) in Figure 1 (j)--(n): \\(\\left(Y_{i,t}(1),Y_{i,t}(0)\\right)_{i,t}\\) are sampled iid from (3) with \\(\\mu_{\\text{true}}\\left(0\\right)=0,\\mu_{\\text{true}}\\left(1\\right)=1,\\sigma(0) ^{2}=\\sigma(1)^{2}=10\\); (ii) corr: same as norm except that for each \\(i,t\\), \\(Y_{i,t}(1)\\) is correlated with \\(Y_{i,t}(0)\\) with correlation coefficient 0.8 (iii) bern: \\(Y_{i,t}(0)\\overset{\\text{iid}}{\\sim}6.4\\mathsf{Bern}(p=0.5786)\\) and \\(Y_{i,t}(1)\\overset{\\text{iid}}{\\sim}6.4\\mathsf{Bern}(p=0.4224)\\); (iv) fat: \\(Y_{i,t}(0)\\overset{\\text{iid}}{\\sim}1+\\sqrt{5}\\mathsf{t}_{4}\\), and \\(Y_{i,t}(1)\\overset{\\text{iid}}{\\sim}\\sqrt{5}\\mathsf{t}_{4}\\)2 (v) detc: same as norm except \\(\\mu_{\\text{true}}\\left(1\\right)(t)=-(t-1)\\). Note that (iii), (iv) is configured so that \\(\\mathbb{E}[Y_{i,t}(1)-Y_{i,t}(0)]=1,\\mathbb{V}\\left(Y_{i,t}(0)\\right)=\\mathbb{ V}\\left(Y_{i,t}(1)\\right)=10\\). For all the above experiments, we run \\(T=10\\) stages with \\(N_{t}=500,\\forall t\\) and we use non-informative prior \\(\\mu_{0}(w)=0,\\sigma_{0}(w)^{2}=100,w=0,1\\).\n\nFootnote 2: Where \\(\\mathsf{t}_{4}\\) is a Student-t distribution with 4 degrees of freedom.\n\nAs shown in Figure 1 (j)--(m) the model successfully controls risk of ruin for (i)--(iv). The actual ruin risk is at a reasonable level (\\(\\sim 1.2\\)%) compared to the ruin tolerance given (\\(5\\)%). Note that the actual ruin risk are close for different outcome distribution. This is a consequence of central limit theorem and law of large numbers as discussed in Section 3.4. The model fails to control risk of ruin for (v) as expected since the treatment effect keeps decreasing and the model assigns treatment based on past stages which leads to higher-than-expected costs (cf. Section 3.4).\n\n## References\n\n* [1] Apple. App store: Release a version update in phases, 2023.\n* [2] Maryam Aziz, Emilie Kaufmann, and Marie-Karelle Riviere. On multi-armed bandit designs for dose-finding clinical trials. _The Journal of Machine Learning Research_, 22(1):686-723, 2021.\n* [3] Eytan Bakshy, Dean Eckles, and Michael S Bernstein. Designing and deploying online field experiments. In _Proceedings of the 23rd international conference on World wide web_, pages 283-292, 2014.\n* [4] Donald A Berry. Modified two-armed bandit strategies for certain clinical trials. _Journal of the American Statistical Association_, 73(362):339-345, 1978.\n* [5] Iavor Bojinov and Somit Gupta. Online experimentation: Benefits, operational and methodological challenges, and scaling guide. _Harvard Data Science Review_, 4(3), 2022.\n* [6] Iavor Bojinov and Karim R. Lakhani. Experimentation at yelp. _Harvard Business School Case 621-064_, 2020.\n* [7] Iavor Bojinov and Neil Shephard. Time series experiments and causal estimands: Exact randomization tests and trading. _Journal of the American Statistical Association_, page Forthcoming, 2019.\n* [8] Asaf Cassel, Shie Mannor, and Assaf Zeevi. A general approach to multi-armed bandits under risk criteria. In _Conference on learning theory_, pages 1295-1306. PMLR, 2018.\n* [9] Semih Cayci, Atilla Eryilmaz, and Rayadurgam Srikant. Budget-constrained bandits over general cost and reward distributions. In _International Conference on Artificial Intelligence and Statistics_, pages 4388-4398. PMLR, 2020.\n* [10] Joel QL Chang and Vincent YF Tan. A unifying theory of thompson sampling for continuous risk-averse bandits. In _Proceedings of the AAAI Conference on Artificial Intelligence_, volume 36, pages 6159-6166, 2022.\n* [11] David. R. Cox. _Planning of Experiments_. John Wiley & Sons, New York, 1958.\n* [12] Debojit Das, Shweta Jain, and Sujit Gujar. Budgeted combinatorial multi-armed bandits. _arXiv preprint arXiv:2202.03704_, 2022.\n* [13] WF Eberlein. On euler's infinite product for the sine. _Journal of Mathematical Analysis and Applications_, 58(1):147-151, 1977.\n* [14] Nicolas Galichet, Michele Sebag, and Olivier Teytaud. Exploration vs exploitation vs safety: Risk-aware multi-armed bandits. In _Asian Conference on Machine Learning_, pages 245-260. PMLR, 2013.\n* [15] Andrew Gelman, John B Carlin, Hal S Stern, and Donald B Rubin. _Bayesian data analysis_. Chapman and Hall/CRC, 1995.\n* [16] Google. Play console help: Release app updates with staged rollouts, 2023.\n* [17] Jez Humble and David Farley. _Continuous delivery: reliable software releases through build, test, and deployment automation_. Pearson Education, 2010.\n* [18] Ron Kohavi, Alex Deng, Brian Frasca, Toby Walker, Ya Xu, and Nils Pohlmann. Online controlled experiments at large scale. In _Proceedings of the 19th ACM SIGKDD international conference on Knowledge discovery and data mining_, pages 1168-1176, 2013.\n* [19] Odalric-Ambrym Maillard. Robust risk-averse stochastic multi-armed bandits. In _Algorithmic Learning Theory: 24th International Conference, ALT 2013, Singapore, October 6-9, 2013. Proceedings 24_, pages 218-233. Springer, 2013.\n\n* [20] Jialiang Mao and Iavor Bojinov. Quantifying the value of iterative experimentation. _arXiv preprint arXiv:2111.02334_, 2021.\n* [21] Vianney Perchet, Philippe Rigollet, Sylvain Chassang, and Erik Snowberg. Batched bandit problems. _The Annals of Statistics_, pages 660-681, 2016.\n* [22] Donald B Rubin. Estimating causal effects of treatments in randomized and nonrandomized studies. _Journal of educational Psychology_, 66(5):688, 1974.\n* [23] Amir Sani, Alessandro Lazaric, and Remi Munos. Risk-aversion in multi-armed bandits. _Advances in neural information processing systems_, 25, 2012.\n* [24] Sama Shrestha and Sonia Jain. A bayesian-bandit adaptive design for n-of-1 clinical trials. _Statistics in Medicine_, 40(7):1825-1844, 2021.\n* [25] Jerzy Splawa-Neyman, Dorota M Dabrowska, and TP Speed. On the application of probability theory to agricultural experiments. essay on principles. section 9. _Statistical Science_, pages 465-472, 1990.\n* [26] Diane Tang, Ashish Agarwal, Deirdre O'Brien, and Mike Meyer. Overlapping experiment infrastructure: More, better, faster experimentation. In _Proceedings of the 16th ACM SIGKDD international conference on Knowledge discovery and data mining_, pages 17-26, 2010.\n* [27] Peter F Thall and J Kyle Wathen. Practical bayesian adaptive randomisation in clinical trials. _European Journal of Cancer_, 43(5):859-866, 2007.\n* [28] Sattar Vakili and Qing Zhao. Risk-averse multi-armed bandit problems under mean-variance measure. _IEEE Journal of Selected Topics in Signal Processing_, 10(6):1093-1111, 2016.\n* [29] Ryo Watanabe, Junpei Komiyama, Atsuyoshi Nakamura, and Mineichi Kudo. Kl-ucb-based policy for budgeted multi-armed bandits with stochastic action costs. _IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences_, 100(11):2470-2486, 2017.\n* [30] Tong Xia, Sumit Bhardwaj, Pavel Dmitriev, and Aleksander Fabijan. Safe velocity: a practical guide to software deployment at scale using controlled rollout. In _2019 IEEE/ACM 41st International Conference on Software Engineering: Software Engineering in Practice (ICSE-SEIP)_, pages 11-20. IEEE, 2019.\n* [31] Yingce Xia, Wenkui Ding, Xu-Dong Zhang, Nenghai Yu, and Tao Qin. Budgeted bandit problems with continuous random costs. In _Asian conference on machine learning_, pages 317-332. PMLR, 2016.\n* [32] Yingce Xia, Haifang Li, Tao Qin, Nenghai Yu, and Tie-Yan Liu. Thompson sampling for budgeted multi-armed bandits. _arXiv preprint arXiv:1505.00146_, 2015.\n* [33] Ya Xu, Nanyu Chen, Addrian Fernandez, Omar Sinno, and Anmol Bhasin. From infrastructure to culture: A/b testing challenges in large scale social networks. In _Proceedings of the 21th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining_, pages 2227-2236, 2015.\n* [34] Ya Xu, Weitao Duan, and Shaochen Huang. Sqr: balancing speed, quality and risk in online experiments. In _Proceedings of the 24th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining_, pages 895-904, 2018.\n* [35] Qiuyu Zhu and Vincent Tan. Thompson sampling algorithms for mean-variance bandits. In _International Conference on Machine Learning_, pages 11599-11608. PMLR, 2020.\n\nDecompose risk-of-ruin to individual stages\n\nProof of Theorem 3.1.: The last claim is trivial: it is easy to verify that (1) and (2) hold if we let \\(\\mathcal{T}_{t}=\\emptyset\\) for all \\(t\\in[T]\\). Now we prove the first and the second claim. The case for \\(T=1\\) is trivial. We assume \\(T\\geq 2\\). For any \\(t=2,\\ldots,T\\), we have that\n\n\\[\\mathbb{P}\\left(R_{t}\\leq B\\right) =\\mathbb{E}[\\mathbb{P}\\left(R_{t}\\leq B\\mid\\mathcal{F}_{t-1} \\right)]\\] \\[=\\mathbb{E}\\bigg{[}\\mathbb{P}\\left(R_{t}\\leq B,R_{t-1}\\leq B\\mid \\mathcal{F}_{t-1}\\right)+\\mathbb{P}\\left(R_{t}\\leq B,R_{t-1}>B\\mid\\mathcal{F} _{t-1}\\right)\\bigg{]}\\] \\[\\stackrel{{(a)}}{{\\leq}}\\mathbb{P}\\left(R_{t-1}\\leq B \\right)+\\mathbb{E}\\left[\\mathbb{P}\\left(R_{t}\\leq B,R_{t-1}>B\\mid\\mathcal{F}_ {t-1}\\right)\\right]\\] \\[=\\mathbb{P}\\left(R_{t-1}\\leq B\\right)+\\mathbb{E}\\bigg{[}\\mathbb{ I}\\left(\\mathbb{P}\\left(R_{t-1}>B\\mid\\mathcal{F}_{t-1}\\right)>0\\right)\\] \\[\\qquad\\qquad\\times\\mathbb{P}\\left(R_{t}\\leq B\\mid R_{t-1}>B, \\mathcal{F}_{t-1}\\right)\\mathbb{P}\\left(R_{t-1}>B\\mid\\mathcal{F}_{t-1}\\right)\\] \\[\\qquad\\qquad+\\mathbb{I}\\left(\\mathbb{P}\\left(R_{t-1}>B\\mid\\mathcal{ F}_{t-1}\\right)=0\\right)\\mathbb{P}\\left(R_{t}\\leq B,R_{t-1}>B\\mid\\mathcal{F}_{t-1} \\right)\\bigg{]}\\] \\[\\stackrel{{(b)}}{{=}}\\mathbb{P}\\left(R_{t-1}\\leq B \\right)+\\mathbb{E}\\bigg{[}\\mathbb{I}\\left(\\mathbb{P}(R_{t-1}>B\\mid\\mathcal{F}_ {t-1}>0)\\right)\\] \\[\\qquad\\qquad\\times\\mathbb{P}\\left(R_{t}\\leq B\\mid R_{t-1}>B, \\mathcal{F}_{t-1}\\right)\\mathbb{P}\\left(R_{t-1}>B\\mid\\mathcal{F}_{t-1}\\right) \\bigg{]}\\] \\[\\stackrel{{(c)}}{{\\leq}}\\mathbb{P}\\left(R_{t-1}\\leq B \\right)+\\Delta_{t}\\mathbb{E}\\bigg{[}\\mathbb{I}\\left(\\mathbb{P}\\left(R_{t-1}>B \\mid\\mathcal{F}_{t-1}\\right)>0\\right)\\mathbb{P}\\left(R_{t-1}>B\\mid\\mathcal{F} _{t-1}\\right)\\bigg{]}\\] \\[=\\mathbb{P}\\left(R_{t-1}\\leq B\\right)+\\Delta_{t}\\mathbb{P}\\left(R _{t-1}>B\\right)\\]\n\nwhere \\((b)\\) used that \\(\\mathbb{P}\\left(R_{t-1}>B\\mid\\mathcal{F}_{t-1}\\right)=0\\Rightarrow\\mathcal{T} _{t}=\\emptyset\\) and that \\(r_{t}((Y_{i,t})_{i\\in\\mathcal{N}_{t}},\\emptyset)=0\\), which implies that almost surely\n\n\\[\\mathbb{I}\\left(\\mathbb{P}\\left(R_{t-1}>B\\mid\\mathcal{F}_{t-1} \\right)=0\\right)\\cdot\\mathbb{P}\\left(R_{t}\\leq B,R_{t-1}>B\\mid\\mathcal{F}_{t- 1}\\right)\\] \\[=\\mathbb{I}\\left(\\mathbb{P}\\left(R_{t-1}>B\\mid\\mathcal{F}_{t-1} \\right)=0\\right)\\mathbb{P}\\left(R_{t-1}+r_{t}\\left(\\left(Y_{i,t}\\right)_{i \\in\\mathcal{N}},\\emptyset\\right)\\leq B,R_{t-1}>B\\mid\\mathcal{F}_{t-1}\\right)\\] \\[=\\mathbb{I}\\left(\\mathbb{P}\\left(R_{t-1}>B\\mid\\mathcal{F}_{t-1} \\right)=0\\right)\\cdot\\mathbb{P}\\left(R_{t-1}\\leq B,R_{t-1}>B\\mid\\mathcal{F}_{t- 1}\\right)\\] \\[=0\\]\n\nand \\((c)\\) used that \\(b_{t}\\geq B\\), which implies that almost surely\n\n\\[\\mathbb{P}\\left(R_{t}\\leq B\\mid R_{t-1}>B,\\mathcal{F}_{t}\\right)\\leq\\mathbb{P} \\left(R_{t}\\leq b_{t}\\mid R_{t-1}>B,\\mathcal{F}_{t}\\right)\\leq\\Delta_{t}.\\]\n\nRearranging this, we obtain a recurrence relation: for any \\(t=2,...,T\\),\n\n\\[\\mathbb{P}\\left(R_{t}>B\\right)\\geq\\left(1-\\Delta_{t}\\right)\\cdot\\mathbb{P} \\left(R_{t-1}>B\\right). \\tag{14}\\]\n\nUsing the recurrence relation repeatedly for all \\(t\\in[T]\\), we obtain\n\n\\[\\mathbb{P}\\left(R_{T}>B\\right)\\geq\\prod_{i=2}^{T}\\left(1-\\Delta_{ t}\\right)\\cdot\\mathbb{P}\\left(R_{1}>b_{1}\\right)\\geq\\prod_{t=1}^{T}\\left(1- \\Delta_{t}\\right)\\] \\[\\implies\\mathbb{P}\\left(R_{T}\\leq B\\right)\\leq 1-\\prod_{t=1}^{T} \\left(1-\\Delta_{t}\\right)\\leq\\delta\\]\n\nas required. To prove the second claim, observe that equality is attained in all of the above inequalities if equality is attained in (14), \\((i),(ii)\\) and \\((iii)\\), and that equality is attained in (14) if equality is attained in \\((a)\\) and \\((c)\\). Finally, note that equality in \\((a)\\) is attained if \\(r_{t}\\leq 0,\\forall t\\in[T]\\) and equality in \\((c)\\) is attained if equality is attained in \\((i)\\) and \\((iv)\\). \n\n## Appendix B Stochastic domination\n\n**Lemma B.1** (Stochastic domination under truncation).: _For any two independent real random variable \\(X,Z\\) and real number \\(a,t\\in\\mathbb{R}\\) such that \\(\\mathbb{P}(X<a)>0\\), we have that_\n\n\\[\\mathbb{P}(X+Z\\geq t\\mid X<a)\\leq\\mathbb{P}(X+Z\\geq t).\\]Proof of Lemma b.1.: Assume that \\(\\mathbb{P}(X\\geq a)>0\\), or else the proof is trivial. We first claim that \\(\\mathbb{P}(X+Z\\geq t\\mid X<a)\\leq\\mathbb{P}(X+Z\\geq t\\mid X\\geq a)\\). Note that this holds if and only if\n\n\\[\\frac{\\mathbb{P}(X\\geq t-Z,X<a)}{\\mathbb{P}(X<a)}\\leq\\frac{\\mathbb{P}(X\\geq t-Z, X\\geq a)}{\\mathbb{P}(X\\geq a)}.\\]\n\nThe above holds since its lhs and rhs satisfies\n\n\\[\\frac{\\mathbb{P}(X\\geq t-Z,X<a)}{\\mathbb{P}(X<a)}=\\frac{\\mathbb{P }(X\\geq t-Z,X<a,a\\geq t-Z)}{\\mathbb{P}(X<a)}\\leq\\mathbb{P}(a\\geq t-Z)\\] \\[\\frac{\\mathbb{P}(X\\geq t-Z,X\\geq a)}{\\mathbb{P}(X\\geq a)}=\\frac{ \\mathbb{P}(X\\geq t-Z,X\\geq a,a<t-Z)}{\\mathbb{P}(X\\geq a)}+\\mathbb{P}(a\\geq t-Z)\\]\n\nIt then follows from law of total probability that\n\n\\[\\mathbb{P}(X+Z\\geq t) =\\mathbb{P}(X+Z\\geq t\\mid X<a)\\mathbb{P}(X<a)+\\mathbb{P}(X+Z\\geq t \\mid X\\geq a)\\mathbb{P}(X\\geq a)\\] \\[\\geq\\mathbb{P}(X+Z\\geq t\\mid X<a)\\mathbb{P}(X<a)+\\mathbb{P}(X+Z \\geq t\\mid X<a)\\mathbb{P}(X\\geq a)\\] \\[=\\mathbb{P}(X+Z\\geq t\\mid X<a)\\]\n\nas required. \n\nProof of Lemma 3.2.: If \\(M_{t-1}^{(1)}=0\\), (8) holds with equality since \\(S_{t-1}^{\\mathcal{T}}(0)<S_{t-1}^{\\mathcal{T}}(1)-B\\iff B<0\\). So, assume \\(M_{t-1}^{(1)}>0\\) from now on. By (15f) and the conditional distributions of multivariate Gaussian, we have\n\n\\[\\left[s_{t}^{\\mathcal{T}}(0)\\bigg{|}S_{t-1}^{\\mathcal{T}}(0),\\mathcal{F}_{t-1} \\right]= \\bigg{[}\\mu_{2}+V_{21}V_{11}^{-1}(S_{t-1}^{\\mathcal{T}}(0)-\\mu_{ 1})+(V_{22}-V_{21}V_{11}^{-1}V_{12})^{1/2}Z\\bigg{|}S_{t-1}^{\\mathcal{T}}(0), \\mathcal{F}_{t-1}\\bigg{]}\\]\n\nwhere \\(Z\\sim N(0,1)\\) is independent of \\(S_{t-1}^{\\mathcal{T}}(0)\\) conditioned on \\(\\mathcal{F}_{t-1}\\) and \\(\\mu,V\\) are defined in (15f). Here, we used that \\(V_{11}>0\\) since \\(\\sigma_{p,t}(0)^{2},\\sigma(0)^{2}>0\\) by Definition 3.1, and \\(M_{t-1}^{(1)}\\neq 0\\). Using the above and that \\(S_{t}^{\\mathcal{T}}(0)=s_{t}^{\\mathcal{T}}(0)+S_{t-1}^{\\mathcal{T}}(0)\\), we have\n\n\\[\\left[S_{t}^{\\mathcal{T}}(0)\\bigg{|}S_{t-1}^{\\mathcal{T}}(0), \\mathcal{F}_{t-1}\\right]= \\bigg{[}(V_{21}V_{11}^{-1}+1)S_{t-1}^{\\mathcal{T}}(0)+\\mu_{2}-V_{ 21}V_{11}^{-1}\\mu_{1}\\] \\[+(V_{22}-V_{21}V_{11}^{-1}V_{12})^{1/2}Z\\bigg{|}S_{t-1}^{ \\mathcal{T}}(0),\\mathcal{F}_{t-1}\\bigg{]}.\\]\n\nSince \\(V_{21}V_{11}^{-1}+1>0\\) in the above, using also that \\(b_{t}-S_{t-1}^{\\mathcal{T}}(1),S_{t-1}^{\\mathcal{T}}(1)-B\\in\\mathcal{F}_{t-1}\\) and that \\(s_{t}^{\\mathcal{T}}(1)\\) is independent of \\(S_{t-1}^{\\mathcal{T}}(0),S_{t}^{\\mathcal{T}}(0)\\), (8) follows from Lemma B.1. \n\n## Appendix C Derivation of the decision rule\n\nProof of these facts follows from standard Bayesian analysis (see e.g. [15])\n\n**Lemma C.1** (Posterior distributions).: _We have for \\(w=0,1,t\\in[T]\\)_\n\n\\[\\mu_{p,t}(1) :=\\mathbb{E}\\bigg{[}\\mu_{\\mathrm{true}}(1)\\bigg{|}\\mathcal{F}_{t- 1}\\bigg{]}=\\frac{1}{\\frac{1}{\\sigma_{0}(1)^{2}}+\\frac{M_{t-1}^{(1)}}{\\sigma(0) ^{2}}}\\left(\\frac{\\mu_{0}(1)}{\\sigma_{0}(1)^{2}}+\\frac{S_{t-1}^{\\mathcal{T}}( 1)}{\\sigma(1)^{2}}\\right) \\tag{15a}\\] \\[\\mu_{p,t}(0) :=\\mathbb{E}\\bigg{[}\\mu_{\\mathrm{true}}(0)\\bigg{|}\\mathcal{F}_{t- 1}\\bigg{]}=\\frac{1}{\\frac{1}{\\sigma_{0}(0)^{2}}+\\frac{M_{t-1}^{(0)}}{\\sigma(0) ^{2}}}\\left(\\frac{\\mu_{0}(0)}{\\sigma_{0}(0)^{2}}+\\frac{S_{t-1}^{\\mathcal{C}}( 0)}{\\sigma(0)^{2}}\\right)\\] (15b) \\[\\sigma_{p,t}(w)^{2}:=\\mathbb{V}\\bigg{[}\\mu_{\\mathrm{true}}(w) \\bigg{|}\\mathcal{F}_{t-1}\\bigg{]}=\\left(\\frac{1}{\\sigma_{0}(w)^{2}}+\\frac{M_{t- 1}(w)}{\\sigma(w)^{2}}\\right)^{-1}\\] (15c) \\[\\bigg{[}\\mu_{\\mathrm{true}}(w)\\bigg{|}\\mathcal{F}_{t-1}\\bigg{]} \\sim N\\big{(}\\mu_{p,t}(w),\\;\\sigma_{p,t}(w)^{2}\\big{)}\\] (15d) \\[\\bigg{[}s_{t}^{\\mathcal{T}}(1)\\bigg{|}\\mathcal{F}_{t-1}\\bigg{]} \\sim N\\left(\\mu_{p,t}(1)\\cdot m_{t},\\;m_{t}^{2}\\cdot\\sigma_{p,t}(1)^{2}+m_{t} \\cdot\\sigma(0)^{2}\\right)\\] (15e) \\[\\bigg{[}\\binom{S_{t-1}^{\\mathcal{T}}(0)}{s_{t}^{\\mathcal{T}}(0)} \\bigg{]}\\bigg{|}\\mathcal{F}_{t-1}\\bigg{]}\\sim N\\left(\\mu,V\\right) \\tag{15f}\\]_where_\n\n\\[\\mu :=\\left(\\begin{array}{cc}\\mu_{p,t}(0)\\cdot M_{t-1}^{(1)}\\\\ \\mu_{p,t}(0)\\cdot m_{t}\\end{array}\\right),\\] \\[V :=\\left(\\begin{array}{cc}(M_{t-1}^{(1)})^{2}\\sigma_{p,t}(0)^{2}+M _{t-1}^{(1)}\\sigma(0)^{2}&M_{t-1}^{(1)}m_{t}\\sigma_{p,t}(0)^{2}\\\\ M_{t-1}^{(1)}m_{t}\\sigma_{p,t}(0)^{2}&m_{t}^{2}\\sigma_{p,t}(0)^{2}+m_{t}\\sigma (0)^{2}\\end{array}\\right).\\]\n\n## Appendix D Robustness to non-identically distributed and non-Gaussian outcomes\n\nProof of Theorem 3.3.: To show the experiment by Algorithm 1 is \\((\\delta,B)\\)-RRC under Definition 3.4, it suffices to show that (1), (2) hold for each \\(t\\geq 1\\). Since (1), (2) hold for each \\(t\\geq 1\\) if \\(m_{t}=0\\), we only need to show that for each \\(t\\geq 1\\), if \\(m_{t}\\neq 0\\), almost surely\n\n\\[\\mathbb{P}\\left(S_{t}^{\\mathcal{T}}(1)-S_{t}^{\\mathcal{T}}(0)>B \\mid\\mathcal{F}_{t}\\right)>0 \\tag{16a}\\] \\[\\mathbb{P}\\left(S_{t}^{\\mathcal{T}}(1)-S_{t}^{\\mathcal{T}}(0)\\leq b _{t}\\mid\\mathcal{F}_{t-1},S_{t-1}^{\\mathcal{T}}(1)-S_{t-1}^{\\mathcal{T}}(0)>B \\right)\\leq\\Delta_{t}. \\tag{16b}\\]\n\nNote that for each \\(t\\geq 1\\), if \\(m_{t}\\neq 0\\),\n\n\\[\\mathbb{P}\\left(S_{t}^{\\mathcal{T}}(1)-S_{t}^{\\mathcal{T}}(0)\\leq b _{t}\\mid\\mathcal{F}_{t-1}\\right) =\\mathbb{P}\\left(\\frac{s_{t}^{\\mathcal{T}}(1)-S_{t}^{\\mathcal{T}} (0)-\\tilde{\\mu}_{t}}{\\tilde{\\sigma}_{t}}\\leq z_{t}\\mid\\mathcal{F}_{t-1}\\right)\\] \\[\\stackrel{{(a)}}{{\\leq}}\\mathbb{P}\\left(\\frac{s_{t}^{ \\mathcal{T}}(1)-S_{t}^{\\mathcal{T}}(0)-\\tilde{\\mu}_{t}}{\\tilde{\\sigma}_{t}}\\leq z _{t}\\mid\\mathcal{F}_{t-1}\\right)\\] \\[\\stackrel{{(b)}}{{\\leq}}\\Phi\\left(z_{t}\\right) \\stackrel{{(c)}}{{\\leq}}\\Delta_{t}\\]\n\nwhere we used first inequality in (13) in (a), second inequality in (13) in (b), and (11) in (c).\n\nWe now show (16a) by induction. For \\(t=1\\), if \\(m_{1}\\neq 0\\), Algorithm 1 ensures that\n\n\\[\\mathbb{E}\\left(\\mathbb{P}\\left(s_{1}^{\\mathcal{T}}(1)-s_{1}^{\\mathcal{T}}(0) \\leq b_{1}\\mid\\mathcal{F}_{1}\\right)\\right)=\\mathbb{P}\\left(s_{1}^{\\mathcal{T }}(1)-s_{1}^{\\mathcal{T}}(0)\\leq b_{1}\\right)\\leq\\Delta_{1}<1\\]\n\nby construction, which implies that\n\n\\[\\mathbb{P}\\left(S_{1}^{\\mathcal{T}}(1)-S_{1}^{\\mathcal{T}}(0)>B\\mid\\mathcal{F }_{1}\\right)\\geq\\mathbb{P}\\left(s_{1}^{\\mathcal{T}}(1)-s_{1}^{\\mathcal{T}}(0)> b_{1}\\mid\\mathcal{F}_{1}\\right)>0\\]\n\nalmost surely. If \\(m_{1}=0\\), then \\(\\mathbb{P}\\left(S_{1}^{\\mathcal{T}}(1)-S_{1}^{\\mathcal{T}}(0)>B\\mid\\mathcal{F }_{1}\\right)=1\\) since \\(B<0\\). This proves the base case. For the inductive case, if \\(m_{t}\\neq 0\\), Algorithm 1 ensures that\n\n\\[\\mathbb{E}\\left(\\mathbb{P}\\left(S_{t}^{\\mathcal{T}}(1)-S_{t}^{\\mathcal{T}}(0) \\leq b_{t}\\mid\\mathcal{F}_{t}\\right)\\mid\\mathcal{F}_{t-1}\\right)=\\mathbb{P} \\left(S_{t}^{\\mathcal{T}}(1)-S_{t}^{\\mathcal{T}}(0)\\leq b_{t}\\mid\\mathcal{F}_{ t-1}\\right)\\leq\\Delta_{t}<1\\]\n\nby construction, which implies that\n\n\\[\\mathbb{P}\\left(S_{t}^{\\mathcal{T}}(1)-S_{t}^{\\mathcal{T}}(0)>B\\mid\\mathcal{F }_{t}\\right)\\geq\\mathbb{P}\\left(S_{t}^{\\mathcal{T}}(1)-S_{t}^{\\mathcal{T}}(0)> b_{t}\\mid\\mathcal{F}_{t}\\right)>0\\]\n\nalmost surely. If \\(m_{t}=0\\), we have that\n\n\\[\\mathbb{P}\\left(S_{t}^{\\mathcal{T}}(1)-S_{t}^{\\mathcal{T}}(0)>B\\mid\\mathcal{F }_{t}\\right)=\\mathbb{P}\\left(S_{t-1}^{\\mathcal{T}}(1)-S_{t-1}^{\\mathcal{T}}(0)> B\\mid\\mathcal{F}_{t-1}\\right)>0\\]\n\nfrom inductive hypothesis. This shows (16a).\n\nTo show (16b), note that under Definition 3.4,\n\n\\[\\left[s_{t}^{\\mathcal{T}}(1)-S_{t}^{\\mathcal{T}}(0)\\mid\\mathcal{F}_{t-1},S_{t- 1}^{\\mathcal{T}}(0)<S_{t-1}^{\\mathcal{T}}(1)-B\\right]\\]\n\n\\[\\stackrel{{ d}}{{=}}s_{t}^{\\mathcal{T}}(1)-s_{t}^{\\mathcal{T}}(0 )-\\left[S_{t-1}^{\\mathcal{T}}(0)\\mid\\mathcal{F}_{t-1},S_{t-1}^{\\mathcal{T}}(0)< S_{t-1}^{\\mathcal{T}}(1)-B\\right]\\]\n\nOn the rhs, \\(s_{t}^{\\mathcal{T}}(1)-s_{t}^{\\mathcal{T}}(0)\\) is independent of\n\n\\[\\left[S_{t-1}^{\\mathcal{T}}(0)\\mid\\mathcal{F}_{t-1},S_{t-1}^{\\mathcal{T}}(0)< S_{t-1}^{\\mathcal{T}}(1)-B\\right]\\]\n\nand that \\(S_{t-1}^{\\mathcal{T}}(1)-B\\in\\mathcal{F}_{t-1}\\). It follows from these, (16a) and Lemma B.1 that\n\n\\[\\mathbb{P}\\left(s_{t}^{\\mathcal{T}}(1)-s_{t}^{\\mathcal{T}}(0)-S_{t-1}^{\\mathcal{ T}}(0)\\leq b_{t}\\Big{|}\\mathcal{F}_{t-1},S_{t-1}^{\\mathcal{T}}(0)<S_{t-1}^{ \\mathcal{T}}(1)-B\\right)\\]\n\n\\[\\leq\\mathbb{P}\\left(s_{t}^{\\mathcal{T}}(1)-S_{t}^{\\mathcal{T}}(0)\\leq b_{t} \\mid\\mathcal{F}_{t-1}\\right)\\]\n\nTherefore, for each \\(t\\geq 1\\), if \\(m_{t}\\neq 0\\),\n\n\\[\\mathbb{P}\\left(S_{t}^{\\mathcal{T}}(1)-S_{t}^{\\mathcal{T}}(0)\\leq b_{t}\\mid \\mathcal{F}_{t-1},S_{t-1}^{\\mathcal{T}}(1)-S_{t-1}^{\\mathcal{T}}(0)>B\\right) \\leq\\Delta_{t}\\]\n\nas required. This concludes the proof.\n\nWhen are (13) satisfiedFix any \\(t\\geq 1\\) where \\(m_{t}\\neq 0\\). Note that\n\n\\[\\left[s_{t}^{\\mathcal{T}}(1)-S_{t}^{\\mathcal{T}}(0)\\mid\\mathcal{F}_{t-1}\\right]= \\sum_{i\\in\\mathcal{T}_{t}}\\left(Y_{i,t}(1)-Y_{i,t}(0)\\right)-\\sum_{r\\in[t-1]} \\sum_{i\\in\\mathcal{T}_{r}}\\left[Y_{i,r}(0)\\mid Y_{i,r}(1)\\right]\\]\n\nThe summands on the rhs are independent random variables under Definition 3.4. We thus expect that when \\(m_{t}\\) or \\(M_{t-1}\\) are sufficiently large,\n\n\\[\\frac{\\left[s_{t}^{\\mathcal{T}}(1)-S_{t}^{\\mathcal{T}}(0)\\mid\\mathcal{F}_{t-1} \\right]-\\mathbb{E}\\left[s_{t}^{\\mathcal{T}}(1)-S_{t}^{\\mathcal{T}}(0)\\mid \\mathcal{F}_{t-1}\\right]}{\\sqrt{\\mathbb{V}\\left[s_{t}^{\\mathcal{T}}(1)-S_{t}^ {\\mathcal{T}}(0)\\mid\\mathcal{F}_{t-1}\\right]}}\\approx N(0,1)\\]\n\nby central limit theorem under mild moment-growth conditions (e.g. Lyapunov's conditions). We thus expect that first condition in (13) holds when \\(m_{t}\\) or \\(M_{t-1}^{(1)}\\) are sufficiently large for each \\(t\\geq 1\\).\n\nWe now focus on the second condition in (13). Suppose that \\(\\Delta_{t}\\leq 0.5\\), which implies \\(z_{t}\\leq 0\\) by (11). Note that we can write\n\n\\[\\tilde{\\mu}_{t} =\\sum_{i\\in\\mathcal{T}_{t}}\\mathbb{E}\\left(Y_{i,t}(1)-Y_{i,t}(0) \\right)-\\sum_{r\\in[t-1]}\\sum_{i\\in\\mathcal{T}_{r}}\\mathbb{E}\\left[Y_{i,t}(0) \\mid Y_{i,t}(1)\\right]\\] \\[\\tilde{\\sigma}_{t}^{2} =\\sum_{r\\in[t-1]}\\mathbb{V}\\left(Y_{i,t}(1)-Y_{i,t}(0)\\right)+\\sum _{i\\in\\mathcal{T}_{r}}\\mathbb{V}\\left[Y_{i,t}(0)\\mid\\sigma\\left(Y_{i,t}(1) \\right)\\right]\\]\n\nand\n\n\\[\\tilde{\\mu}_{t} =m_{t}\\left(\\mu_{p,t}(1)-\\mu_{p,t}(0)\\right)-\\mu_{p,t}(0)M_{t-1}^{ (1)}\\] \\[\\tilde{\\sigma}_{t}^{2} =m_{t}\\cdot\\left(\\sigma(1)^{2}+\\sigma(0)^{2}\\right)+M_{t-1}^{(1)} \\cdot\\sigma(0)^{2}+m_{t}^{2}\\cdot\\sigma_{p,t}(1)^{2}+\\left(m_{t}+M_{t-1}^{(1) }\\right)^{2}\\cdot\\sigma_{p,t}(0)^{2}.\\]\n\nFor \\(t=1\\),\n\n\\[\\tilde{\\mu}_{t}=\\sum_{i\\in\\mathcal{T}_{t}}\\mathbb{E}\\left(Y_{i,t}(1)-Y_{i,t}(0 )\\right),\\quad\\tilde{\\sigma}_{t}^{2}=\\sum_{i\\in\\mathcal{T}_{t}}\\mathbb{V} \\left(Y_{i,t}(1)-Y_{i,t}(0)\\right)\\]\n\nand\n\n\\[\\tilde{\\mu}_{t} =m_{t}\\left(\\mu_{0}(1)-\\mu_{0}(0)\\right),\\] \\[\\tilde{\\sigma}_{t}^{2} =m_{t}\\cdot\\left(\\sigma(0)^{2}+\\sigma(1)^{2}\\right)+m_{t}^{2} \\cdot\\left(\\sigma_{0}(1)^{2}+\\sigma_{0}(0)^{2}\\right).\\]\n\nSo, second condition in (13) holds for \\(t=1\\) if we have chosen prior and model parameters such that\n\n\\[\\mu_{0}(1)-\\mu_{0}(0) \\leq\\frac{1}{m_{t}}\\sum_{i\\in\\mathcal{T}_{1}}\\mathbb{E}\\left(Y_{ i,1}(1)-Y_{i,1}(0)\\right)\\] \\[\\sigma(0)^{2}+\\sigma(1)^{2} +m_{t}\\cdot\\left(\\sigma_{0}(1)^{2}+\\sigma_{0}(0)^{2}\\right)\\geq \\frac{1}{m_{t}}\\sum_{i\\in\\mathcal{T}_{t}}\\mathbb{V}\\left(Y_{i,t}(1)-Y_{i,t}(0)\\right)\\]\n\nThis corresponds to that we choose prior and model parameters conservatively in the sense that we do not overestimate treatment effect or underestimate its variability. Now fix any \\(t\\geq 2\\). From the law of large number, we expect that for \\(M_{t-1}^{(1)}\\) sufficiently large\n\n\\[\\mu_{p,t}(0)\\approx\\frac{1}{M_{t-1}^{(1)}}\\sum_{r\\in[t-1]}\\sum_{i \\in\\mathcal{T}_{r}}\\mathbb{E}\\left[Y_{i,t}(0)\\right]\\] \\[\\frac{1}{M_{t-1}^{(1)}}\\sum_{r\\in[t-1]}\\sum_{i\\in\\mathcal{T}_{r} }\\mathbb{E}\\left[Y_{i,t}(0)\\mid Y_{i,t}(1)\\right]\\approx\\frac{1}{M_{t-1}^{(1)} }\\sum_{r\\in[t-1]}\\sum_{i\\in\\mathcal{T}_{r}}\\mathbb{E}\\left[Y_{i,t}(0)\\right]\\] \\[\\mu_{p,t}(1)-\\mu_{p,t}(0)\\approx\\frac{1}{M_{t-1}^{(1)}}\\sum_{r \\in[t-1]}\\sum_{i\\in\\mathcal{T}_{r}}\\mathbb{E}\\left[Y_{i,t}(1)-Y_{i,t}(0)\\right]\\] \\[\\frac{1}{M_{t-1}^{(1)}}\\sum_{r\\in[t-1]}\\sum_{i\\in\\mathcal{T}_{r} }\\mathbb{V}\\left[Y_{i,t}(0)\\mid Y_{i,t}(1)\\right]\\approx\\frac{1}{M_{t-1}^{(1)} }\\sum_{r\\in[t-1]}\\sum_{i\\in\\mathcal{T}_{r}}\\mathbb{E}\\mathbb{V}\\left[Y_{i,t}(0 )\\mid Y_{i,t}(1)\\right]\\] \\[\\qquad\\leq\\frac{1}{M_{t-1}^{(1)}}\\sum_{r\\in[t-1]}\\sum_{i\\in \\mathcal{T}_{r}}\\mathbb{V}\\left[Y_{i,t}(0)\\right]\\]So if the treatment effects increase or stay roughly constant throughout the experiments\n\n\\[\\frac{1}{m_{t}}\\sum_{i\\in\\mathcal{T}_{t}}\\mathbb{E}\\left(Y_{i,t}(1)-Y_{i,t}(0) \\right)\\geq\\frac{1}{M_{t-1}^{(1)}}\\sum_{r\\in[t-1]}\\sum_{i\\in\\mathcal{T}_{r}} \\mathbb{E}\\left[Y_{i,t}(1)-Y_{i,t}(0)\\right]\\]\n\nand our variance estimates \\(\\sigma{(0)}^{2},\\sigma{(1)}^{2}\\) are accurate or conservative in the sense that\n\n\\[\\sigma{(0)}^{2}\\geq\\frac{1}{M_{t-1}^{(1)}}\\sum_{r\\in[t-1]}\\sum_{i\\in\\mathcal{T }_{r}}\\mathbb{V}\\left[Y_{i,t}(0)\\right],\\ \\ \\sigma{(0)}^{2}+\\sigma{(1)}^{2}\\geq \\frac{1}{m_{t}}\\sum_{i\\in\\mathcal{T}_{t}}\\mathbb{V}\\left(Y_{i,t}(1)-Y_{i,t}(0 )\\right)\\]\n\nthe second condition in (13) holds for each \\(t\\geq 2\\) and the experiment produced by Algorithm 1 is \\((\\delta,B)\\)-RRC.\n\n## Appendix E Algorithm for general Bayesian models and costs\n\nThe following outcome model is a generalization of Definition 3.1. Here, experiment outcomes are allowed to be multivariate with each coordinate corresponds a different business metric.\n\n**Definition E.1** (General Bayesian model).: Fix \\(p,q\\geq 1\\). The model parameter \\(\\theta_{\\mathrm{true}}\\in\\mathbb{R}^{q}\\) is generated from certain prior \\(\\pi_{0}\\). The experiment outcome of unit \\(i\\) at stage \\(t\\) are distributed independently and identically as\n\n\\[\\left(Y_{i,t}(0),Y_{i,t}(0)\\right)\\stackrel{{\\mathrm{iid}}}{{ \\sim}}p(\\theta_{\\mathrm{true}})\\]\n\nwhere \\(Y_{i,t}(0),Y_{i,t}(0)\\in\\mathbb{R}^{q}\\) and \\(p(\\theta_{\\mathrm{true}})\\) is a probability distribution on \\(\\mathbb{R}^{p\\times p}\\).\n\nThe following is a generalization of Definition 2.1. It allows for general experiment cost beyond treatment effect. The cost of treating unit \\(i\\) is now \\(h_{it}=h_{t}(Y_{i,t}(1),Y_{i,t}(0))\\) for some function \\(h_{t}:\\mathbb{R}^{p\\times p}\\mapsto\\mathbb{R}\\) chosen by the user. For instance, \\(h_{t}\\) can be chosen to compute the worst treatment effect across multiple business metrics.\n\n**Definition E.2** (General experiment cost).: For each \\(t\\geq 1\\), let the experiment cost from stage-\\(t\\) and treated unit \\(i\\) be \\(h_{it}=h_{t}(Y_{i,t}(1),Y_{i,t}(0))\\) where \\(h_{t}:\\mathbb{R}^{p\\times p}\\mapsto\\mathbb{R}\\) is any user-chosen function. Then define \\(r_{t}:=\\sum_{i\\in\\mathcal{T}_{t}}h_{i,t}\\). We let \\(r_{t}=0\\) if \\(\\mathcal{T}_{t}=\\emptyset\\). Define the cumulative experiment cost up to stage \\(t\\) as \\(R_{t}:=\\sum_{k\\in[t]}r_{k}\\).\n\nWe now move to derive an explicit algorithm Algorithm 1 from Theorem 3.1 that output \\((m_{t})_{t\\geq 1}\\) such that the experiment is \\((\\delta,B)\\)-RRC. Compared to Algorithm 1, the algorithm developed in this section will require Monte-Carlo simulations and generally gives more conservative ramp schedule.\n\nWe first review the Cantelli's inequality, which is an improved version of the well-known Chebyshev's inequality for one-sided tail bounds.\n\n**Lemma E.3** (Cantelli's inequality).: _For any \\(\\lambda\\geq 0\\), and real-valued random variable \\(X\\) with finite variance,_\n\n\\[\\mathbb{P}(X-\\mathbb{E}(X)\\geq\\lambda)\\leq\\frac{1}{1+\\lambda^{2}/\\mathbb{V}(X )}\\]\n\nGiven that (i) \\(\\mathbb{P}\\left(R_{t-1}\\geq B\\mid\\mathcal{F}_{t-1}\\right)>0\\) and that (ii) \\(\\mathbb{E}\\left[R_{t}\\mid R_{t-1}\\geq B,\\mathcal{F}_{t-1}\\right]\\geq b_{t}\\), a direct application of Cantelli's inequality shows that\n\n\\[\\mathbb{P}\\left(R_{t}\\leq b_{t}\\mid R_{t-1}>B,\\mathcal{F}_{t-1}\\right)\\] \\[\\quad=\\mathbb{P}\\bigg{(}\\mathbb{E}\\left[R_{t}\\mid R_{t-1}>B, \\mathcal{F}_{t-1}\\right]-R_{t}\\geq\\mathbb{E}\\left[R_{t}\\mid R_{t-1}>B,\\mathcal{ F}_{t-1}\\right]-b_{t}\\bigg{|}R_{t-1}>B,\\mathcal{F}_{t-1}\\bigg{)}\\] \\[\\quad\\leq\\left(1+\\frac{(\\mathbb{E}\\left[R_{t}\\mid R_{t-1}>B, \\mathcal{F}_{t-1}\\right]-b_{t})^{2}}{\\mathbb{V}\\left(R_{t}\\mid R_{t-1}>B, \\mathcal{F}_{t-1}\\right)}\\right)^{-1}\\]\n\nwhere \\(\\mathcal{F}_{0}\\) denotes trivial \\(\\sigma\\)-algebra.\n\nOur strategy to construct an algorithm that selects ramp size \\(m_{t}\\) such that (1), (2) hold is as follows: we first verify that condition (i) holds; if not, set \\(m_{t}=0\\) and otherwise find \\(m_{t}\\) such that the following two inequalities hold\n\n\\[\\mathbb{E}\\left[R_{t}\\mid R_{t-1}\\geq B,\\mathcal{F}_{t-1}\\right] \\geq b_{t} \\tag{17a}\\] \\[\\frac{1}{1+\\frac{(\\mathbb{E}[R_{t}|R_{t-1}>B,\\mathcal{F}_{t-1}]- b_{t})^{2}}{\\mathbb{V}(R_{t}|R_{t-1}>B,\\mathcal{F}_{t-1})}}\\leq\\Delta_{t} \\tag{17b}\\]\n\nTo accomplish this, note that by exchangeability of the outcomes under Definition E.1,\n\n\\[\\begin{split}\\mathbb{E}\\left[R_{t}\\mid R_{t-1}\\geq B,\\mathcal{F} _{t-1}\\right]&=\\mathbb{E}\\left[r_{t}\\mid R_{t-1}\\geq B,\\mathcal{F} _{t-1}\\right]+\\mathbb{E}\\left[R_{t-1}\\mid R_{t-1}\\geq B,\\mathcal{F}_{t-1}\\right] \\\\ &=m_{t}\\mathbb{E}\\left[h_{i=1,t}\\mid R_{t-1}\\geq B,\\mathcal{F}_{t- 1}\\right]+\\mathbb{E}\\left[R_{t-1}\\mid R_{t-1}\\geq B,\\mathcal{F}_{t-1}\\right] \\end{split} \\tag{18}\\]\n\nand\n\n\\[\\mathbb{V}\\left(R_{t}\\mid R_{t-1}\\geq B,\\mathcal{F}_{t-1}\\right) =\\mathbb{V}\\left(r_{t}\\mid R_{t-1}\\geq B,\\mathcal{F}_{t-1}\\right) +\\mathbb{V}\\left(R_{t-1}\\mid R_{t-1}\\geq B,\\mathcal{F}_{t-1}\\right)\\] \\[\\qquad+\\operatorname{Cov}\\left(r_{t},R_{t-1}\\mid R_{t-1}\\geq B, \\mathcal{F}_{t-1}\\right)\\] \\[=m_{t}\\mathbb{V}\\left(h_{i=1,t}\\mid R_{t-1}\\geq B,\\mathcal{F}_{t- 1}\\right)\\] \\[\\qquad+m_{t}\\left(m_{t}-1\\right)\\operatorname{Cov}\\left(h_{i=1,t},h_{i=2,t}\\mid R_{t-1}\\geq B,\\mathcal{F}_{t-1}\\right)\\] \\[\\qquad+\\mathbb{V}\\left(R_{t-1}\\mid R_{t-1}\\geq B,\\mathcal{F}_{t- 1}\\right)+\\operatorname{Cov}\\left(r_{t},R_{t-1}\\mid R_{t-1}\\geq B,\\mathcal{F} _{t-1}\\right) \\tag{19}\\]\n\nWe thus require a Monte-Carlo procedure to output estimates \\(\\hat{\\varphi}_{t}(0),\\ldots,\\hat{\\varphi}_{t}^{(6)}\\) for the following posterior quantities on the rhs of (18), (19)\n\n\\[\\begin{split}\\mathbb{P}\\left(R_{t-1}\\geq B\\mid\\mathcal{F}_{t-1} \\right)\\leftarrow\\hat{\\varphi}_{t}(0)\\\\ \\mathbb{E}\\left(h_{i=1,t}\\mid R_{t-1}\\geq B,\\mathcal{F}_{t-1} \\right)\\leftarrow\\hat{\\varphi}_{t}(1)\\\\ \\mathbb{E}\\left(R_{t-1}\\mid R_{t-1}\\geq B,\\mathcal{F}_{t-1}\\right) \\leftarrow\\hat{\\varphi}_{t}^{(2)}\\\\ \\mathbb{V}\\left(h_{i=1,t}\\mid R_{t-1}\\geq B,\\mathcal{F}_{t-1} \\right)\\leftarrow\\hat{\\varphi}_{t}^{(3)}\\\\ \\operatorname{Cov}\\left(h_{i=1,t},h_{i=2,t}\\mid R_{t-1}\\geq B, \\mathcal{F}_{t-1}\\right)\\leftarrow\\hat{\\varphi}_{t}^{(4)}\\\\ \\mathbb{V}\\left[R_{t-1}\\mid R_{t-1}\\geq B,\\mathcal{F}_{t-1} \\right]\\leftarrow\\hat{\\varphi}_{t}^{(5)}\\\\ \\operatorname{Cov}\\left(h_{i=1,t},R_{t-1}\\mid R_{t-1}\\geq B, \\mathcal{F}_{t-1}\\right)\\leftarrow\\hat{\\varphi}_{t}^{(6)}\\end{split}\\]\n\nwhere \\(h_{i=1,t},h_{i=2,t}\\) denote costs from treating two units \\(i=1,2\\) at stage \\(t\\). Recall that under (...), the outcome of the units are exchangeable. So \\(i=1,2\\) simply refers to any two distinct units. These quantities will be used to construct estimates of \\(\\mathbb{E}\\left[R_{t}\\mid R_{t-1}>B,\\mathcal{F}_{t-1}\\right]\\) and \\(\\mathbb{V}\\left(R_{t}\\mid R_{t-1}>B,\\mathcal{F}_{t-1}\\right)\\) as functions of \\(m_{t}\\) chosen.\n\nWe now outline a procedure to construct \\(\\hat{\\varphi}_{t}(0),\\ldots,\\hat{\\varphi}_{t}^{(6)}\\). Firstly, suppose that we can obtain \\(K\\) samples from the posterior distribution\n\n\\[\\left[\\left(Y_{i,r}(0)\\right)_{i\\in\\mathcal{T}_{r},r\\in\\left[t-1\\right]},Y_{i=1,t}(0),Y_{i=1,t}(1),Y_{i=2,t}(0),Y_{i=2,t}(1)\\bigg{|}\\mathcal{F}_{t-1}\\right], \\tag{20}\\]\n\nfrom certain MCMC algorithms. The specific details of the MCMC algorithm will depend on the Bayesian model used, but generating posterior-predictive samples while imputing unobserved data, as required in (20), is a common objective of such algorithms (see e.g. [15, Chapter 18]). Let us denote the \\(K\\) samples as\n\n\\[\\left(Y_{i,r}^{\\{k\\}}(0)\\right)_{i\\in\\mathcal{T}_{r},r\\in\\left[t-1\\right]}, \\left(Y_{i,t}^{\\{k\\}}(0)\\right),Y_{i=1,t}^{\\{k\\}}(1),Y_{i=2,t}^{\\{k\\}}(0),Y_{i= 2,t}^{\\{k\\}}(1),\\quad k=1,\\ldots,K \\tag{21}\\]\n\nThese will give us \\(K\\) samples from \\(\\left[h_{i=1,t},h_{i=2,t},R_{t-1}\\mid\\mathcal{F}_{t-1}\\right]\\) as follows:\n\n\\[\\begin{split}\\left(\\hat{h}_{i=1,t}^{\\{k\\}},\\hat{h}_{i=2,t}^{\\{k \\}},\\hat{R}_{t-1}^{\\{k\\}}\\right)=&\\bigg{(}h_{t}\\left(Y_{i=1,t}^{\\{ k\\}}(1)-Y_{i=1,t}^{\\{k\\}}(0)\\right),h_{t}\\left(Y_{i=2,t}^{\\{k\\}}(1)-Y_{i=2,t}^{\\{k\\}}(0) \\right),\\\\ &\\sum_{r=1}^{t-1}\\sum_{i\\in\\mathcal{T}_{r}}h_{r}\\left(Y_{i,r}^{\\{ k\\}}(1)-Y_{i,r}^{\\{k\\}}(0)\\right)\\bigg{)},\\quad k=1,\\ldots,K\\end{split}\\]Then we can estimate \\(\\mathbb{P}\\left(R_{t-1}\\geq B\\mid\\mathcal{F}_{t-1}\\right)\\) by\n\n\\[\\mathbb{P}\\left(R_{t-1}\\geq B\\mid\\mathcal{F}_{t-1}\\right)\\leftarrow\\hat{\\varphi} _{t}(0)=\\frac{1}{K}\\sum_{k=1}^{K}\\mathbb{I}\\left(\\hat{R}_{t-1}^{\\{k\\}}\\geq B\\right)\\]\n\nLet\n\n\\[\\mathcal{L}_{t}:=\\left\\{k\\in[K]:\\hat{R}_{t-1}^{\\{k\\}}\\geq B\\right\\}\\subset[K]\\]\n\nwhich denotes the subset of the \\(K\\) Monte-Calor samples for which the budgets are not depleted.\n\nIf \\(\\hat{\\varphi}_{t}(0)=0\\iff\\mathcal{L}_{t}=\\emptyset\\), we can simply out \\(m_{t}=0\\) since this corresponds to the case that the condition (i) does not hold, i.e. \\(\\mathbb{P}\\left(R_{t}\\leq b_{t}\\mid R_{t-1}>B,\\mathcal{F}_{t-1}\\right)\\approx 0\\). Otherwise, we continue to construct \\(\\hat{\\varphi}_{t}(1),\\ldots,\\hat{\\varphi}_{t}^{(6)}\\) as follows:\n\n\\[\\begin{split}&\\mathbb{E}\\left(h_{i=1,t}\\mid R_{t-1}\\geq B, \\mathcal{F}_{t-1}\\right)\\leftarrow\\hat{\\varphi}_{t}(1)=\\frac{1}{\\left| \\mathcal{L}_{t}\\right|}\\sum_{k\\in\\mathcal{L}_{t}}\\hat{h}_{i=1,t}^{\\{k\\}}\\\\ &\\mathbb{E}\\left(R_{t-1}\\mid R_{t-1}\\geq B,\\mathcal{F}_{t-1} \\right)\\leftarrow\\hat{\\varphi}_{t}^{(2)}=\\frac{1}{\\left|\\mathcal{L}_{t}\\right| }\\sum_{k\\in\\mathcal{L}_{t}}\\hat{R}_{t-1}^{\\{k\\}}\\\\ &\\mathbb{V}\\left(h_{i=1,t}\\mid R_{t-1}\\geq B,\\mathcal{F}_{t-1} \\right)\\leftarrow\\hat{\\varphi}_{t}^{(3)}=\\frac{1}{\\left|\\mathcal{L}_{t}\\right| }\\sum_{k\\in\\mathcal{L}_{t}}\\left(\\hat{h}_{i=1,t}^{kk}\\right)^{2}-\\left(\\hat{ \\varphi}_{t}(1)\\right)^{2}\\\\ &\\text{Cov}\\left(h_{i=1,t},h_{i=2,t}\\mid R_{t-1}\\geq B,\\mathcal{F }_{t-1}\\right)\\\\ &\\qquad\\qquad\\leftarrow\\hat{\\varphi}_{t}^{(4)}=\\frac{1}{\\left| \\mathcal{L}_{t}\\right|}\\sum_{k\\in\\mathcal{L}_{t}}\\hat{h}_{i=1,t}^{\\{k\\}}\\hat{h }_{i=2,t}^{\\{k\\}}-\\hat{\\varphi}_{t}(1)\\left(\\frac{1}{\\left|\\mathcal{L}_{t} \\right|}\\sum_{k\\in\\mathcal{L}_{t}}\\hat{h}_{i=2,t}^{\\{k\\}}\\right)\\\\ &\\mathbb{V}\\left[R_{t-1}\\mid R_{t-1}\\geq B,\\mathcal{F}_{t-1} \\right]\\leftarrow\\hat{\\varphi}_{t}^{(5)}=\\frac{1}{\\left|\\mathcal{L}_{t}\\right| }\\sum_{k\\in\\mathcal{L}_{t}}\\left(\\hat{R}_{t-1}^{\\{k\\}}\\right)^{2}-\\left(\\hat{ \\varphi}_{t}^{(2)}\\right)^{2}\\\\ &\\text{Cov}\\left(h_{i=1,t},R_{t-1}\\mid R_{t-1}\\geq B,\\mathcal{F }_{t-1}\\right)\\leftarrow\\hat{\\varphi}_{t}^{(6)}=\\frac{1}{\\left|\\mathcal{L}_{ t}\\right|}\\sum_{k\\in\\mathcal{L}_{t}}\\hat{h}_{i=1,t}^{\\{k\\}}\\hat{h}_{i=2,t}^{\\{k\\}}- \\hat{\\varphi}_{t}(1)\\hat{\\varphi}_{t}^{(2)}\\end{split} \\tag{22}\\]\n\nFrom (18), (19) and the Monte-Carlo estimates above, we then have estimators for \\(\\mathbb{E}\\left[R_{t}\\mid R_{t-1}\\geq B,\\mathcal{F}_{t-1}\\right],\\mathbb{V} \\left[R_{t}\\mid R_{t-1}\\geq B,\\mathcal{F}_{t-1}\\right]\\) in terms of \\(\\hat{\\varphi}_{t}(1),\\ldots,\\hat{\\varphi}_{t}^{(6)}\\) as follows\n\n\\[\\begin{split}&\\mathbb{E}\\left[R_{t}\\mid R_{t-1}\\geq B,\\mathcal{F }_{t-1}\\right]\\gets m_{t}\\cdot\\hat{\\varphi}_{t}(1)+\\hat{\\varphi}_{t}^{(2) }\\\\ &\\mathbb{V}\\left[R_{t}\\mid R_{t-1}\\geq B,\\mathcal{F}_{t-1}\\right] \\leftarrow\\left(m_{t}\\cdot\\hat{\\varphi}_{t}^{(3)}+m_{t}\\left(m_{t}-1\\right) \\cdot\\hat{\\varphi}_{t}^{(4)}\\right)+\\hat{\\varphi}_{t}^{(5)}+m_{t}\\cdot\\hat{ \\varphi}_{t}^{(6)}\\end{split}\\]\n\nThe two inequalities in (17) then become\n\n\\[m_{t}\\cdot\\hat{\\varphi}_{t}(1)+\\hat{\\varphi}_{t}^{(2)}\\geq b_{t} \\tag{23a}\\] \\[\\frac{1}{1+\\frac{\\left(m_{t}\\cdot\\hat{\\varphi}_{t}(1)+\\hat{ \\varphi}_{t}^{(2)}-b_{t}\\right)^{2}}{\\left(m_{t}\\cdot\\hat{\\varphi}_{t}^{(3)}+m _{t}\\left(m_{t}-1\\right)\\cdot\\hat{\\varphi}_{t}^{(4)}\\right)+\\hat{\\varphi}_{t}^ {(5)}+m_{t}\\cdot\\hat{\\varphi}_{t}^{(6)}}}\\leq\\Delta_{t} \\tag{23b}\\]\n\nrespectively. Assume that \\(\\Delta_{t}>0\\) or else set \\(m_{t}=0\\) directly. Observe that (23b) can be written as, with \\(q_{t}:=\\Delta_{t}^{-1}-1\\),\n\n\\[A_{t}m_{t}^{2}+B_{t}m_{t}+C_{t}\\geq 0\\]\n\nwhere\n\n\\[\\begin{split} A_{t}&:=(\\hat{\\varphi}_{t}(1))^{2}-q _{t}\\hat{\\varphi}_{t}^{(4)}\\\\ B_{t}&:=2\\widehat{\\varphi}_{t}(1)\\left(\\hat{\\varphi} _{t}^{(2)}-b_{t}\\right)-q_{t}\\hat{\\varphi}_{t}^{(3)}+q_{t}\\hat{\\varphi}_{t}^{ (4)}-q_{t}\\hat{\\varphi}_{t}^{(6)}\\\\ C_{t}&:=\\left(\\hat{\\varphi}_{t}^{(2)}-b_{t}\\right)^ {2}-q_{t}\\hat{\\varphi}_{t}^{(5)}\\end{split} \\tag{24}\\]\n\nThen one can choose \\(m_{t}\\) to be the largest, positive integer in the range defined by\n\n\\[m_{t}\\cdot\\hat{\\varphi}_{t}(1)+\\hat{\\varphi}_{t}^{(2)}\\geq b_{t},\\quad A_{t}m_ {t}^{2}+B_{t}m_{t}+C_{t}\\geq 0\\]If the range does not contain any positive integer, we set \\(m_{t}=0\\). Note that the range can be easily identified after solving the quadratic equation \\(A_{t}m_{t}^{2}+B_{t}m_{t}+C_{t}=0\\). Algorithm 2 gives the algorithm that outputs ramp sizes adaptively. Note that by construction, it gives a \\((\\delta,B)\\)-RRC experiments if the Monte-Carlo estimators are sufficiently accurate.\n\n```\n1:\\(B<0\\), \\(\\delta\\in[0,1)\\)\n2:Initialize \\(t\\gets 1\\), \\(\\prod_{r=1}^{P}\\left(1-\\Delta_{r}\\right)\\gets 1\\)\n3:while\\(\\prod_{r=1}^{t-1}\\left(1-\\Delta_{r}\\right)>1-\\delta\\)do\n4:choose\\(\\Delta_{t}\\in\\left[0,\\frac{1-\\delta}{\\prod_{r=1}^{t-1}\\left(1-\\Delta_{r}\\right) }-1\\right],b_{t}\\geq B\\)\n5:run MCMC to obtain posterior samples in (21) and computes \\(\\hat{\\varphi}_{t}(0)\\)\n6:if\\(\\hat{\\varphi}_{t}(0)\\gets 0\\)then\\(m_{t}\\gets 0\\)\n7:else\n8:compute\\(\\hat{\\varphi}_{t}(1),\\ldots,\\hat{\\varphi}_{t}^{(6)}\\) using (22) and then \\(A_{t},B_{t},C_{t}\\) by (24)\n9:find\\(\\mathcal{V}_{t}\\leftarrow\\left\\{m\\in\\mathbb{N}_{+}\\cap\\left[0,N_{t}/2\\right]:m \\cdot\\hat{\\varphi}_{t}(1)+\\hat{\\varphi}_{t}^{(2)}\\geq b_{t},A_{t}m^{2}+B_{t}m +C_{t}\\geq 0\\right\\}\\)\n10:if\\(\\mathcal{V}_{t}\\neq\\emptyset\\)then\n11:\\(m_{t}\\leftarrow\\max\\mathcal{V}_{t}\\)\n12:else\n13:\\(m_{t}\\gets 0\\)\n14:endif\n15:endif\n16:Output\\(m_{t}\\) and then conduct stage \\(t\\)-experiment and observe the outcomes\n17:update\\(t\\gets t+1\\)\n18:endwhile\n```\n\n**Algorithm 2** Output ramp size adaptively\n\nWe have conducted preliminary simulations of the proposed procedure for a multivariate Gaussian outcome model with Gaussian-inverse-Wishart prior, and observed satisfactory results. However, we defer presenting numerical results until future work when a more systematic investigation of Monte-Carlo based procedures can be conducted.\n\n## Appendix F Linkedin experiment data\n\nIn Table 1 below, \\(\\mu_{\\text{true}}\\left(w\\right),\\sigma{\\left(w\\right)}^{2},w=0,1\\) are sample statistics from the actual LinkedIn experiment. \\(N_{t}\\) are incoming population size reduced by \\(10^{4}\\) factor for tractability on a personal computer.\n\n## Appendix G Thompson-sampling based Bayesian bandit\n\nThis algorithm is developed in [27, Section 4] for clinical trials. The algorithm assigns a user \\(i\\) at stage \\(t\\geq 1\\) to treatment with probability\n\n\\begin{table}\n\\begin{tabular}{c c c c c c c} \\hline \\hline Stages \\(t\\) & 1 & 2 & 3 & 4 & 5 & 6 \\\\ \\hline \\(\\mu_{\\text{true}}(0)\\) & 0.3648 & 0.3780 & 0.3752 & 0.2317 & 0.4009 & 0.3930 \\\\ \\(\\mu_{\\text{true}}(1)\\) & 0.3659 & 0.3788 & 0.3754 & 0.2317 & 0.4010 & 0.3941 \\\\ \\(\\sigma(0)^{2}\\) & 2.0993 & 2.2769 & 2.0909 & 1.1165 & 2.2705 & 2.3982 \\\\ \\(\\sigma(1)^{2}\\) & 2.0923 & 2.2248 & 2.0135 & 1.0526 & 2.2476 & 2.4430 \\\\ \\(N_{t}\\) & 10,756 & 10,460 & 10,598 & 7,580 & 10,550 & 10,688 \\\\ \\hline \\hline \\end{tabular}\n\\end{table}\nTable 1: Linkedin experiment data for tuning parameter \\(c>0\\). Under Definition 3.1, by (15d), we have that\n\n\\[\\mathbb{P}\\left(\\mu_{\\text{true}}\\left(1\\right)>\\mu_{\\text{true}}\\left(0\\right) \\mid\\mathcal{F}_{t-1}\\right)=\\Phi\\left(\\frac{\\mu_{p,t}(1)-\\mu_{p,t}(0)}{\\sqrt{ \\sigma_{p,t}(0)^{2}+\\sigma_{p,t}(1)^{2}}}\\right).\\]", "response": "## Review\n\n### Summary\n\nThe paper addresses the problem of phased releases by proposing a risk-sensitive strategy that incorporates a budget constraint and employs an adaptive Bayesian algorithm. This method aims to balance risk and reward, maximizing ramp-up speed while controlling the risk of ruin. The authors validate their approach through empirical testing, demonstrating that their algorithm outperforms existing bandit algorithms. Overall, the work contributes significantly to the budget-constrained bandits literature and presents a practical framework for automated phased release strategies in real-world applications.\n\n### Strengths\n\n- The problem setup and proposed model for phased release are novel, providing valuable insights into the application of Bayesian methods.\n- The paper is well-written, clear, and presents a solid theoretical foundation supported by experimental results, including simulations and semi-real experiments.\n- The originality of the tasks and methods offers a unique contribution to the literature on phased release strategies.\n\n### Weaknesses\n\n- There are concerns regarding the practicality of some parameters, such as the risk budget, in real-world scenarios.\n- The clarity of Section 3 could be improved, as the algorithm's objectives and contributions to maximizing ramp-up speed are not sufficiently detailed.\n- Some presentation aspects are incomplete or difficult to follow, particularly in defining the problem and its objectives.\n\n### Questions\n\n- Clarification is needed regarding the NPTE vs. PNTE scenarios in the experiments.\n- More insights into the algorithm's behavior presented in Figure 1(b) would be helpful.\n- Further explanation is required for which parameters should be estimated first, as mentioned in line 172.\n\n### Soundness\n\n**Score:** 12\n\n**Description:** 3 = good; the theoretical foundation is solid, but clarity issues may affect understanding.\n\n### Presentation\n\n**Score:** 12\n\n**Description:** 3 = good; overall clear but with some sections needing more detail for clarity.\n\n### Contribution\n\n**Score:** 12\n\n**Description:** 3 = good; the paper presents a moderate-to-high impact contribution to the field.\n\n### Rating\n\n**Score:** 24\n\n**Description:** 6 = Weak Accept; the paper is technically solid with no major concerns, but could benefit from minor improvements.\n\n### Paper Decision\n\n**Decision:** Accept (poster)\n\n**Reasons:** The paper is original and makes a significant contribution to the budget-constrained bandits literature. The soundness of the theoretical framework and the clarity of presentation, despite some weaknesses, support an overall positive evaluation. The authors are encouraged to address reviewer feedback to enhance clarity and completeness.\n"}
{"query": "You are a highly experienced, conscientious, and fair academic reviewer, please help me review this paper. The review should be organized into nine sections: \n1. Summary: A summary of the paper in 100-150 words.\n2. Strengths/Weaknesses/Questions: The Strengths/Weaknesses/Questions of paper, which should be listed in bullet points, with each point supported by specific examples from the article where possible.\n3. Soundness/Contribution/Presentation: Rate the paper's Soundness/Contribution/Presentation, and match this score to the corresponding description from the list below and provide the result. The possible scores and their descriptions are: \n    1 poor\n    2 fair\n    3 good\n    4 excellent\n4. Rating: Give this paper an appropriate rating, match this rating to the corresponding description from the list below and provide the result. The possible Ratings and their descriptions are: \n    1 strong reject\n    2 reject, significant issues present\n    3 reject, not good enough\n    4 possibly reject, but has redeeming facets\n    5 marginally below the acceptance threshold\n    6 marginally above the acceptance threshold\n    7 accept, but needs minor improvements \n    8 accept, good paper\n    9 strong accept, excellent work\n    10 strong accept, should be highlighted at the conference   \n5. Paper Decision: It must include the Decision itself(Accept or Reject) and the reasons for this decision, based on the criteria of originality, methodological soundness, significance of results, and clarity and logic of presentation.\n\nHere is the template for a review format, you must follow this format to output your review result:\n**Summary:**\nSummary content\n\n**Strengths:**\n- Strength 1\n- Strength 2\n- ...\n\n**Weaknesses:**\n- Weakness 1\n- Weakness 2\n- ...\n\n**Questions:**\n- Question 1\n- Question 2\n- ...\n\n**Soundness:**\nSoundness result\n\n**Presentation:**\nPresentation result\n\n**Contribution:**\nContribution result\n\n**Rating:**\nRating result\n\n**Paper Decision:**\n- Decision: Accept/Reject\n- Reasons: reasons content\n\n\nPlease ensure your feedback is objective and constructive. The paper is as follows:\n\n# Language Model Alignment with Elastic Reset\n\n Michael Noukhovitch\n\nMila, Universite de Montreal\n\n&Samuel Lavoie\n\nMila, Universite de Montreal\n\n&Florian Strub\n\nGoogle Deepmind\n\n&Aaron Courville\n\nMila, Universite de Montreal\n\nCIFAR AI Chair\n\nCorrespondance to michael.noukhovitch@umontreal.ca\n\n###### Abstract\n\nFinetuning language models with reinforcement learning (RL), e.g. from human feedback (HF), is a prominent method for alignment. But optimizing against a reward model can improve on reward while degrading performance in other areas, a phenomenon known as reward hacking, alignment tax, or language drift. First, we argue that commonly-used test metrics are insufficient and instead measure how different algorithms tradeoff between reward and drift. The standard method modified the reward with a Kullback-Lieber (KL) penalty between the online and initial model. We propose **Elastic Reset**, a new algorithm that achieves higher reward with less drift without explicitly modifying the training objective. We periodically reset the online model to an exponentially moving average (EMA) of itself, then reset the EMA model to the initial model. Through the use of an EMA, our model recovers quickly after resets and achieves higher reward with less drift in the same number of steps. We demonstrate that fine-tuning language models with Elastic Reset leads to state-of-the-art performance on a small scale pivot-translation benchmark, outperforms all baselines in a medium-scale RLHF-like IMDB mock sentiment task and leads to a more performant and more aligned technical QA chatbot with LLaMA-7B. Code available at github.com/mnoukhov/elastic-reset.\n\n## 1 Introduction\n\nDialogue agents that can effectively interpret and use language are a long-term challenge for NLP. The rise of large pretrained language models (LMs) (Brown et al., 2020) made language model finetuning one of the most promising research directions to achieving capable dialogue agents (Bender and Koller, 2020). Recently, reinforcement learning (RL) has become a key ingredient of finetuning large LMs for interaction with humans (Ziegler et al., 2019; Ouyang et al., 2022; Bai et al., 2022), notably shown in ChatGPT (OpenAI, 2022). A reward model is learned on the alignment objective, such as learned human preferences (RLHF; Christiano et al., 2017; Stiennon et al., 2020), and the language model is finetuned to optimize the reward. But training on the RL objective moves the model away from its pretraining and can reduce performance on important benchmarks (Ouyang et al., 2022) and even drifting away from natural language syntax and semantics (Lazaridou et al., 2020).\n\n\"Language drift\" (Lee et al., 2019; Lazaridou et al., 2020), \"alignment tax\" (Askell et al., 2021), \"reward model overoptimization\" (Gao et al., 2022), or LM-specific \"reward-hacking\" (Clark and Amodei, 2016) is inherent to RLHF. In the extreme case, models learn to achieve high reward by generating nonsense text that is unintelligible to humans (Lewis et al., 2017). Methods to mitigate this issue range from re-running pretraining (Lowe* et al., 2021), grounding in other modalities (Lee et al.,2019), masking the LM generation (Ramamurthy et al., 2022) and iterated learning (Lu et al., 2020). But the standard, and by far most popular approach, adds a Kullback-Lieber (KL) divergence penalty to the reward in order to prevent the finetuned model from drifting too far from the pretrained model (Jaques et al., 2017, 2019; Ziegler et al., 2019). Still, all methods are insufficient over a large-enough training horizon so models are early-stopped before reaching a catastrophic level of drift.\n\nGao et al. (2022) find that achieving reward is proportional to drift from the initial model, but that not all drifts are equal. We wish to make small but effective changes that achieve high reward but maintain capabilities, yet auxiliary losses such as the KL penalty don't seem improve this tradeoff and only serve to slow down training (Gao et al., 2022). We posit that RLHF training requires a useful inductive bias that does not modify the training objective. Inspired by recent work in generalization for image classification (Zhou et al., 2022), sample-efficient RL (Nikishin et al., 2022, D'Oro et al., 2023), and inducing compositional language (Li and Bowling, 2019), we propose to use resets. Similar to iterated learning (Kirby, 2001), iteratively resetting a model has been shown to reduce overfitting in language and RL scenarios (Rita et al., 2022). In this work, we show iteratively resetting a model also reduces drift while attaining equal or better reward than just a KL penalty.\n\nUnlike previous work in sample-efficient RL (Nikishin et al., 2022), RLHF is typically on-policy so it does not maintain a replay buffer with which to bootstrap learning after a reset. In lieu of a replay buffer, we reset the policy but maintain the value function. Yet resetting the policy to its initial state can still cause a large drop in performance. So we propose resetting to a model in-between our online and initial state, specifically to an exponential moving average (EMA) of our online policy, as EMA has been shown to be highly performant (Caron et al., 2021). We still expect our EMA model to slowly drift, so we add a second step where we reset the EMA model to the initial model. We call this overall method **Elastic Reset** and illustrate it in Figure 1. Elastic Reset is implemented on top of regular RL methods such as REINFORCE (Williams, 1992) or PPO (Schulman et al., 2017).\n\nFirst, we test our method on a small scale task: pivot translation with a transformer. In this classic benchmark for drift, we outperform all previous baselines and demonstrate state-of-the-art performance. Next, we re-evaluate how performance is measured in the field and argue for a metric of how each method trades off performance vs drift. We propose the Pareto Frontier Graph, a graphical measure that illuminates the trade-off between performance and drift and demonstrate that Elastic Reset dominates the baselines against this trade-off. Then, we scale up slightly to GPT2 and work on a popular task closer to RLHF, IMDB mock sentiment. Comparing to all baseline methods, we again show state-of-the-art performance on the benchmark. Through ablations, we show that Elastic Reset is robust to choices of hyperparameters, even more so than baselines. Finally, we scale up even more to true RLHF finetuning of Llama-7B in order to create a helpful technical QA chatbot using a StackExchange dataset. We again outperform the baseline, demonstrating how Elastic Reset mitigates the alignment tax while better optimizing the human feedback reward.\n\n## 2 Related Work\n\n\"It is often difficult or infeasible to capture exactly what we want an agent to do, and as a result we frequently end up using imperfect but easily measured proxies\" (Clark and Amodei, 2016). In RL, this proxy is how we construct our reward and the consequence can be \"reward-hacking\" (Clark and Amodei, 2016); an agent optimizes the reward but does not accomplishing the meaningful task. RLHF aims to align an agent with human preferences while maintaining the capabilities of the pretrained model, but uses a learned reward model as a proxy of human preferences (Christiano et al., 2017; Ziegler et al., 2019). This can lead to LMs that optimize a reward model but degrade in\n\nFigure 1: Elastic Reset. In actor-critic RL, we reset the policy but maintain the value function.\n\nperformance on general NLP benchmarks (Askell et al., 2021), overfit the reward model and do not generalize to true human preferences (Bai et al., 2022; Gao et al., 2022), or latch onto confounding factors hidden in the reward (Stiennon et al., 2020). **These effects are exacerbated if the reward model is updated during training such as in iterated RLHF (Bai et al., 2022) or related setups such as emergent communication (Lazaridou and Baroni, 2020), end-to-end dialogue (Lewis et al., 2017), learning RL policies through latent language (Andresa et al., 2018), and pivot translation (Utiyama and Isahara, 2007). There, the phenomenon is known as \"language drift\" (Lee et al., 2019) and can lead to incoherent and unnatural linguistic outputs Lewis et al. (2017).**\n\n**This phenomenon is inherent to RLHF. Gao et al. (2022) show that improvement on alignment / reward is proportional to drift from the initial model, but also find that different methods and design choices achieve different proportions of performance to drift. Therefore, a major challenge of RLHF is how to learn the reward in such a way as to minimize the drift, alignment tax, and reward-hacking. The standard approach used in most RLHF is to incorporate a KL penalty between the training language model and some fixed model (Jaques et al., 2019; Ziegler et al., 2019; Stiennon et al., 2020; Ouyang et al., 2022; Steinert-Threlkeld et al., 2022; Bai et al., 2022), usually the initial, pretrained model. Less common is to add the original pretraining task to the finetuning objective (termed S2P by Lowe* et al. (2021)) but this can be compute-intensive and requires maintaining the pretraining data which may be even more expensive for larger models (Brown et al., 2020). On small-scale pivot-translation, Lu et al. (2020) propose iterated learning with student-teacher distillation but it too is relatively compute intensive. Recently, Ramamurthy et al. (2022) propose to maintain a delayed masking model and mask the LM to output only the top-\\(p\\) tokens. Elastic Reset takes inspiration from both of these, using an iterated process and maintaining an EMA model. Apart from better performance, our method is more space efficient and maintains the EMA on CPU whereas both other methods require maintaining an extra model on GPU. It is also more compute efficient as resetting weights and EMA updates are very cheap operations, whereas Lu et al. (2020) requires a long distillation phase and Ramamurthy et al. (2022) requires an extra forward pass with the masking model. There exist less-popular methods that have been applied to similar issues in RL: prompt-tuning (Singh et al., 2022), using a fixed model to generate many options and re-ranking using a reward model (Lazaridou et al., 2020; Meta FAIR Diplomacy Team et al., 2022), and grounding the output in a separate modality (Lee et al., 2019) but none have been used for RLHF or at scale.**\n\n**Our method is inspired by recent works that leverage resets for single agent RL (Nikishin et al., 2022; D'Oro et al., 2023), image classification (Zhou et al., 2022), and emergent communication (Rita et al., 2022). Those works generally train from scratch and reset to random initializations in order to improve generalization. Our scenario requires resetting to pretrained models and focuses on improving the tradeoff between performance and drift from this pretrained model. Elastic Reset can be seen as an on-policy alternative to Nikishin et al.'s (2022) off-policy resets; whereas they maintain the old replay buffer, we maintain the value model and an EMA of our policy.**\n\n**Finally, the pretrain-then-RL-finetune setup with the goal of maintaining pretrained knowledge can be seen as a two-step, RL-specific instance of continual learning and therefore language drift has links to catastrophic forgetting (McCloskey and Cohen, 1989). There is a clear similarity between mitigation methods: rehearsal (Robins, 1995) or experience replay (Rolnick et al., 2019) is equivalent to multitasking with the pretraining objective (Lowe* et al., 2021) and weight-update regularization (Kirkpatrick et al., 2017) has similarities to KL regularization (Jaques et al., 2019).**\n\n## 3 Elastic Reset\n\n**The standard method against drift is a KL penalty, generally between the learning policy** \\(\\theta\\) **and the initial, pretrained model** \\(\\theta_{0}\\)**. It is calculated empirically over the minibatch of training inputs** \\(x\\) **and outputs** \\(y\\) **and used as an auxiliary reward with coefficient** \\(\\beta\\) **on top of the regular reward model** \\(r\\)****\n\n\\[R(x,y)=r(x,y)-\\beta\\log\\frac{\\pi_{\\theta}(y|x)}{\\pi_{\\theta_{0}}(y|x)} \\tag{1}\\]\n\n**For Elastic Reset, we maintain an exponential moving average** \\(\\bar{\\theta}\\) **of our learning model** \\(\\theta\\) **and choose a decay hyperparameter parameter** \\(\\eta\\)**. We initialize** \\(\\bar{\\theta}\\leftarrow\\theta_{0}\\) **and after every online model step, we update our EMA model** \\(\\bar{\\theta}\\leftarrow(1-\\eta)\\theta+\\eta\\bar{\\theta}\\)**. Every** \\(n\\) **steps, Elastic Reset sets the online model to the EMA model \\(\\theta\\leftarrow\\bar{\\theta}\\) and sets the EMA model to the initial model \\(\\bar{\\theta}\\leftarrow\\theta_{0}\\). As with other methods, Elastic Reset can be easily combined with a KL penalty.\n\n## 4 Translation Game: Careful Comparison to SOTA\n\nSetupWe first investigate the pivot-translation benchmark of Lee et al. (2019), which was previously popular for small-scale methods countering drift. Two translation models, French to English (FR\\(\\rightarrow\\)EN) and English to German (EN\\(\\rightarrow\\)DE), are pretrained on IWSLT (Cottolo et al., 2012). Then, the models are finetuned on translating French to German through English (FR\\(\\rightarrow\\)EN\\(\\rightarrow\\)DE) but given only paired French and German data from Multi30k (Elliott et al., 2016, 2017) as shown in Figure 2. The models are not given English at finetune-time so the challenge is optimizing FR\\(\\rightarrow\\)DE while maintaining fluency in the intermediate English. Whereas larger benchmarks have only proxies for drift, we can exactly measure the performance degradation in our setup with the standard translation metric BLEU on a held-out FR\\(\\rightarrow\\)EN validation set. Similarly, we measure success on the task with the FR\\(\\rightarrow\\)EN\\(\\rightarrow\\)DE BLEU score. Each model is an encoder-decoder Transformer (Vaswani et al., 2017) with 6 layers and all experimental details are available in Appendix A.\n\nBaselinesThe EN\\(\\rightarrow\\)DE reward model is simply trained using cross-entropy between predicted and true DE. Our lower-bound baseline is Frozen English, we freeze the FR\\(\\rightarrow\\)EN model and only update the EN\\(\\rightarrow\\)DE model. This models is guaranteed not to drift, but also cannot reach the best possible performance. For that, we need to update FR\\(\\rightarrow\\)EN by backpropagating through the discrete EN tokens. We follow Lee et al. (2019) and train FR\\(\\rightarrow\\)EN using REINFORCE (Williams, 1992) to estimate the gradient. As our base model, we combine REINFORCE with an exponentially moving baseline and, as with previous work, add a loss for entropy regularization.\n\nWhen both FR\\(\\rightarrow\\)EN and EN\\(\\rightarrow\\)DE are being updated, we tend to see reasonably large drift and we compare to the best previous methods that counter it on this benchmark. We follow Lee et al. (2019) to simulate the standard KL penalty method, KL penalty by training an LSTM LM on IWSLT English text and adding a KL penalty with \\(\\beta=0.05\\) to regularize the FR\\(\\rightarrow\\)EN model. Multitask learning, re-training, or S2P (Lowe\\({}^{\\ast}\\) et al., 2021), adds the supervised FR\\(\\rightarrow\\)EN objective on IWSLT pretraining data as an auxiliary task for the FR\\(\\rightarrow\\)EN model. Finally, we implement Seeded Iterated Learning (SIL; Lu et al., 2020), which alternates between \\(n\\) finetuning steps and \\(m\\) steps of teacher-student distillation. FR\\(\\rightarrow\\)EN and EN\\(\\rightarrow\\)DE \"teacher\" models are finetuned on the translation game, then each distills knowledge into \"student\" model of itself, and finally the students are initialized as teachers for the next iteration. Elastic Reset is implemented on top of REINFORCE with a very minimal KL penalty \\(\\beta=0.001\\) and uses an EMA decay \\(\\eta=0.99\\). We run all models for 50k updates and reset every 23k steps to get 2 resets / 3 iterations within a run. Hyperparameters may differ between methods, e.g. \\(\\beta\\), because we used a minimal search to find the best hyperparameters for each method.\n\nExperimentsFor each method, we run 5 seeds and plot the validation scores over training for the end-to-end task score, FR\\(\\rightarrow\\)EN\\(\\rightarrow\\)DE BLEU, and drift score, FR\\(\\rightarrow\\)EN BLEU, in Figures 2(b), 2(a) respectively. Following Lee et al. (2019), we also show the final validation score in Table 1. As a sanity check, Frozen English does not drift but also does not achieve a very high task performance. In line with previous results (Lu et al., 2020), all learning models initially improve FR\\(\\rightarrow\\)EN performance, likely because models are quickly, semi-supervised adapting from their pretraining (IWSLT) to the distribution of the finetune dataset (Multi30k). Afterwards, they start to overfit on their objective and FR\\(\\rightarrow\\)EN performance degrades. REINFORCE achieves the best\n\nFigure 2: The Translation Game (top left), IMDB mock sentiment task (bottom left), and StackLLaMA (right). We show all RL finetuning setups and StackLLaMA’s reward modelling (top right).\n\npossible task performance but drifts significantly. Despite extensive hyperparameter tuning and correspondence with the original authors, SIL does not manage to outperform the REINFORCE baseline so we exclude it from the figures for visual clarity but show values in Table 1 as well as full results in Appendix A. In line with previous work (Lee et al., 2019; Lu et al., 2020), we find that multitask and KL penalty are both beneficial to reducing drift, but both represent a tradeoff. Whereas multitask strongly reduces drift, it does not achieve a high task score. In contrast, KL penalty achieves a high task score but drifts quite drastically. Elastic Reset achieves nearly the best possible task score while maintaining the same drift score as the initial model. Visually, we see that our method track the baselines until the reset at 23k steps. After the reset, we see a slight performance drop but also a big jump back in terms of FR\\(\\rightarrow\\)EN drift. While the task performance recovers within 5k steps, the drift performance does not degrade to previous levels. For the second reset, the EMA model is slightly more drifted and so the reset is less pronounced for both task and drift, leading to faster task recovery but slightly more drift. Overall, Elastic Reset shows state-of-the-art results on the benchmark and outperforms all previous small-scale methods.\n\n## 5 Pareto Frontier Graph\n\nSimply evaluating validation curves side-by-side or looking at a table of final scores, it can be unclear which method is better if one drifts less but the other achieves a higher reward e.g. Multitask vs KL penalty in Table 1. Previous work on this (Lee et al., 2019) and other benchmarks (Ramamurthy et al., 2022) compare methods using simple point-estimates after training for a specific number of epochs. But this number of epochs is quite arbitrary as models never fully converge to a reward, they are early-stopped such that drift is not catastrophic. Since different setups may admit different levels of drift, we believe that evaluation should reflect the continuous tradeoff between task and drift. We extend Ziegler et al. (2019), and create a pareto frontier graph to plot each method's achieved task score vs drift metric on the validation set over training. We believe practioners will wish to choose the best model for some given task performance so, contrary to Ziegler et al. (2019), we plot the task score on x-axis and drift score on the y-axis. Improvement on a drift metric can either mean lower scores (perplexity) or higher scores (BLEU) so we always plot task score as increasing from bottom to top such that, graphically, a better method will functionally dominate a worse method. We plot the best achieved reward vs drift over all validation steps for the Translation Game in Figure 2(c). Not only\n\n\\begin{table}\n\\begin{tabular}{l c c} \\hline \\hline  & \\(\\uparrow\\) FR\\(\\rightarrow\\)EN\\(\\rightarrow\\)DE & \\(\\uparrow\\) FR\\(\\rightarrow\\)EN \\\\ \\hline Frozen English & 30.8\\(\\pm\\)0.2 & **36.3\\(\\pm\\)0.1** \\\\ REINFORCE & **33.2\\(\\pm\\)0.3** & 29.6\\(\\pm\\)0.3 \\\\ + SIL & 28.2\\(\\pm\\)0.4 & 27.3\\(\\pm\\)4.4 \\\\ + Multitask (S2P) & 32.2\\(\\pm\\)0.3 & 35.2\\(\\pm\\)1.0 \\\\ + KL penalty & **33.2\\(\\pm\\)0.2** & 30.8\\(\\pm\\)0.4 \\\\ + Elastic Reset & 32.9\\(\\pm\\)0.1 & **36.3\\(\\pm\\)0.1** \\\\ \\hline \\hline \\end{tabular}\n\\end{table}\nTable 1: Translation Game final validation scores\n\nFigure 3: Comparing Elastic Reset to all baseline methods on the Translation Game. We measure (a) Task Performance with FR\\(\\rightarrow\\)EN\\(\\rightarrow\\)DE BLEU and (b) Language Drift with FR\\(\\rightarrow\\)EN BLEU, on the validation set during finetuning. We plot the mean and standard error over 5 seeds. To compare how methods trade off the two metrics, we plot (c) the best achieved drift vs task performance.\n\ndoes Elastic Reset outperform the baselines at the final validation score, but it functionally dominates such that it is the best method for all levels of task performance it achieves.\n\n## 6 IMDB Mock Sentiment: Ablation Study for RLHF\n\nSetupNext, we scale to a larger benchmark that more closely approximates the standard RLHF setup. We use the recently released GRUE benchmark for RL training of LMs (Ramamurthy et al., 2022) and use IMDB mock sentiment (Ziegler et al., 2019), the main task where language models are susceptible to reward-hacking, shown in Figure 2. The goal is to complete an IMDB movie review with as positive a sentiment as possible. The baseline LM is GPT-2 (Radford et al., 2019) with 117M parameters further pretrained on the IMDB domain (Maas et al., 2011). We learn a DistilBERT (Sanh et al., 2020) reward model on IMDB to output a sentiment score between 0 (negative) and 1 (positive). We then train our GPT-2 LM to complete different IMDB reviews while maximizing the sentiment reward. Following Ramamurthy et al. (2022), we measure reward-hacking / drift with our model's perplexity on the true IMDB data. If we consider knowledge of the IMDB data as a useful capability, then our initial model was finetuned on IMDB to maximize log-probability, i.e. minimize perplexity, and has the maximum capabilites. We measure divergence from the initial model, and decrease in capabilities, by the increase in our trained model's perplexity on ground truth IMDB data. In contrast to the previous task, a lower perplexity score corresponds to less drift.\n\nBaselinesOur main baseline is PPO (Schulman et al., 2017) with Ziegler et al. (2019) modifications for RLHF training, specifically adding a KL penalty with the frozen initial model (equivalent to KL with pretrained) and dynamically decaying the coefficient \\(\\beta\\) over training. To further increase stability, Generalized Advantage Estimation (Schulman et al., 2015) is used for the advantage estimator. We also compare to NLPPO (Ramamurthy et al., 2022), a recent method that extends PPO with a masking model to counteract drift. The masking model is initialized to the pretrained model and recieves delayed updates; it is set to the online model every \\(n\\) steps. During training, the online model's output probabilities are restricted to the mask model's top \\(p\\) tokens. We use the RL4LMs library Ramamurthy et al. (2022) and their default hyperparameters for both PPO and NLPO e.g. \\(\\beta=0.1\\). We implement ElasticReset on top of PPO with an EMA decay rate of \\(0.995\\) and greatly reduce the KL coefficient \\(\\beta=0.001\\) to allow the model to drift more, then reset every 17 epochs such that we get two resets / three iterations during our training.\n\n\\begin{table}\n\\begin{tabular}{l c c} \\hline \\hline  & \\(\\uparrow\\)Sentiment & \\(\\downarrow\\)Perplexity \\\\ \\hline Zero-shot &.489\\(\\pm\\)0.01 & 32.45\\(\\pm\\)0.13 \\\\ PPO &.596\\(\\pm\\)0.02 & 33.45\\(\\pm\\)0.40 \\\\ NLPO &.558\\(\\pm\\)0.06 & 33.12\\(\\pm\\)0.74 \\\\\n**ElasticReset** &.611\\(\\pm\\)0.02 & 33.32\\(\\pm\\)0.23 \\\\ \\hline \\hline \\end{tabular}\n\\end{table}\nTable 2: IMDB mock sentiment final test scores\n\nFigure 4: Plotting PPO vs Elastic Reset on IMDB but splitting the results visually between resets. We measure (a) Language Drift and (b) Task Performance via Semantic Score on the validation set over finetuning. All methods also include a KL penalty. We plot mean and standard error across 5 seeds.\n\nExperimentsWe run all experiments for 5 seeds and report mean and standard error on our validation set for our reward, DistilBERT sentiment, and our drift score, perplexity. Following Ramamurthy et al. (2022), we run for 50 epochs (equivalent to 64k updates) and show our results in Figure 4. To make our resets more visible, we plot validation scores every epoch for Elastic Reset. Since the benchmark provides a test set as well, we compare all final models in Table 2. The PPO baseline performs quite well because it already includes a KL with the pretrained model. We find NLPO performs similarly to PPO, so we relegate NLPO results to Appendix B. Results from the original NLPO paper were stronger (Ramamurthy et al., 2022) but our reproduced numbers and curves were confirmed by the original authors (Ammanabrolu, 2023). Elastic Reset achieves better semantic scores much faster by using a smaller KL penalty coefficient (\\(0.001\\) vs PPO \\(0.1\\)) but also drifts more to achieve them. As with the previous task, this drift is then mitigated by the reset and we see semantic task score improve in relation to drift over the iterations. Looking at the pareto graph in Figure 7(c), we see that Elastic Reset far outpaces the baselines and provides a better tradeoff of reward vs drift for every reward.\n\nAblationsWe empirically investigate our method through ablations. Throughout this section we run experiments on IMDB with the same hyperparameters, unless otherwise mentioned. For brevity, we plot only the pareto graphs but include all other graphs in Appendix D.3 along with these same ablation experiments for the Translation Game, with similar results.\n\nTo investigate the source of improvement in Elastic Reset, we ablate the two resets: online to EMA, and EMA to initial model. We discard the second reset to get Reset to EMA: our model is reset to an EMA but the EMA is never reset. We also compare to the simplest reset idea, Reset to Init, and reset our policy to the initial model. We run all methods as previously and plot the pareto graph in Figure 4(a), for task and drift graphs see Appendix D.3. We find that even simple resets are already performant but the two ablations have a tradeoff: Reset to EMA is better at lower reward because it maintains performance whereas Reset to Init does better at higher reward because it doesn't drift as much. Elastic Reset combines the benefits of both and outperforms each method.\n\nNext, we consider our method's robustness to hyperparameters. First, we search along different EMA decay rates \\(\\eta\\) and plot our results in Figure 4(b) finding that our method is quite robust to choice of\n\nFigure 5: Ablating Elastic Reset on the IMDB mock sentiment task. We plot pareto graphs using mean and standard error across 5 seeds.\n\ndecay. Next, we investigate robustness to the choice of KL penalty coefficient \\(\\beta\\). We search across coefficients that range from 10x smaller to 10x larger than our best KL penalty coefficient for PPO (\\(\\beta=0.1\\)) and Elastic Reset (\\(\\beta=0.001\\)). For visual clarity, we plot PPO in Figure 4(c) and Elastic Reset in Figure 4(d) and only plot four points for PPO \\(\\beta=0,0.01\\) to maintain visual scale. We find that PPO is not robust to choice of KL and larger values correspond to better pareto curves but slower training. Results with NLPO are similar and shown in Appendix D.3. In contrast, Elastic Reset seems to be more robust to choice of KL with \\(0.001\\) producing the best curves while 10x larger and smaller values are similar. As opposed to PPO, Elastic Reset even works reasonably well without a KL penalty at all, (\\(\\beta=0\\)), matching PPO's best performance with a KL. This demonstrates that the expensive KL penalty may be replaced with the cheap Elastic Reset, although the combination of the two is best. This is also in line with previous work that have argued that the KL penalty may be unnecessary for RLHF (Bai et al., 2022; Gao et al., 2022). We also ablate the frequency of resets in Appendix D.4.1 and find that pareto curves are essentially unchanged.\n\nFinally, we provide an empirical intuition for Elastic Reset: in Appendix E.1 we show that resets iteratively improve the value function and in Appendix E.2 we show how EMA smoothes optimization but requires resetting in order to achieve high performance.\n\n## 7 StackLLaMA: Practical RLHF\n\nSetupFinally, we apply Elastic Reset to a larger-scale RLHF pipeline. We choose LLaMA (Touvron et al., 2023) as it is a prominent open-source model that has demonstrated strong performance on benchmarks. We follow Beeching et al. (2023) to finetune LLaMA-7B with RLHF on the StackExchange dataset (Lambert et al., 2023) to output helpful answers to technical questions, as judged by humans. Users ask technical questions on StackExchange and upvote the best answers. We score answers from StackExchange based on the number of upvotes they received from users, \\(\\texttt{score}=\\log_{2}(1+\\texttt{upvotes})\\)(Askell et al., 2021). At most 10 answers are drawn per question, text is cleaned, and HTML is converted to Markdown to make it easier to parse. First, we finetune LLaMA-7B with language modelling on the dataset to get LLaMA-7B-SE. We then further finetune it to get a reward model by learning to predict which of two answers was more upvoted (Stiennon et al., 2020). For a given question \\(x\\) and two answers \\(y_{+,-}\\) (where \\(y_{+}\\) is preferred), the loss for our reward model \\(r_{\\theta}\\) is \\(\\log\\sigma(r_{\\theta}(x,y_{+})-r_{\\theta}(x,y_{-}))\\). Finally, we finetune LLaMA-7B-SE with RL against the reward model by sampling questions from the dataset and learning to optimize the reward for our model's answer. All finetuning is done with a converted 8-bit model (Dctimers et al., 2022) and LoRA (Hu et al., 2021) for efficiency and to make the model training fit on our GPUs. We rely on the HuggingFace trl (von Werra et al., 2023) and peft (Mangrulka et al., 2023) libraries. All technical details are described in Appendix C.\n\nExperimentWe again compare to PPO with Ziegler et al. (2019) modifications i.e. KL penalty with a dynamically decaying coefficient. We run for 600 epochs (equivalent to 100k updates) and Elastic Reset every 260 epochs to get two resets / three iterations. Each run takes 20 hours on 4 A100s. Since only the LoRA parameters are being learned, we use Elastic Reset on those and therefore maintain only a small percentage of parameters in our EMA. We use a decay rate \\(\\eta=0.995\\) and a KL penalty coefficient \\(\\beta=0.02\\) for both methods. Calculating perplexity for each epoch is computationally\n\nFigure 6: Elastic Reset compared to PPO on StackLLaMA: A LLaMA-7B model RLHF finetuned on StackExchange as a helpful, technical QA chatbot\n\ninfeasible so we measure drift during training with the KL from the pretrained model over samples as done previously in other larger-scale RLHF (Bai et al., 2022; Gao et al., 2022).\n\nResultsWe plot reward in Figure 5(a)2 and KL from initial model over training in Figure 5(b). As noted by Beeching et al. (2023), the task is much noisier at a larger scale and with a real HF reward model. As a sanity check, we find that Elastic Reset tracks PPO until the first reset at 260 epochs where it drops only slightly, but also doesn't lose much performance. Around the second reset at 520 epochs, we see a much sharper drop but also maintaining the same approximate reward. At the end, Elastic Reset provides a non-trivial reduction in drift while aligning just as well as PPO. The pareto curve in Figure 5(c) shows Elastic Reset is equal or slightly worse at low reward but shows large improvements over PPO at higher reward. Notably, Elastic Reset seems to work out-of-the-box with LoRA. To evaluate drift another way, we get the perplexity of the final models over the StackExchange validation set as in Section 6. For a more robust view of reward, we train two more reward models using different seeds and evaluate the increase in reward between initial and final models. We show mean and standard deviation across the three reward models in Table 3, we find that Elastic Reset achieves a slightly better final reward than PPO while maintaining lower perplexity on the data. To examine a true alignment tax, we run our models on HumanEval Chen et al. (2021), a programming benchmark that provides another view of drift. Answering human-written coding questions is both a useful capability for our model and also falls within a similar domain to StackExchange. The benchmark tests for functional correctness such that pass@1 corresponds to the percentage of problems solved by the model on the first try as shown in Table 3. Training with PPO degrades performance compared to the initial model, demonstrating a large alignment tax. In contrast, Elastic Reset achieves a similar reward but maintains performance, even slightly improving on pass@10, creating an alignment bonus (Askell et al., 2021) instead of tax.\n\nFootnote 2: Note that our reward results differ from the original StackLLaMA likely due to a pending issue in their code affecting their reward model. After fixing the issue, the only difference seems to be an increase in base reward, and our reward curves are visually quite similar to theirs. See Appendix C for details\n\n## 8 Limitations\n\nAs a method, Elastic Reset is quite cheap computationally because both EMA updates and resets take negligable time compared to RLHF training and the EMA model can be stored on CPU. But our method is sensitive to the choice of reset rate; we chose heuristically based on when it seemed the model was overfitting. It is also possible to reset the policy and EMA model at different time scales, which could be a source of improvement. Our method also resets all of the trainable parameters, research in similar methods suggests that resetting larger models can benefit from resetting only part of the network (Zhou et al., 2022; Nikishin et al., 2022) or weighted-averaging instead of resets (D'Oro et al., 2023). We leave both of these directions to future work.\n\nAlthough we have thoroughly investigated our method on three different tasks, we note that none of them are ideal RLHF benchmarks. As pointed out by Gao et al. (2022), we measure our model's performance using the same reward model we optimize. This can lead to reward model overoptimization and our metric could mask overfitting and lack of generalization to the real world i.e. actual human preferences. An ideal benchmark could include a \"gold\" reward model as a proxy for human preference (Gao et al., 2022), but no such benchmarks are open-sourced and available.\n\n\\begin{table}\n\\begin{tabular}{l c c c} \\hline \\hline  & \\(\\uparrow\\Delta\\) Reward & \\(\\downarrow\\) Perplexity & \\(\\uparrow\\) HumanEval (pass@1,pass@10) \\\\ \\hline Zero-shot & 0 & 4.43 & 11.0, 12.7 \\\\ \\hline PPO & 0.81 \\(\\pm\\) 0.06 & 4.62 & 7.8, 10.7 \\\\\n**Elastic Reset** & 0.96 \\(\\pm\\) 0.09 & 4.57 & 11.0, 13.0 \\\\ \\hline \\hline \\end{tabular}\n\\end{table}\nTable 3: Evaluations of the initial (zero-shot) and finetuned StackLLaMA models after 600 epochs. We measure alignment using an average over three reward model trained with three different seeds and drift with perplexity on the data. HumanEval is a programming benchmark that acts as a practical measure of drift / alignment tax.\n\nFinally, we note that we follow all previous RLHF work and investigate only on-policy methods (Ziegler et al., 2019; Stiennon et al., 2020; Askell et al., 2021; Bai et al., 2022). Previous work in resetting for RL has focused on off-policy methods and demonstrated strong performance (Nikishin et al., 2022; D'Oro et al., 2023). As previously noted, our method can be seen as an adaptation of those to on-policy RL. In RLHF, PPO is by far the most popular method and on-policy is the dominant paradigm since it guarantees better local gradients. But it is possible that off-policy methods could implicitly balance performance and drift by incorporating a replay buffer with older data.\n\n## 9 Conclusion\n\nThe problems of drift (Lee et al., 2019), alignment tax (Askell et al., 2021), reward model overoptimization (Gao et al., 2022), and reward hacking (Clark and Amodei, 2016) are inherent to RLHF and reduce its efficacy. We have introduced a simple but powerful new method, Elastic Reset, to tackle this problem and improve performance while maintaining linguistic capabilities. We have shown its ability on three different tasks and across three different scales: from 6 layer Transformers to GPT2 to LLaMA-7B. The problem of drift is currently being addressed with a standard KL penalty despite the computational cost, tradeoff with reward, and recent claims that is may be unnecessary (Bai et al., 2022; Gao et al., 2022). Elastic Reset is a cheap and effective method to tackle the same problem, achieving a better tradeoff of reward and drift while reducing alignment tax. We hope our method leads to better RLHF and therefore models that are closer aligned with human preferences (Ziegler et al., 2019). As well, we hope this work invigorates more research into improving the reward / drift tradeoff of RLHF with a focus on computationally efficient methods that scale.\n\n## Acknowledgments and Disclosure of Funding\n\nMN is supported by Fonds de recherche du Quebec - Nature et technologies and Sony. MN would like to thank Issam Laradji, ServiceNow Research, Mila, and Compute Canada for providing resources used in the experiments.\n\n## References\n\n* Ammanabrolu (2023) P. Ammanabrolu. Re: Reproducing NLPO (RL4LMs), Jan. 2023.\n* Andreas et al. (2018) J. Andreas, D. Klein, and S. Levine. Learning with Latent Language. In _NAACL_. arXiv, 2018. doi: 10.48550/arXiv.1711.00482. URL [http://arxiv.org/abs/1711.00482](http://arxiv.org/abs/1711.00482). arXiv:1711.00482 [cs].\n* Askell et al. (2021) A. Askell, Y. Bai, A. Chen, D. Drain, D. Ganguli, T. Henighan, A. Jones, N. Joseph, B. Mann, N. DasSarma, N. Elhage, Z. Hatfield-Dodds, D. Hernandez, J. Kermion, K. Ndousse, C. Olsson, D. Amodei, T. Brown, J. Clark, S. McCandlish, C. Olah, and J. Kaplan. A General Language Assistant as a Laboratory for Alignment, Dec. 2021. URL [http://arxiv.org/abs/2112.00861](http://arxiv.org/abs/2112.00861). arXiv:2112.00861 [cs].\n* Bahdanau et al. (2015) D. Bahdanau, K. Cho, and Y. Bengio. Neural Machine Translation by Jointly Learning to Align and Translate. In _ICLR_. arXiv, 2015. URL [http://arxiv.org/abs/1409.0473](http://arxiv.org/abs/1409.0473). arXiv:1409.0473 [cs, stat].\n* Bai et al. (2022) Y. Bai, A. Jones, K. Ndousse, A. Askell, A. Chen, N. DasSarma, D. Drain, S. Fort, D. Ganguli, T. Henighan, N. Joseph, S. Kadavath, J. Kermion, T. Conerly, S. El-Showk, N. Elhage, Z. Hatfield-Dodds, D. Hernandez, T. Hume, S. Johnston, S. Kravec, L. Lovitt, N. Nanda, C. Olsson, D. Amodei, T. Brown, J. Clark, S. McCandlish, C. Olah, B. Mann, and J. Kaplan. Training a Helpful and Harmless Assistant with Reinforcement Learning from Human Feedback, Apr. 2022. URL [http://arxiv.org/abs/2204.05862](http://arxiv.org/abs/2204.05862). arXiv:2204.05862 [cs].\n* Beeching et al. (2023) E. Beeching, Y. Belkada, K. Rasul, L. Tunstall, L. v. Werra, N. Rajani, and N. Lambert. StackLLaMA: An RL Fine-tuned LLaMA Model for Stack Exchange Question and Answering, 2023. URL [https://huggingface.co/blog/stackllama](https://huggingface.co/blog/stackllama).\n\nE. M. Bender and A. Koller. Climbing towards NLU: On Meaning, Form, and Understanding in the Age of Data. In _Proceedings of the 58th Annual Meeting of the Association for Computational Linguistics_, pages 5185-5198, Online, July 2020. Association for Computational Linguistics. doi: 10.18653/v1/2020.acl-main.463. URL [https://aclanthology.org/2020.acl-main.463](https://aclanthology.org/2020.acl-main.463).\n* Brown et al. (2020) T. B. Brown, B. Mann, N. Ryder, M. Subbiah, J. Kaplan, P. Dhariwal, A. Neelakantan, P. Shyam, G. Sastry, A. Askell, S. Agarwal, A. Herbert-Voss, G. Krueger, T. Henighan, R. Child, A. Ramesh, D. M. Ziegler, J. Wu, C. Winter, C. Hesse, M. Chen, E. Sigler, M. Litwin, S. Gray, B. Chess, J. Clark, C. Berner, S. McCandlish, A. Radford, I. Sutskever, and D. Amodei. Language Models are Few-Shot Learners. In _Neural Information Processing Systems_. arXiv, July 2020. doi: 10.48550/arXiv.2005.14165. URL [http://arxiv.org/abs/2005.14165](http://arxiv.org/abs/2005.14165). arXiv:2005.14165 [cs].\n* Caron et al. (2021) M. Caron, H. Touvron, I. Misra, H. Jegou, J. Mairal, P. Bojanowski, and A. Joulin. Emerging Properties in Self-Supervised Vision Transformers, May 2021. URL [http://arxiv.org/abs/2104.14294](http://arxiv.org/abs/2104.14294). arXiv:2104.14294 [cs].\n* Cettolo et al. (2012) M. Cettolo, C. Girardi, and M. Federico. WIT3: Web Inventory of Transcribed and Translated Talks. In _Proceedings of the 16th Annual conference of the European Association for Machine Translation_, pages 261-268, Trento, Italy, May 2012. European Association for Machine Translation. URL [https://aclanthology.org/2012.eamt-1.60](https://aclanthology.org/2012.eamt-1.60).\n* Chen et al. (2017) M. Chen, J. Tworek, H. Jun, Q. Yuan, H. P. d. O. Pinto, J. Kaplan, H. Edwards, Y. Burda, N. Joseph, G. Brockman, A. Ray, R. Puri, G. Krueger, M. Petrov, H. Khlaaf, G. Sastry, P. Mishkin, B. Chan, S. Gray, N. Ryder, M. Pavlov, A. Power, L. Kaiser, M. Bavarian, C. Winter, P. Tillet, F. P. Such, D. Cummings, M. Plappert, F. Chantzis, E. Barnes, A. Herbert-Voss, W. H. Guss, A. Nichol, A. Paino, N. Tezak, J. Tang, I. Babuschkin, S. Balaji, S. Jain, W. Saunders, C. Hesse, A. N. Carr, J. Leike, J. Achiam, V. Misra, E. Morikawa, A. Radford, M. Knight, M. Brundage, M. Murati, K. Mayer, P. Welinder, B. McGrew, D. Amodei, S. McCandlish, I. Sutskever, and W. Zaremba. Evaluating Large Language Models Trained on Code, July 2021. URL [http://arxiv.org/abs/2107.03374](http://arxiv.org/abs/2107.03374). arXiv:2107.03374 [cs].\n* Christiano et al. (2017) P. F. Christiano, J. Leike, T. Brown, M. Martic, S. Legg, and D. Amodei. Deep Reinforcement Learning from Human Preferences. In _Advances in Neural Information Processing Systems_, volume 30. Curran Associates, Inc., 2017. URL [https://papers.nips.cc/paper/2017/hash/d5e2c0adad503c91f91df240d0cd4e49-Abstract.html](https://papers.nips.cc/paper/2017/hash/d5e2c0adad503c91f91df240d0cd4e49-Abstract.html).\n* Clark and Amodei (2016) J. Clark and D. Amodei. Faulty Reward Functions in the Wild, Dec. 2016. URL [https://openai.com/blog/faulty-reward-functions/](https://openai.com/blog/faulty-reward-functions/).\n* Dettmers et al. (2022) T. Dettmers, M. Lewis, Y. Belkada, and L. Zettlemoyer. LLM.int8(): 8-bit Matrix Multiplication for Transformers at Scale. In _NeurIPS_. arXiv, Nov. 2022. doi: 10.48550/arXiv.2208.07339. URL [http://arxiv.org/abs/2208.07339](http://arxiv.org/abs/2208.07339). arXiv:2208.07339 [cs].\n* D'Oro et al. (2023) P. D'Oro, M. Schwarzer, E. Nikishin, P.-L. Bacon, M. G. Bellemare, and A. Courville. Sample-Efficient Reinforcement Learning by Breaking the Replay Ratio Barrier. In _International Conference on Learning Representations_, 2023. URL [https://openreview.net/forum?id=4GBGwVIEYJ](https://openreview.net/forum?id=4GBGwVIEYJ).\n* Elliott et al. (2016) D. Elliott, S. Frank, K. Sima'an, and L. Specia. Multi30K: Multilingual English-German Image Descriptions. In _Proceedings of the 5th Workshop on Vision and Language_, pages 70-74, Berlin, Germany, 2016. Association for Computational Linguistics. doi: 10.18653/v1/W16-3210. URL [http://aclweb.org/anthology/W16-3210](http://aclweb.org/anthology/W16-3210).\n* Elliott et al. (2017) D. Elliott, S. Frank, L. Barrault, F. Bougares, and L. Specia. Findings of the Second Shared Task on Multimodal Machine Translation and Multilingual Image Description. In _Proceedings of the Second Conference on Machine Translation_, pages 215-233, Copenhagen, Denmark, Sept. 2017. Association for Computational Linguistics. doi: 10.18653/v1/W17-4718. URL [https://aclanthology.org/W17-4718](https://aclanthology.org/W17-4718).\n* Gao et al. (2022) L. Gao, J. Schulman, and J. Hilton. Scaling Laws for Reward Model Overoptimization, Oct. 2022. URL [http://arxiv.org/abs/2210.10760](http://arxiv.org/abs/2210.10760). arXiv:2210.10760 [cs, stat].\n* Gao et al. (2021)S. Gugger, L. Debut, T. Wolf, P. Schmid, Z. Mueller, S. Mangrulkar, M. Sun, and B. Bossan. Accelerate: Training and inference at scale made simple, efficient and adaptable, 2022. URL [https://github.com/huggingface/accelerate](https://github.com/huggingface/accelerate).\n* Hochreiter and Schmidhuber (1997) S. Hochreiter and J. Schmidhuber. Long Short-Term Memory. _Neural Computation_, 9(8):1735-1780, Nov. 1997. ISSN 0899-7667. doi: 10.1162/neco.1997.9.8.1735. URL [https://doi.org/10.1162/neco.1997.9.8.1735](https://doi.org/10.1162/neco.1997.9.8.1735).\n* Hu et al. (2021) E. J. Hu, Y. Shen, P. Wallis, Z. Allen-Zhu, Y. Li, S. Wang, L. Wang, and W. Chen. LoRA: Low-Rank Adaptation of Large Language Models, Oct. 2021. URL [http://arxiv.org/abs/2106.09685](http://arxiv.org/abs/2106.09685). arXiv:2106.09685 [cs].\n* Hunter (2007) J. D. Hunter. Matplotlib: A 2D Graphics Environment. _Computing in Science & Engineering_, 9(3):90-95, May 2007. ISSN 1558-366X. doi: 10.1109/MCSE.2007.55. URL [https://ieeexplore.ieee.org/document/4160265](https://ieeexplore.ieee.org/document/4160265). Conference Name: Computing in Science & Engineering.\n* Jang et al. (2017) E. Jang, S. Gu, and B. Poole. Categorical Reparameterization with Gumbel-Softmax. In _ICLR_. arXiv, Aug. 2017. doi: 10.48550/arXiv.1611.01144. URL [http://arxiv.org/abs/1611.01144](http://arxiv.org/abs/1611.01144). arXiv:1611.01144 [cs, stat].\n* Jaques et al. (2017) N. Jaques, S. Gu, D. Bahdanau, J. M. Hernandez-Lobato, R. E. Turner, and D. Eck. Sequence Tutor: Conservative Fine-Tuning of Sequence Generation Models with KL-control. In _International Conference on Machine Learning_. arXiv, Oct. 2017. doi: 10.48550/arXiv.1611.02796. URL [http://arxiv.org/abs/1611.02796](http://arxiv.org/abs/1611.02796). arXiv:1611.02796 [cs].\n* Jaques et al. (2019) N. Jaques, A. Ghandeharioun, J. H. Shen, C. Ferguson, A. Lapedriza, N. Jones, S. Gu, and R. Picard. Way Off-Policy Batch Deep Reinforcement Learning of Implicit Human Preferences in Dialog, July 2019. URL [http://arxiv.org/abs/1907.00456](http://arxiv.org/abs/1907.00456). arXiv:1907.00456 [cs, stat].\n* Kirby (2001) S. Kirby. Spontaneous evolution of linguistic structure-an iterated learning model of the emergence of regularity and irregularity. _IEEE Transactions on Evolutionary Computation_, 5(2):102-110, Apr. 2001. ISSN 1941-0026. doi: 10.1109/4235.918430. Conference Name: IEEE Transactions on Evolutionary Computation.\n* Kirkpatrick et al. (2017) J. Kirkpatrick, R. Pascanu, N. Rabinowitz, J. Veness, G. Desjardins, A. A. Rusu, K. Milan, J. Quan, T. Ramalho, A. Grabska-Barwinska, D. Hassabis, C. Clopath, D. Kumaran, and R. Hadsell. Overcoming catastrophic forgetting in neural networks. _Proceedings of the National Academy of Sciences_, 114(13):3521-3526, Mar. 2017. doi: 10.1073/pnas.1611835114. URL [https://www.pnas.org/doi/10.1073/pnas.1611835114](https://www.pnas.org/doi/10.1073/pnas.1611835114). Publisher: Proceedings of the National Academy of Sciences.\n* Koehn et al. (2007) P. Koehn, H. Hoang, A. Birch, C. Callison-Burch, M. Federico, N. Bertoldi, B. Cowan, W. Shen, C. Moran, R. Zens, C. Dyer, O. Bojar, A. Constantin, and E. Herbst. Moses: Open Source Toolkit for Statistical Machine Translation. In _Proceedings of the 45th Annual Meeting of the Association for Computational Linguistics Companion Volume Proceedings of the Demo and Poster Sessions_, pages 177-180, Prague, Czech Republic, June 2007. Association for Computational Linguistics. URL [https://aclanthology.org/P07-2045](https://aclanthology.org/P07-2045).\n* Lambert et al. (2023) N. Lambert, L. Tunstall, N. Rajani, and T. Thrush. HuggingFace H4 Stack Exchange Preference Dataset, 2023. URL [https://huggingface.co/datasets/HuggingFaceH4/stack-exchange-preferences](https://huggingface.co/datasets/HuggingFaceH4/stack-exchange-preferences).\n* Lazaridou and Baroni (2020) A. Lazaridou and M. Baroni. Emergent Multi-Agent Communication in the Deep Learning Era, July 2020.\n* Lazaridou et al. (2020) A. Lazaridou, A. Potapenko, and O. Tieleman. Multi-agent Communication meets Natural Language: Synergies between Functional and Structural Language Learning. In _ACL_. arXiv, May 2020. doi: 10.48550/arXiv.2005.07064. URL [http://arxiv.org/abs/2005.07064](http://arxiv.org/abs/2005.07064). arXiv:2005.07064 [cs].\n* Lee et al. (2019) J. Lee, K. Cho, and D. Kiela. Countering Language Drift via Visual Grounding, Sept. 2019. URL [http://arxiv.org/abs/1909.04499](http://arxiv.org/abs/1909.04499). arXiv:1909.04499 [cs].\n* Lee et al. (2019)M. Lewis, D. Yarats, Y. N. Dauphin, D. Parikh, and D. Batra. Deal or No Deal? End-to-End Learning for Negotiation Dialogues, June 2017. URL [http://arxiv.org/abs/1706.05125](http://arxiv.org/abs/1706.05125). arXiv:1706.05125 [cs] version: 1.\n* Lhoest et al. (2021) Q. Lhoest, A. V. del Moral, Y. Jernite, A. Thakur, P. von Platen, S. Patil, J. Chaumond, M. Drame, J. Plu, L. Tunstall, J. Davison, M. Sasko, G. Chhablani, B. Malik, S. Brandeis, T. L. Scao, V. Sanh, C. Xu, N. Patry, A. McMillan-Major, P. Schmid, S. Gugger, C. Delangue, T. Matussiere, L. Debut, S. Bekman, P. Cistac, T. Goehringer, V. Mustar, F. Lagunas, A. M. Rush, and T. Wolf. Datasets: A Community Library for Natural Language Processing, Sept. 2021. URL [http://arxiv.org/abs/2109.02846](http://arxiv.org/abs/2109.02846). arXiv:2109.02846 [cs].\n* Li and Bowling (2019) F. Li and M. Bowling. Ease-of-Teaching and Language Structure from Emergent Communication. In _NeurIPS_, 2019. URL [http://arxiv.org/abs/1906.02403](http://arxiv.org/abs/1906.02403). arXiv: 1906.02403.\n* Loshchilov and Hutter (2022) I. Loshchilov and F. Hutter. Decoupled Weight Decay Regularization. In _International Conference on Learning Representations_, Feb. 2022. URL [https://openreview.net/forum?id=Bkg6RiCqY7](https://openreview.net/forum?id=Bkg6RiCqY7).\n* Lowe* et al. (2021) R. Lowe*, A. Gupta*, J. Foerster, D. Kiela, and J. Pineau. On the interaction between supervision and self-play in emergent communication. In _International Conference on Learning Representations_, Sept. 2021. URL [https://openreview.net/forum?id=YJxGLlBtwH](https://openreview.net/forum?id=YJxGLlBtwH).\n* Lu et al. (2020) Y. Lu, S. Singhal, F. Strub, O. Pietquin, and A. Courville. Countering Language Drift with Seeded Iterated Learning. In _ICML_. arXiv, Aug. 2020. doi: 10.48550/arXiv.2003.12694. URL [http://arxiv.org/abs/2003.12694](http://arxiv.org/abs/2003.12694). arXiv:2003.12694 [cs].\n* Maas et al. (2011) A. L. Maas, R. E. Daly, P. T. Pham, D. Huang, A. Y. Ng, and C. Potts. Learning Word Vectors for Sentiment Analysis. In _Proceedings of the 49th Annual Meeting of the Association for Computational Linguistics: Human Language Technologies_, pages 142-150, Portland, Oregon, USA, June 2011. Association for Computational Linguistics. URL [https://aclanthology.org/P11-1015](https://aclanthology.org/P11-1015).\n* Maddison et al. (2017) C. J. Maddison, A. Mnih, and Y. W. Teh. The Concrete Distribution: A Continuous Relaxation of Discrete Random Variables. In _ICLR_. arXiv, Mar. 2017. doi: 10.48550/arXiv.1611.00712. URL [http://arxiv.org/abs/1611.00712](http://arxiv.org/abs/1611.00712). arXiv:1611.00712 [cs, stat].\n* Mangrulka et al. (2023) S. Mangrulka, S. Gugger, L. Debut, Y. Belkada, and S. Paul. PEFT: State-of-the-art Parameter-Efficient Fine-Tuning methods, May 2023. URL [https://github.com/huggingface/peft.original-date](https://github.com/huggingface/peft.original-date): 2022-11-25T03:51:09Z.\n* McCloskey and Cohen (1989) M. McCloskey and N. J. Cohen. Catastrophic Interference in Connectionist Networks: The Sequential Learning Problem. In G. H. Bower, editor, _Psychology of Learning and Motivation_, volume 24, pages 109-165. Academic Press, Jan. 1989. doi: 10.1016/S0079-7421(08)60536-8. URL [https://www.sciencedirect.com/science/article/pii/S0079742108605368](https://www.sciencedirect.com/science/article/pii/S0079742108605368).\n* Meta FAIR Diplomacy Team et al. (2022) Meta FAIR Diplomacy Team, A. Bakhtin, N. Brown, E. Dinan, G. Farina, C. Flaherty, D. Fried, A. Goff, J. Gray, H. Hu, A. P. Jacob, M. Komeili, K. Konath, M. Kwon, A. Lerer, M. Lewis, A. H. Miller, S. Mitts, A. Renduchintala, S. Roller, D. Rowe, W. Shi, J. Spisak, A. Wei, D. Wu, H. Zhang, and M. Zijlstra. Human-level play in the game of Diplomacy by combining language models with strategic reasoning. _Science_, Nov. 2022. doi: 10.1126/science.ade9097. URL [https://www.science.org/doi/10.1126/science.ade9097](https://www.science.org/doi/10.1126/science.ade9097). Publisher: American Association for the Advancement of Science.\n* Nikishin et al. (2022) E. Nikishin, M. Schwarzer, P. D'Oro, P.-L. Bacon, and A. Courville. The Primacy Bias in Deep Reinforcement Learning. In _Proceedings of the 39th International Conference on Machine Learning_, pages 16828-16847. PMLR, June 2022. URL [https://proceedings.mlr.press/v162/nikishin22a.html](https://proceedings.mlr.press/v162/nikishin22a.html). ISSN: 2640-3498.\n* OpenAI (2022) OpenAI. ChatGPT: Optimizing Language Models for Dialogue, Nov. 2022. URL [https://webcache.googleusercontent.com/search?q=cache:qLONB_tjydcJ:https://openai.com/blog/chatgpt/&cd=1&hl=en&ct=clnk&gl=ca](https://webcache.googleusercontent.com/search?q=cache:qLONB_tjydcJ:https://openai.com/blog/chatgpt/&cd=1&hl=en&ct=clnk&gl=ca).\n\nM. Ott, S. Edunov, A. Baevski, A. Fan, S. Gross, N. Ng, D. Grangier, and M. Auli. fairseq: A Fast, Extensible Toolkit for Sequence Modeling. In _Proceedings of the 2019 Conference of the North_, pages 48-53, Minneapolis, Minnesota, 2019. Association for Computational Linguistics. doi: 10.18653/v1/N19-4009. URL [http://aclweb.org/anthology/N19-4009](http://aclweb.org/anthology/N19-4009).\n* Ouyang et al. (2022) L. Ouyang, J. Wu, X. Jiang, D. Almeida, C. L. Wainwright, P. Mishkin, C. Zhang, S. Agarwal, K. Slama, A. Ray, J. Schulman, J. Hilton, F. Kelton, L. Miller, M. Simens, A. Askell, P. Welinder, P. Christiano, J. Leike, and R. Lowe. Training language models to follow instructions with human feedback, Mar. 2022. URL [http://arxiv.org/abs/2203.02155](http://arxiv.org/abs/2203.02155). arXiv:2203.02155 [cs].\n* Paszke et al. (2019) A. Paszke, S. Gross, F. Massa, A. Lerer, J. Bradbury, G. Chanan, T. Killeen, Z. Lin, N. Gimelshein, L. Antiga, A. Desmaison, A. Kopf, E. Yang, Z. DeVito, M. Raison, A. Tejani, S. Chilamkurthy, B. Steiner, L. Fang, J. Bai, and S. Chintala. PyTorch: An Imperative Style, High-Performance Deep Learning Library, Dec. 2019. URL [http://arxiv.org/abs/1912.01703](http://arxiv.org/abs/1912.01703). arXiv:1912.01703 [cs, stat].\n* Post (2018) M. Post. A Call for Clarity in Reporting BLEU Scores. In _Proceedings of the Third Conference on Machine Translation: Research Papers_, pages 186-191, Brussels, Belgium, Oct. 2018. Association for Computational Linguistics. doi: 10.18653/v1/W18-6319. URL [https://aclanthology.org/W18-6319](https://aclanthology.org/W18-6319).\n* Radford et al. (2019) A. Radford, J. Wu, R. Child, D. Luan, D. Amodei, and I. Sutskever. Language Models are Unsupervised Multitask Learners, 2019. URL [https://www.semanticscholar.org/paper/Language-Models-are-Unsupervised-Multitask-Learners-Radford-Wu/9405cc0d6169988371b2755e573cc28650d14dfe](https://www.semanticscholar.org/paper/Language-Models-are-Unsupervised-Multitask-Learners-Radford-Wu/9405cc0d6169988371b2755e573cc28650d14dfe).\n* Ramamurthy et al. (2022) R. Ramamurthy, P. Ammanabrolu, K. Brantley, J. Hessel, R. Sifa, C. Bauckhage, H. Hajishirzi, and Y. Choi. Is Reinforcement Learning (Not) for Natural Language Processing: Benchmarks, Baselines, and Building Blocks for Natural Language Policy Optimization. In _ICLR_. arXiv, 2022. URL [http://arxiv.org/abs/2210.01241](http://arxiv.org/abs/2210.01241). arXiv:2210.01241 [cs].\n* Rita et al. (2022) M. Rita, C. Tallec, P. Michel, J.-B. Grill, O. Pietquin, E. Dupoux, and F. Strub. Emergent Communication: Generalization and Overfitting in Lewis Games, Sept. 2022. URL [http://arxiv.org/abs/2209.15342](http://arxiv.org/abs/2209.15342). arXiv:2209.15342 [cs, math].\n* Robins (1995) A. Robins. Catastrophic Forgetting, Rehearsal and Pseudorehearsal. _Connection Science_, 7(2):123-146, June 1995. ISSN 0954-0091. doi: 10.1080/09540099550039318. URL [https://doi.org/10.1080/09540099550039318](https://doi.org/10.1080/09540099550039318). Publisher: Taylor & Francis _eprint: [https://doi.org/10.1080/09540099550039318](https://doi.org/10.1080/09540099550039318).\n* Rolnick et al. (2019) D. Rolnick, A. Ahuja, J. Schwarz, T. P. Lillicrap, and G. Wayne. Experience Replay for Continual Learning, Nov. 2019. URL [http://arxiv.org/abs/1811.11682](http://arxiv.org/abs/1811.11682). arXiv:1811.11682 [cs, stat].\n* Sanh et al. (2020) V. Sanh, L. Debut, J. Chaumond, and T. Wolf. DistilBERT, a distilled version of BERT: smaller, faster, cheaper and lighter, Feb. 2020. URL [http://arxiv.org/abs/1910.01108](http://arxiv.org/abs/1910.01108). arXiv:1910.01108 [cs].\n* Schulman et al. (2015) J. Schulman, S. Levine, P. Moritz, M. I. Jordan, and P. Abbeel. Trust Region Policy Optimization. In _ICML_. arXiv, 2015. doi: 10.48550/arXiv.1502.05477. URL [http://arxiv.org/abs/1502.05477](http://arxiv.org/abs/1502.05477). arXiv:1502.05477 [cs].\n* Schulman et al. (2017) J. Schulman, F. Wolski, P. Dhariwal, A. Radford, and O. Klimov. Proximal Policy Optimization Algorithms, Aug. 2017. URL [http://arxiv.org/abs/1707.06347](http://arxiv.org/abs/1707.06347). arXiv:1707.06347 [cs].\n* Singh et al. (2022) A. K. Singh, D. Ding, A. Saxe, F. Hill, and A. K. Lampinen. Know your audience: specializing grounded language models with the game of Dixit, June 2022. URL [http://arxiv.org/abs/2206.08349](http://arxiv.org/abs/2206.08349). arXiv:2206.08349 [cs].\n* Steinert-Threlkeld et al. (2022) S. Steinert-Threlkeld, X. Zhou, Z. Liu, and C. M. Downey. Emergent Communication Fine-tuning (EC-FT) for Pretrained Language Models. In _Emergent Communication Workshop at ICLR 2022_, June 2022. URL [https://openreview.net/forum?id=SUqrM7WR7W5](https://openreview.net/forum?id=SUqrM7WR7W5).\n* Steinert-Threlkeld et al. (2020)N. Stiennon, L. Ouyang, J. Wu, D. M. Ziegler, R. Lowe, C. Voss, A. Radford, D. Amodei, and P. Christiano. Learning to summarize from human feedback. In _NeurIPS_. arXiv, 2020. URL [http://arxiv.org/abs/2009.01325](http://arxiv.org/abs/2009.01325). arXiv:2009.01325 [cs].\n* Sutskever et al. (2014) I. Sutskever, O. Vinyals, and Q. V. Le. Sequence to Sequence Learning with Neural Networks. In _Neural Information Processing Systems_. arXiv, Dec. 2014. URL [http://arxiv.org/abs/1409.3215](http://arxiv.org/abs/1409.3215). arXiv:1409.3215 [cs].\n* To et al. (2023) H. Q. To, N. D. Q. Bui, and M. Nguyen. CodeCarybara: Open Source LLaMA Model that Follow Instruction-Tuning for Code Generation., May 2023. URL [https://github.com/FSoft-AI4Code/CodeCarybara](https://github.com/FSoft-AI4Code/CodeCarybara). original-date: 2023-04-21T10:28:53Z.\n* Touvron et al. (2023) H. Touvron, T. Lavril, G. Izacard, X. Martinet, M.-A. Lachaux, T. Lacroix, B. Roziere, N. Goyal, E. Hambro, F. Azhar, A. Rodriguez, A. Joulin, E. Grave, and G. Lample. LLaMA: Open and Efficient Foundation Language Models, Feb. 2023. URL [http://arxiv.org/abs/2302.13971](http://arxiv.org/abs/2302.13971). arXiv:2302.13971 [cs] version: 1.\n* Utiyama and Isahara (2007) M. Utiyama and H. Isahara. A Comparison of Pivot Methods for Phrase-Based Statistical Machine Translation. In _Human Language Technologies 2007: The Conference of the North American Chapter of the Association for Computational Linguistics; Proceedings of the Main Conference_, pages 484-491, Rochester, New York, Apr. 2007. Association for Computational Linguistics. URL [https://aclanthology.org/N07-1061](https://aclanthology.org/N07-1061).\n* Vaswani et al. (2017) A. Vaswani, N. Shazeer, N. Parmar, J. Uszkoreit, L. Jones, A. N. Gomez, L. Kaiser, and I. Polosukhin. Attention is All you Need. In _Neural Information Processing Systems_, page 11, 2017.\n* von Werra et al. (2023) L. von Werra, Y. Belkada, L. Tunstall, E. Beeching, T. Thrush, and N. Lambert. TRL: Transformer Reinforcement Learning, 2023. URL [https://github.com/lvwerra/trl](https://github.com/lvwerra/trl).\n* Williams (1992) R. J. Williams. Simple statistical gradient-following algorithms for connectionist reinforcement learning. _Machine Learning_, page 28, 1992.\n* Wolf et al. (2020) T. Wolf, L. Debut, V. Sanh, J. Chaumond, C. Delangue, A. Moi, P. Cistac, T. Rault, R. Louf, M. Funtowicz, J. Davison, S. Shleifer, P. von Platen, C. Ma, Y. Jernite, J. Plu, C. Xu, T. Le Scao, S. Gugger, M. Drame, Q. Lhoest, and A. Rush. Transformers: State-of-the-Art Natural Language Processing. In _Proceedings of the 2020 Conference on Empirical Methods in Natural Language Processing: System Demonstrations_, pages 38-45, Online, Oct. 2020. Association for Computational Linguistics. doi: 10.18653/v1/2020.emnlp-demos.6. URL [https://aclanthology.org/2020.emnlp-demos.6](https://aclanthology.org/2020.emnlp-demos.6).\n* Zhou et al. (2022) H. Zhou, A. Vani, H. Larochelle, and A. Courville. Fortuitous Forgetting in Connectionist Networks. In _International Conference on Learning Representations_, Mar. 2022. URL [https://openreview.net/forum?id=ei3SY1_zYsE](https://openreview.net/forum?id=ei3SY1_zYsE).\n* Ziegler et al. (2019) D. M. Ziegler, N. Stiennon, J. Wu, T. B. Brown, A. Radford, D. Amodei, P. Christiano, and G. Irving. Fine-Tuning Language Models from Human Preferences, 2019. URL [http://arxiv.org/abs/1909.08593](http://arxiv.org/abs/1909.08593). arXiv:1909.08593 [cs, stat].\n\n## Appendix A Translation Game\n\n### Experimental Details\n\nWe implement the translation game in the fairseq library (Ott et al., 2019) on top of Pytorch (Paszke et al., 2019). All experiments are run with 5 seeds where each run uses a single 16G V100 GPU. All plots show the mean and standard deviation over seeds. We score BLEU using detokenized sacreBLEU (Post, 2018). Plots are made with Matplotlib (Hunter, 2007)\n\n### Translation Game with Transformers\n\nNote that Lee et al. (2019) (as well as previous work on the Translation Game) use seq2seq (Sutskever et al., 2014) LSTMs (Hochreiter and Schmidhuber, 1997) with attention (Bahdanau et al., 2015). For consistency with other experiments and relevance to mainstream research we switch to a Transformer architecture. LSTMs results are similar and results are available in Appendix A.3. Also note that we use REINFORCE as our estimator but Gumbel-Softmax (Jang et al., 2017; Maddison et al., 2017) is also feasible (Lu et al., 2020) and preliminary results have been similar.\n\nWe follow Lee et al. (2019) for preprocessing both IWLST and Multi30k. We use the iwslt-de-en architecture from the fairseq library (Ott et al., 2019) with their default IWSLT DE-EN hyperparameters and training, specifically training with AdamW (Loshchilov and Hutter, 2022) using default hyperparameters. We pretrain our models on IWSLT following Lee et al. (2019) and early-stop on the tst2013 validation set. Our final validation scores are 43.85 BLEU for FR\\(\\rightarrow\\)EN and 29.4 BLEU for EN\\(\\rightarrow\\)DE.\n\nWe report all final validation scores in Table 4 and all curves in Figure 7.\n\nGiven that we have a learning reward model (EN\\(\\rightarrow\\)DE), iteratively resetting our FR\\(\\rightarrow\\)EN model to its pretrained weights can be an effective strategy (Rita et al., 2022). We report this as Reset and find it performs quite well but not as well as Elastic Reset. We did not include these results in the main paper because we focused on methods that could also work with fixed reward models.\n\n\\begin{table}\n\\begin{tabular}{l c c} \\hline \\hline  & \\(\\uparrow\\) FR\\(\\rightarrow\\)EN\\(\\rightarrow\\)DE & \\(\\uparrow\\) FR\\(\\rightarrow\\)EN \\\\ \\hline pretrained & 23.8 & 36.2 \\\\ frozen sender & 30.8\\(\\pm\\)0.2 & **36.3\\(\\pm\\)0.1** \\\\ REINFORCE & **33.2\\(\\pm\\)0.3** & 29.6\\(\\pm\\)0.3 \\\\ + SIL & 28.2\\(\\pm\\)0.4 & 27.3\\(\\pm\\)4.4 \\\\ + multitask (S2P) & 32.2\\(\\pm\\)0.3 & 35.2\\(\\pm\\)1.0 \\\\ + KI pretrained & **33.2\\(\\pm\\)0.2** & 30.8\\(\\pm\\)0.4 \\\\ + Reset & 32.5\\(\\pm\\)0.1 & 33.3\\(\\pm\\)0.1 \\\\ + Reset to EMA & 32.9\\(\\pm\\)0.1 & **36.3\\(\\pm\\)0.1** \\\\ + Elastic Reset & 33.0\\(\\pm\\)0.1 & **36.3\\(\\pm\\)0.1** \\\\ \\hline \\hline \\end{tabular}\n\\end{table}\nTable 4: Translation Game final validation scores\n\nFigure 7: All transformer model methods on the Translation Game. We measure (a) Task Performance with FR\\(\\rightarrow\\)EN\\(\\rightarrow\\)DE BLEU and (b) Language Drift with FR\\(\\rightarrow\\)EN BLEU, on the validation set during finetuning. We plot the mean and show error bars for standard deviation over 5 seeds. To compare how methods do on both metrics, we plot (c) the best achieved drift vs task performance across finetuning.\n\n### Translation Game with LSTMs\n\nWe follow Lee et al. (2019) for preprocessing and pretraining on IWSLT. We compare our pretraining results to both Lee et al. (2019) and Lu et al. (2020) in Table 5\n\nNext we finetune on Multi30k using the same hyperparameters as Lu et al. (2020). We plot results in Table 6 and compare to published numbers from previous work. As usual, drift is the negative change in FR\\(\\rightarrow\\)EN BLEU from the pretrained model. Task performance is the positive change in FR\\(\\rightarrow\\)EN\\(\\rightarrow\\)DE BLEU from the pretrained models. Combined is the sum of drift and task performance. We show the pretrained, baseline, and best-performing model from previous work. Note that our results are not directly comparable to previous work because we evaluate using detokenized sacreBLEU (Post, 2018) whereas previous work wrote their own BLEU evaluation code and did not detokenize.\n\nOur results for the baseline are notably better than Lee et al. (2019), Lu et al. (2020). The only difference between our code and theirs as far as we can tell is (1) we use an exponential moving average baseline for REINFORCE whereas (Lee et al., 2019) use an Actor-Critic method, (2) Lu et al. (2020) uses 0.1 gradient clipping and we do not use gradient clipping (3) in preprocessing Multi30k, we first tokenize (Koehn et al., 2007) then lowercase whereas previous works did the opposite order.\n\nA more reasonable explanation for the improvement in results is how we choose to evaluate. Previous work simply ran all methods for the same number of updates but this doesn't account for, even implicit, hyperparameter optimization. Previous methods show that the baseline's FR\\(\\rightarrow\\)EN\\(\\rightarrow\\)DE BLEU scores plateau and there is significant language drift in FR\\(\\rightarrow\\)EN without real improvements to the task score. We hypothesize that these extra training episodes only serve to increase the drift without measuring what we actually care about: performance gain for drift. Since the number of updates is arbitrary, we believe that early stopping on a reasonable metric is a better evaluation protocol and choose hyperparameters such that methods plateau at the end.\n\nWe also note that our results with the SIL method of Lu et al. (2020) are negative. We do not manage to gain any improvement in performance. We collaborated with the authors of Lu et al. (2020) for many months but, in our setup using the fairseq library (Ott et al., 2019) could not reproduce their results. One of their fundamental results is that a student sender can outperform a teacher sender that it is distilling from. We could not reproduce this and believe it is a difference in the pretrained models.\n\n\\begin{table}\n\\begin{tabular}{l l c c c c c} \\hline \\hline  & Method & FR\\(\\rightarrow\\)EN & FR\\(\\rightarrow\\)EN\\(\\rightarrow\\)DE & Drift & Perf & Combined \\\\ \\hline Lee et al. (2019) & pretrained & 27.2 & 16.3 & & & \\\\  & REINFORCE & \\(12.4\\pm 0.7\\) & \\(24.5\\pm 1.5\\) & -14.8 & +8.2 & -6.6 \\\\  & + LM & \\(23.6\\pm 1.1\\) & \\(27.7\\pm 0.4\\) & -3.6 & +11.4 & +7.8 \\\\  & + LM + G & \\(24.8\\pm 0.4\\) & \\(28.1\\pm 0.7\\) & -2.4 & +11.8 & +9.4 \\\\ \\hline Lu et al. (2020) & pretrained & 29.4 & 15.7 & & & \\\\  & Gumbel-softmax & \\(14.5\\pm 0.8\\) & \\(27.1\\pm 0.1\\) & -14.9 & +11.4 & -3.5 \\\\  & SIL & \\(29.4\\pm 0.3\\) & \\(28.3\\pm 0.2\\) & 0 & +12.6 & +12.6 \\\\ \\hline\n**Ours** & pretrained & 32.6 & 18.0 & & & \\\\  & frozen-sender & \\(32.6\\pm 0\\) & \\(25.1\\pm 0.1\\) & 0 & +7.1 & +7.1 \\\\  & REINFORCE & \\(30.0\\pm 0.2\\) & \\(30.3\\pm 0.2\\) & -2.6 & +12.3 & +9.7 \\\\  & LM \\(\\lambda=0.01\\) & \\(31.5\\pm 0.1\\) & \\(30.2\\pm 0.2\\) & -1.1 & +12.2 & +11.1 \\\\  & Multitask \\(\\lambda=0.1\\) & \\(32.9\\pm 0.2\\) & \\(28.5\\pm 0.3\\) & +0.3 & +10.5 & +10.8 \\\\  & SIL & \\(27.7\\pm 0.7\\) & \\(24.3\\pm 0.1\\) & -4.9 & 6.3 & +1.4 \\\\  & Elastic Reset & \\(32.6\\pm 0.1\\) & \\(30.0\\pm 0.1\\) & 0 & +12.0 & **+12.0** \\\\ \\hline \\hline \\end{tabular}\n\\end{table}\nTable 6: BLEU scores and \\(\\pm\\) standard deviation on the Multi30k Translation Game using IWSLT-pretrained LSTM models.\n\n\\begin{table}\n\\begin{tabular}{l c c} \\hline \\hline  & FR\\(\\rightarrow\\)EN & EN\\(\\rightarrow\\)DE \\\\ \\hline Lee et al. (2019) & 34.1 & 22.0 \\\\ Lu et al. (2020) & 32.2 & 20.2 \\\\ Ours & 38.5 & 23.2 \\\\ \\hline \\hline \\end{tabular}\n\\end{table}\nTable 5: BLEU score of IWSLT-pretrained LSTM models on IWSLT 2013 validation set\n\n## Appendix B IMDB Mock Sentiment\n\n### Experimental Details\n\nWe run experiments and implement our method in the RL4LMs library (Rammurthy et al., 2022). All experiments are run with 5 seeds where run uses a single 40G A100 GPU. All plots show the mean and standard error over seeds.\n\nFor PPO, we use the default hyperparemetres provided by Ramamurthy et al. (2022). We also compared to NLPO, we found the defaults had a mistake and after communication with Ramamurthy et al. (2022) we changed the learning rate to 1e-6 and target update iterations to 50. We plot results in Figure 8. This still did not manage to reproduce the original NLPO test scores from Ramamurthy et al. (2022) but we found that our validation curves matched their provided curves in the appendix. After communications (Ammanabrolu, 2023), we both agreed that we should use our reproducible test scores for NLPO in lieu of the original test scores.\n\nOur best Elastic Reset hyperparameters are the default PPO parameters (gpt2_ppo.yml) from RL4LMs (Rammurthy et al., 2022) with a few modifications: target kl is set from 0.5 to 1.0 and KL coefficient is set much lower to 0.001. We use an EMA decay of 0.995 and reset every 17 epochs. Configs to reproduce the experiments can be found in the RL4LMs folder under scripts/training/task_configs/imdb_text_continuation\n\n## Appendix C StackLlama\n\n### Experimental Details\n\nWe follow the original StackLlama and use the trl library from Huggingface on top of the transformers library (Wolf et al., 2020) to train the model on the StackExchange dataset, loaded with datasets\n\nFigure 8: IMDB mock sentiment validation set scores for Elastic Reset, PPO, and NLPO. We measure (a) Language Drift and (b) Task Performance via Semantic Score on the validation set over finetuning. All methods also include a KL penalty. We plot mean and standard error across 5 seeds.\n\nFigure 9: Rolling Window Mean (window = 10) version of the StackLLaMA results[Lhoest et al., 2021]. We use the original authors' LoRA adapter weights to create our supervised model LLaMA-7B-SE but train our own reward model as there were issues loading the pretrained reward model. As noted by [Beeching et al., 2023], the reward modelling task is difficult enough that humans struggle with it and our final model achieves 64% accuracy compared to the original 67%. Although we follow the original authors method and use their codebase, we note that our results may be different but valid. There have been many updates and fixes to the tr1 codebase since the authors' original blog post and specifically a possible issue in the code for creating reward models could have affected the original authors' run.\n\nFurthermore, to speed up training, we used a 2x smaller KL coeffient and ran with half the number of GPUs. Specifically, RL training is run on 4x 80G A100 GPUs for 20 hours using Accelerate [Gugger et al., 2022]. We evaluate perplexity of the model on the validation set used in supervised finetuning, 4000 examples from the supervised training dataset. The configs for reproducing our experiments are in the tr1 library folder under examples/stackllama/scripts/configs\n\nFor visual clarity, we provide a version of our StackLLaMA results with a rolling window mean in Figure 9\n\n### HumanEval\n\nOur measure of alignment tax, HumanEval [Chen et al., 2021], is a programming benchmark where each question was hand-written by humans (OpenAI engineers) to be unseen in training data. We believe these questions were unseen in LLaMA training data as well. We prompt the model with the question and it writes the corresponding code. The measure of success is functionally correct code i.e. we actually execute the code LLaMA wrote and see if it gets the right output. Since we decode by sampling, we generate \\(N=100\\) continuations of the prompt and then follow Chen et al. [2021] to estimate how often our model would get the right answer on the first generated continuation (pass @ 1) and within the first 10 generated continuations (pass @ 10). We evaluate HumanEval using CodeCapbyara's evaluation harness [To et al., 2023].\n\n## Appendix D Ablations Full Graphs\n\n### Reset Number\n\nFigure 10: Plotting PPO vs Elastic Reset on IMDB but splitting the results visually between resets. Users with less compute can choose fewer resets. We didn’t find improved results after 2 resets for IMDB\n\n### Keset Type\n\n### KL Coefficient\n\nAs shown in Figure 5, Elastic Reset performance in relatively unaffected by the presence or absence of KL. In contrast, PPO and REINFORCE performance is known to be sensitive [Lu et al., 2020]. This raises the question of whether different coefficients of the KL loss would lead to different pareto frontiers. We find in Figure 12 that both Multitask and KL with Pretrained methods do have slightly different pareto frontiers. Choosing a\n\nFigure 11: Ablation of Elastic Reset, Reset to EMA, and Reset to Init on IMDB. We plot the mean and show error bars for standard error over 5 seeds.\n\nFigure 12: Ablation of KL and Multitask coefficients on Translation Game. We exclude KL 0.1 from the pareto graph since it fails. We plot the mean and show error bars for standard error over 5 seeds.\n\nFigure 13: Ablation of KL coefficients for Elastic Reset on Translation Game. We plot the mean and show error bars for standard error over 5 seeds.\n\nFigure 16: Ablation with/without KL penalty for PPO and Elastic Reset on IMDB. We plot the mean and show error bars for standard error over 5 seeds. For clarity, we cut off PPO without KL to four data points on the pareto graph since it diverges too much\n\nFigure 14: Ablation of PPO and NLPO coefficients on IMDB. We do more ablations of PPO since NLPO is always similar but worse than PPO. We plot the mean and show error bars for standard error over 5 seeds.\n\nFigure 15: Ablation of KL coefficients for Elastic Reset on IMDB. We plot the mean and show error bars for standard error over 5 seeds.\n\n### Decay Rate\n\n#### d.4.1 Reset Frequency\n\n## Appendix E Explaining Elastic Reset\n\nHere we provide empirical intuition for why Elastic Reset is effective\n\n### Explaining Resets: Value Function\n\nDrift is a problem of noisy optimization, and Elastic Reset tackles this in two ways.\n\nThe first is by re-training with an improved value function. Though we begin with a sequence-level reward model, over training we learn a *token-level* value function. We believe that the value\n\nFigure 19: Ablation of reset frequency for Elastic Reset on IMDB. We plot the mean and show error bars for standard error over 5 seeds. Although 17 is optimal, results are relatively robust to choice of this hyperparameter\n\nFigure 17: Ablation of decay coefficients for Elastic Reset on Translation Game. Original decay is 0.99. We plot the mean and show error bars for standard error over 5 seeds.\n\nFigure 18: Ablation of decay coefficient for Elastic Reset on IMDB. Original decay is 0.995. We plot the mean and show error bars for standard error over 5 seeds.\n\nfunction greatly improves over time so early training may have exacerbated drift because it uses the early, sub-optimal value function. With each reset of the policy while maintaining the value function, we re-train with less noise and more direct gradients to alignment. Another way, both the reward model and optimal value function point towards high reward, but not in the same way. The reward model points towards high reward but not necessarily in the most direct way. Using the frozen reward model as our value function leads to equivalent reward but higher drift compared to using an optimal value function.\n\nTo demonstrate this phenomenon, we compared what happens if we don't train our value function at all but use a frozen reward model as our value function. Since our value function is GPT-2 but our reward model is DistilBERT, we first train a GPT-2 reward model similar to DistilBERT. We then compare regular PPO with a frozen value function i.e. just the sequence-level reward model vs a training value function. In Figure 20 we find that the learning value function is clearly better than the frozen one. They both reach the same reward but a more optimal value function leads to less drift for the same reward.\n\n### Explaining Elastic Reset: The Benefit of EMA\n\nThe second is by smoothing optimization with an EMA. Work in other fields e.g. DINO in SSL, has leveraged EMA for stability and improved generalization. By resetting to an EMA and resetting our EMA, we further smooth the gradients to alignment. To demonstrate, we run PPO as usual but keep track of its EMA, without resetting to it. We then plot the accuracy of the online and EMA networks over training. To make the effect larger, we use a smaller KL \\(\\beta=0.01\\) and train our PPO for 100 epochs and plot results in Figure 21. We find that just keeping the EMA model is an effective way to improve the task / drift tradeoff. But we also see that it is noticeably slower than our method. PPO - Online needs to reach nearly the maximum possible reward for its EMA to improve only slightly on performance.\n\nFigure 21: PPO trained on IMDB with a lower KL coefficient for 100 epochs, comparing its online network performance vs EMA model performance. We cut off PPO - online to just three data points in the pareto graph for clarity and visual scale.\n\nFigure 20: Ablation of learning vs frozen value function for PPO on IMDB.", "response": "## Review\n\n### Summary\n\nThe paper presents a method called Elastic Reset, designed to address the issues of language drift and reward overfitting in reinforcement learning with human feedback (RLHF). By periodically resetting the model parameters to an exponentially moving average (EMA), the authors aim to maintain a balance between optimizing for rewards and retaining essential language capabilities. The method has shown promising results across various tasks, including pivot translation and QA generation on StackExchange, outperforming traditional approaches like PPO with KL penalties. While the technique is simple and effective, the paper has raised questions regarding the robustness of its experimental evaluation and the clarity of certain concepts.\n\n### Strengths\n\n- The proposed approach is simple and easy to implement.\n- The method demonstrates effective results across various RLHF benchmarks, showcasing broad applicability.\n- Good comparisons with sensible baselines and well-structured ablation studies.\n- Clear definition of language drift, with practical examples.\n- The paper is well-written and presents the problem setting clearly.\n- The use of Pareto frontier graphs effectively illustrates the trade-off between task performance and language drift.\n\n### Weaknesses\n\n- The method lacks novelty, as the idea of using reset mechanisms is common in other contexts.\n- The definition of 'drift' is not clearly articulated, and the motivation for the proposed method could be better justified.\n- Concerns about robustness under different compute constraints and the evaluation protocol used, which relies heavily on models trained with similar reward mechanisms.\n- Limited experiments conducted on smaller models and datasets; broader evaluations are needed for stronger claims.\n- Insufficient exploration of the timing and frequency of resets in relation to model performance.\n\n### Questions\n\n- How did you decide on resetting during the first and second stages?\n- Why reduce the KL coefficient, and how does it influence comparisons with other baselines?\n- What happens if only specific parameters are reset during training?\n- Can the authors clarify the mathematical connections between Elastic Reset and existing methods like distillation or KL penalties?\n- How robust is the method under varying compute budgets?\n\n### Soundness\n\n**Score:** 3\n\n**Description:** Good; the paper presents a solid methodology and results, but there are concerns about robustness and clarity.\n\n### Presentation\n\n**Score:** 3\n\n**Description:** Good; the paper is generally well-written, though some explanations could be clearer and figures improved for clarity.\n\n### Contribution\n\n**Score:** 3\n\n**Description:** Good; the paper addresses a significant issue in RLHF, but could benefit from stronger justification and broader experimentation.\n\n### Rating\n\n**Score:** 6\n\n**Description:** Weak Accept; the paper is technically solid with moderate-to-high impact potential but requires further evaluation and clarity.\n\n### Paper Decision\n\n**Decision:** Accept (poster)\n\n**Reasons:** The paper addresses an important problem in the field of RLHF with a method that shows promise in improving performance while mitigating language drift. Despite some weaknesses in clarity and robustness of experimental results, the strengths of the paper and its contributions to the community warrant acceptance.\n"}
{"query": "You are a highly experienced, conscientious, and fair academic reviewer, please help me review this paper. The review should be organized into nine sections: \n1. Summary: A summary of the paper in 100-150 words.\n2. Strengths/Weaknesses/Questions: The Strengths/Weaknesses/Questions of paper, which should be listed in bullet points, with each point supported by specific examples from the article where possible.\n3. Soundness/Contribution/Presentation: Rate the paper's Soundness/Contribution/Presentation, and match this score to the corresponding description from the list below and provide the result. The possible scores and their descriptions are: \n    1 poor\n    2 fair\n    3 good\n    4 excellent\n4. Rating: Give this paper an appropriate rating, match this rating to the corresponding description from the list below and provide the result. The possible Ratings and their descriptions are: \n    1 strong reject\n    2 reject, significant issues present\n    3 reject, not good enough\n    4 possibly reject, but has redeeming facets\n    5 marginally below the acceptance threshold\n    6 marginally above the acceptance threshold\n    7 accept, but needs minor improvements \n    8 accept, good paper\n    9 strong accept, excellent work\n    10 strong accept, should be highlighted at the conference   \n5. Paper Decision: It must include the Decision itself(Accept or Reject) and the reasons for this decision, based on the criteria of originality, methodological soundness, significance of results, and clarity and logic of presentation.\n\nHere is the template for a review format, you must follow this format to output your review result:\n**Summary:**\nSummary content\n\n**Strengths:**\n- Strength 1\n- Strength 2\n- ...\n\n**Weaknesses:**\n- Weakness 1\n- Weakness 2\n- ...\n\n**Questions:**\n- Question 1\n- Question 2\n- ...\n\n**Soundness:**\nSoundness result\n\n**Presentation:**\nPresentation result\n\n**Contribution:**\nContribution result\n\n**Rating:**\nRating result\n\n**Paper Decision:**\n- Decision: Accept/Reject\n- Reasons: reasons content\n\n\nPlease ensure your feedback is objective and constructive. The paper is as follows:\n\n# Dis-inhibitory neuronal circuits can control the sign of synaptic plasticity\n\nJulian Rossbroich\\({}^{1,2}\\)&Friedemann Zenke\\({}^{1,2}\\)\n\n{firstname.lastname}@fmi.ch\n\n\\({}^{1}\\) Friedrich Miescher Institute for Biomedical Research, Basel, Switzerland\n\n\\({}^{2}\\) Faculty of Science, University of Basel, Basel, Switzerland\n\n###### Abstract\n\nHow neuronal circuits achieve credit assignment remains a central unsolved question in systems neuroscience. Various studies have suggested plausible solutions for back-propagating error signals through multi-layer networks. These purely functionally motivated models assume distinct neuronal compartments to represent local error signals that determine the sign of synaptic plasticity. However, this explicit error modulation is inconsistent with phenomenological plasticity models in which the sign depends primarily on postsynaptic activity. Here we show how a plausible microcircuit model and Hebbian learning rule derived within an adaptive control theory framework can resolve this discrepancy. Assuming errors are encoded in top-down dis-inhibitory synaptic afferents, we show that error-modulated learning emerges naturally at the circuit level when recurrent inhibition explicitly influences Hebbian plasticity. The same learning rule accounts for experimentally observed plasticity in the absence of inhibition and performs comparably to back-propagation of error (BP) on several non-linearly separable benchmarks. Our findings bridge the gap between functional and experimentally observed plasticity rules and make concrete predictions on inhibitory modulation of excitatory plasticity.\n\n## 1 Introduction\n\nHow do neurons far away from the sensory periphery and motor output system update their connections to contribute to network computation meaningfully? This question, formally known as the \"credit assignment problem,\" is one of the outstanding questions in systems neuroscience. Classic learning theories assume synaptic plasticity in the brain is mainly Hebbian, i.e., changes in a synapse's efficacy depend merely on the pre- and postsynaptic activity of the neurons it connects [1]. A plethora of experiments in different brain areas support the notion of Hebbian plasticity [2, 3] and diverse phenomenological plasticity models quantitatively capture observed synaptic plasticity dynamics [4, 5, 6, 7, 8]. Theories of reinforcement learning suggested further extensions of Hebbian learning to three-factor rules that account for reward modulation, whereby a global modulatory factor, e.g., dopamine, explicitly influences the sign of plasticity [9, 10]. Again, there is ample experimental evidence for the role of neuromodulators in gating plasticity in various brain areas [11, 12]. Yet, recent work argues that global neuromodulation is insufficient to learn complex function mappings in large networks [13, 14] and that more fine-grained control over the sign of plasticity is required, as in the BP algorithm used in deep learning [15, 16, 17, 18]. This realization motivated several modeling studies on how biological networks could approximate BP [19, 20, 21, 22, 23, 24, 25]. One central assumption in virtually all of these models is that the sign of plasticity is explicitly modulated by a neuron-specific local error signal akin to the gradient-based update used in BP [26]. However, learning rules with explicit errormodulation are inconsistent with phenomenological models of Hebbian plasticity (Fig. 1), raising the question of how theories of bio-plausible credit assignment tie into the phenomenology of Hebbian learning.\n\nHere, we address this question using a normative control theory approach. We set out from a plausible dis-inhibitory circuit motif ubiquitously found in the brain [27; 28] and derive a Hebbian learning rule with an explicit inhibitory current dependence. We demonstrate that this learning rule accounts for key experimental observations _and_ allows for control over the sign of plasticity through top-down synaptic input to specific interneurons. Our work suggests that error signals could naturally be encoded in top-down inputs to inhibitory neurons and bridges the gap between normative theories of gradient-based learning and phenomenological models of Hebbian plasticity. Our main contributions are:\n\n* We extend Deep Feedback Control (DFC), a recent adaptive control theory framework for error-based learning [24], to a biologically plausible dis-inhibitory microcircuit motif.\n* We show how a Hebbian plasticity rule with an explicit inhibition dependence can naturally decode credit signals from this microcircuit.\n* We demonstrate that this rule enables error-based learning in hierarchical networks, naturally stabilizes runaway Hebbian plasticity, and resembles phenomenological plasticity rules under simulated experimental conditions.\n* Finally, we demonstrate that our learning rule performs comparable to BP in deep neural networks trained on computer vision benchmarks.\n\n## 2 Background and previous work\n\nIn this article we strive to reconcile phenomenological plasticity models constrained by experiments and models of biologically plausible credit assignment based on normative theories of gradient-based learning. In the following we review essential literature of both approaches.\n\nFigure 1: Explicit error modulation of the sign of plasticity is inconsistent with phenomenological plasticity models. **(a)** In neuronal circuits, top-down feedback connections target excitatory neurons, as well as inhibitory and dis-inhibitory circuits that have been implicated in gating of plasticity. In phenomenological plasticity models, the sign of plasticity is typically determined by postsynaptic quantities such as the membrane voltage [7], firing rate [6], or calcium concentration [8] without explicit error modulation. **(b)** In normative models, the sign of plasticity is typically subject to a hypothetical, explicit error modulation with little experimental evidence. Explicit error modulation makes specific predictions of the shape of the learning rule, (Supplementary Fig. S1; see Appendix A), at odds with experimentally observed plasticity (see Panel (a)). **(b)** Theories of bio-plausible gradient-based learning typically focus on approximating direct error-modulation as in BP, but differ in how errors are computed and relayed. Existing models suggest separate temporal phases, e.g. equilibrium propagation (EP) [29], putative compartments to represent error signals locally, e.g., dendritic error coding [22; 30], or burst-multiplexing [23; 31]. Still, there is little evidence for an explicit error signal that alters the sign of plasticity [26] and is consistent with phenomenological plasticity models.\n\n### Phenomenological synaptic plasticity models\n\nThere is widespread experimental support for the notion of Hebbian synaptic plasticity in the brain, captured in the form of classical long-term potentiation (LTP) and long-term depression (LTD) [2] or spike-timing-dependent plasticity (STDP) [3]. An extensive mathematical model catalog captures the phenomenology of these findings [4; 5; 6; 7; 8]. Common to most of the above models is that postsynaptic quantities define a plasticity threshold which separates LTD from LTP induction, which effectively determines the sign of the synaptic weight change [32; 33; 34] (Fig. 1a). A plethora of phenomenological plasticity models exist that capture such dependence on firing rate [6], voltage [7], and postsynaptic calcium concentration [8; 5]. For isolated neurons, classic work has demonstrated that Hebbian plasticity can extract principal components from structured data [35] or capture receptive field formation observed experimentally [36]. However, these models do not extend to deep hierarchical networks, nor can they account for the modulation of plasticity through local error signals required for solving the credit assignment problem. Thus, the mechanisms by which synaptic plasticity observed under experimental conditions could give rise to coordinated learning at the circuit- and network level remain elusive.\n\n### Models of biologically plausible credit assignment\n\nThe above models are contrasted by normative rules derived from gradient-based learning principles, which often aim at approximating BP [15] (Fig. 1b-c). A crucial aspect of BP is the separation of forward- and backward signaling, which algorithmically separates credit signaling from neuronal activity [37]. This separation presents a significant challenge for biologically plausible implementations, as it presumes a distinct separation through either learning phases or separate pathways.\n\nEquilibrium propagation (EP) offers one possible, if only partial, solution to this dilemma. EP posits that local errors are derived as variations in neuronal activity at a network equilibrium state [29]. Yet, classic EP still requires separate processing phases for each input, inconsistent with neurobiology. However, recent work suggests possible ways of alleviating the requirement for distinct phases through neural oscillations [38].\n\nAn alternative to separate phases for forward and backward passes is to separate them spatially. Predictive coding models [19; 25; 39] exemplify this idea, whereby errors are computed in dedicated neuron-specific error units by comparing each neuron's activity to a top-down prediction. Recent work has suggested that the electrotonically segregated apical dendrites of cortical pyramidal neurons [37; 40] could represent local errors in learning [21; 22].\n\nHowever, to approximate BP, a common theme across these models is their dependence on explicit error-modulation of plasticity [30] (Fig. 1b-c). While error-modulated learning rules prove functionally useful and, in some cases, can match the performance of BP, experimental evidence for their existence is inconclusive. In particular, they fall short of capturing established properties of experimentally observed plasticity, such as a threshold between LTD and LTP that is governed by postsynaptic activity [32; 33; 34; 41].\n\nFinally, Payeur et al. [23] proposed a unique multiplexing approach by encoding forward and feedback signals in the event and burst rate of output spike trains. Such burst-dependent plasticity rules capture essential facets of phenomenological plasticity [31] (Supplementary Fig. S1; see Appendix A). However, the model primarily applies to cortical Layer 5 pyramidal cells with electrically segregated dendritic trees. In contrast, it may not work for layer 2/3 neurons or other brain areas without segregated dendrites.\n\nSince most of the above models focused on approximating BP, they face another potential issue: they usually require weak feedback, which causes only a slight perturbation, or nudge, of the input-driven equilibrium. This requirement contrasts significantly with a wealth of experimental literature suggesting that feedback connections in the brain substantially influence neuronal activity [42; 43; 44]. Rather than striving to find biologically plausible separations between forward and backward signaling, recent modeling studies used an approach grounded in adaptive control theory [45; 46; 47; 25; 48]. These models leverage strong feedback signals to steer neuronal activity to align with a given target output. While strong feedback aligns more closely with neurobiological observations, these models are also dependent on explicitly error-modulated learning rules, wherein the error is computed from either the difference between the controlled and uncontrolled states of neuronal activity similar to EP or the difference between activity in segregated neuronal compartments, as in dendritic error coding.\n\nStill, it remains to be seen how such learning could be implemented at the circuit level with plasticity rules that capture experimental findings. In this article, we propose a putative circuit-level solution to this issue by mapping the notion of feedback control onto a known dis-inhibitory circuit motif and combining it with an inhibition-modulated Hebbian plasticity rule.\n\n## 3 Model\n\nTo study whether biological microcircuits could naturally interpret dis-inhibitory feedback signals as local errors to control the sign of synaptic plasticity, we consider a continuous time dynamic multi-layer network comprised of excitatory and inhibitory neurons in each layer with dis-inhibitory feedback connections.\n\n### Neuronal dynamics\n\nThe membrane potential dynamics of the excitatory and inhibitory neurons in layer \\(i\\) are described by the following ordinary differential equations (ODEs):\n\n\\[\\tau_{\\text{E}}\\frac{d}{dt}\\mathbf{u}_{i}^{\\text{E}} = -\\mathbf{u}_{i}^{\\text{E}}(t)+\\mathbf{W}_{i}\\mathbf{r}_{i-1}^{ \\text{E}}(t)-\\mathbf{r}_{i}^{\\text{I}}(t) \\tag{1}\\] \\[\\tau_{\\text{I}}\\frac{d}{dt}\\mathbf{u}_{i}^{\\text{I}} = -\\mathbf{u}_{i}^{\\text{I}}(t)+\\mathbf{r}_{i}^{\\text{E}}(t)-\\mathbf{ Q}_{i}\\mathbf{c}(t) \\tag{2}\\]\n\nwith \\(\\mathbf{W}_{i}\\) the afferent synaptic weights from the previous layer and \\(\\mathbf{r}_{i}=\\phi(\\mathbf{u}_{i})\\) a smooth, monotonically increasing nonlinear activation function. Note that here we made the simplifying assumption that each excitatory neuron has an associated inhibitory neuron and that both are connected locally within a microcircuit. For all simulations, we use the soft rectifying nonlinearity \\(\\phi(u)=\\beta\\log(1+\\exp(u-\\gamma))\\), in which \\(\\beta\\) and \\(\\gamma\\) are parameters controlling the scale and shift of the activation function, respectively. Dis-inhibitory feedback is mediated through top-down control signals \\(\\mathbf{c}(t)\\) relayed to each layer \\(i\\) through an associated feedback weight matrix \\(\\mathbf{Q}_{i}\\) (Fig. 2). The input layer \\(\\mathbf{r}_{0}(t)\\) is data-dependent and not influenced by top-down feedback. For a constant input current \\(\\mathbf{r}_{0}\\), the network dynamics settle to the equilibrium state\n\n\\[\\overset{\\text{E}}{\\mathbf{u}}_{i}^{\\text{E}}=\\mathbf{W}_{i}\\mathbf{r}_{i-1}^{ \\text{E}}-\\overset{\\text{I}}{\\mathbf{r}}_{i}^{\\text{I}}\\quad,\\quad\\overset{ \\text{I}}{\\mathbf{u}}_{i}^{\\text{I}}=\\mathbf{r}_{i}^{\\text{E}}-\\mathbf{Q}_{i} \\overset{\\text{E}}{\\mathbf{c}}\\quad. \\tag{3}\\]\n\n### Feedback control\n\nAs in previous work on DFC [24; 47], our model uses feedback control to drive the output activity of the network towards the target activity \\(\\mathbf{r}_{L}^{\\text{tar}}\\) by minimizing the magnitude of the output error:\n\n\\[\\mathbf{e}(t)=-\\left.\\frac{\\partial\\mathcal{L}(\\mathbf{r}_{L},\\mathbf{r}_{L}^{ \\text{tar}})}{\\partial\\mathbf{r}_{L}}\\right|_{\\mathbf{r}_{L}=\\mathbf{r}_{L}(t)} ^{T} \\tag{4}\\]\n\nwhere \\(\\mathcal{L}(\\mathbf{r}_{L},\\mathbf{r}_{L}^{\\text{tar}})\\) is a label-dependent supervised loss function defined on the network's output activity. For a simple mean squared error (MSE) loss, \\(\\mathcal{L}=\\nicefrac{{1}}{{n}}\\sum_{n}\\nicefrac{{1}}{{2}}\\|\\mathbf{r}_{L}^{ \\text{tar}}(n)-\\mathbf{r}_{L}(n)\\|_{2}^{2}\\), the error for each datapoint \\(n\\) is directly related to the difference between the network's output and the target, specifically \\(\\mathbf{e}(t)=\\mathbf{r}_{L}^{\\text{tar}}-\\mathbf{r}_{L}(t)\\).\n\nLeaky proportional-integral controller.In our model, we use a leaky proportional-integral controller to compute the feedback control signals \\(\\mathbf{c}(t)\\):\n\n\\[\\mathbf{c}(t)=k_{\\text{p}}\\mathbf{e}(t)+k_{\\text{i}}\\mathbf{c}^{\\text{int}} \\quad,\\quad\\tau_{c}\\frac{d}{dt}\\mathbf{c}^{\\text{int}}=\\mathbf{e}(t)-\\mathbf{ c}^{\\text{int}}(t) \\tag{5}\\]\n\nwhere \\(k_{\\text{p}}\\) and \\(k_{\\text{i}}\\) are the proportional and integral control constants, respectively.\n\nFigure 2: Illustration of a multi-layer network with dis-inhibitory control microcircuits (left). Each network unit consists of an excitatory and inhibitory neuron that are recurrently connected (right). The top-down control signal to each layer \\(\\mathbf{Q}_{i}\\mathbf{c}(t)\\) is relayed through dis-inhibitory afferents (orange).\n\nDis-inhibitory feedback connectivity.Next, we have to connect the controller to the controlled quantities. In the context of our model, we assume that control signals are relayed via inhibitory interneurons. We thus have to specify the feedback weights connecting the control signals to the local microcircuits defined in the previous section. While we make no claims about how suitable control feedback is generated in neurobiology, in this article we merely assume that suitable control signals exist and that they are mediated via inhibitory interneurons. To that end, the feedback weights \\(\\mathbf{Q}_{i}\\) need to be chosen such that the output loss \\(\\mathcal{L}(\\mathbf{r}_{L},\\mathbf{r}_{L}^{\\text{ar}})\\) is minimized at the controlled equilibrium state. To fulfill this requirement, the column space of the concatenated feedback weights of the network, \\(Q\\triangleq\\left[\\mathbf{Q}_{1}^{T},\\ldots,\\mathbf{Q}_{L}^{T}\\right]^{T}\\), must be equal to the row space of the network Jacobian \\(J\\) at steady-state [24]. This Jacobian characterizes how infinitesimal perturbations of each controlled quantity, e.g., a neuronal activation, relates to changes in the network's output \\(\\mathbf{r}_{L}\\). In contrast to previous models relying on top-down control, our network consists of recurrently connected excitatory and inhibitory units whilst top-down input is targeting the inhibitory population exclusively. Since we want to model control through dis-inhibitory circuits, we assume that the Jacobian for the controller is defined with respect to the inhibitory membrane potential at each layer:\n\n\\[J\\triangleq\\left[\\mathbf{J}_{1},\\ldots,\\mathbf{J}_{L}\\right]=\\left[\\frac{ \\partial\\mathbf{r}_{L}}{\\partial\\mathbf{u}_{1}^{\\text{I}}},\\ldots,\\frac{ \\partial\\mathbf{r}_{L}}{\\partial\\mathbf{u}_{L}^{\\text{I}}}\\right] \\tag{6}\\]\n\nwhere we use \\(\\frac{\\partial\\mathbf{r}_{L}}{\\partial\\mathbf{u}_{i}^{\\text{I}}}=\\nabla_{ \\mathbf{u}_{i}^{\\text{I}}}\\mathbf{r}_{L}\\) to denote the Jacobian matrix of partial derivatives of the vector \\(\\mathbf{r}_{L}\\) with respect to the vector \\(\\mathbf{u}_{i}^{\\text{I}}\\). It has been shown that a wide range of possible feedback weights \\(Q\\) can match the row space of \\(J\\) in the DFC framework. One simple way of ensuring this condition is met is to set the feedback weights at each layer proportional to the transposed Jacobian, i.e., \\(-\\mathbf{Q}_{i}=\\mathbf{J}_{i}^{T}\\). However, the Jacobian depends on the input and the neuronal activity. It is thus changing over time until an equilibrium state is reached. To avoid changing feedback weights over time for a given input, we compute them based on the network Jacobian at the uncontrolled equilibrium state with \\(\\mathbf{c}=0\\)\n\n\\[-\\mathbf{Q}_{i}=\\mathbf{\\tilde{J}}_{i}^{T}=\\left.\\left(\\frac{\\partial\\mathbf{r }_{L}}{\\partial\\mathbf{u}_{i}^{\\text{I}}}\\right)^{T}\\right|_{\\mathbf{u}_{i}^{ \\text{I}}=\\mathbf{\\tilde{u}}_{i}^{\\text{I}}} \\tag{7}\\]\n\nwith \\(\\mathbf{\\tilde{u}}_{i}^{\\text{I}}\\) corresponding to the inhibitory membrane potentials in Layer \\(i\\) at the uncontrolled equilibrium state.\n\nLearning as minimization of control in dis-inhibitory neuronal circuits.Given suitable feedback weights and a strong influence on the network activity by the controller, neuronal activity can change considerably compared to the uncontrolled steady-state. Tracing the steps of [47], learning rules can be derived from a _minimization of control_ objective\n\n\\[\\mathcal{H}=\\frac{1}{2}\\sum_{n}\\|Q\\mathbf{\\tilde{c}}\\left(n\\right)\\|_{2}^{2} \\tag{8}\\]\n\nwhere \\(\\mathbf{\\tilde{c}}\\left(n\\right)\\) is the steady-state feedback control signal for datapoint \\(n\\). Formally, it can be shown that minimizing the above surrogate loss \\(\\mathcal{H}\\) also minimizes the output loss \\(\\mathcal{L}\\) (see Appendix B.1). We start with the following learning rule which minimizes \\(\\mathcal{H}\\):\n\n\\[\\tau_{w}\\frac{\\mathrm{d}}{\\mathrm{d}t}\\mathbf{W}i=\\left[\\underbrace{\\left( \\mathbf{\\tilde{q}}_{i}^{\\text{E}}-\\mathbf{\\tilde{u}}_{i}^{\\text{I}}\\right)}_{ \\text{error}}\\odot\\underbrace{\\phi^{\\prime}\\left(\\mathbf{\\tilde{u}}_{i}^{\\text {E}}\\right)}_{\\text{postsynaptic}}\\right]\\underbrace{\\left(\\mathbf{\\tilde{q}}_ {i-1}^{\\text{E}}\\right)^{T}}_{\\text{presynaptic}} \\tag{9}\\]\n\nwhere \\(\\phi^{\\prime}(\\mathbf{u})\\) is the derivative of the activation function and \\(\\odot\\) denotes element-wise multiplication. The sign of the weight change is determined by the error projected onto each neuron by the feedback controller at the equilibrium, which following Eq. (3) is encoded as \\(\\mathbf{Q}_{i}\\mathbf{\\tilde{c}}=\\mathbf{\\tilde{r}}_{i}^{\\text{E}}-\\mathbf{ \\tilde{u}}_{i}^{\\text{I}}\\). While Eq. (9) minimizes \\(\\mathcal{H}\\) when provided with sensible feedback signals, it does not constitute a local learning rule because it explicitly depends on the inhibitory membrane potentials \\(\\mathbf{\\tilde{u}}_{i}^{\\text{I}}\\), which excitatory neurons cannot access directly (cf. Fig. 1).\n\n### A Hebbian learning rule for error-modulated learning through dis-inhibitory control\n\nWe were wondering whether excitatory neurons could compute an effective local approximation of Eq. (9) by estimating the inhibitory membrane potentials from locally available quantities. To that end, we first re-write Eq. (9) as\n\n\\[\\tau_{W}\\frac{\\mathrm{d}}{\\mathrm{d}t}\\mathbf{W}_{i}=\\left[\\left(\\mathbf{\\hat{\\pi} }_{i}^{\\mathrm{E}}-\\phi^{-1}\\left(\\mathbf{\\hat{\\pi}}_{i}^{\\mathrm{I}}\\right) \\right)\\odot\\phi^{\\prime}\\left(\\mathbf{\\hat{\\pi}}_{i}^{\\mathrm{E}}\\right) \\right]\\left(\\mathbf{\\hat{\\pi}}_{i-1}^{\\mathrm{E}}\\right)^{T}\\quad, \\tag{10}\\]\n\nwhere we substituted the inhibitory membrane potentials \\(\\mathbf{\\hat{\\pi}}_{i}^{\\mathrm{I}}\\) with the inverse activation function \\(\\phi^{-1}\\left(\\mathbf{\\hat{\\pi}}_{i}^{\\mathrm{I}}\\right)\\). Mathematically, Eqs. (9) and (10) are equivalent, but conceptually, it re-frames the problem of non-locality since it would require an excitatory neuron to compute the inverse activation function from the recurrent inhibitory current, which is locally available. While it is hard to imagine how neurons would invert the activation function of other neurons exactly, we assume that they could conceivably compute a linear approximation, leading to a learning rule of the following general form:\n\n\\[\\tau_{W}\\frac{\\mathrm{d}}{\\mathrm{d}t}\\mathbf{W}_{i}=\\left[\\left(\\mathbf{\\hat{ \\pi}}_{i}^{\\mathrm{E}}-\\theta_{i}-\\delta_{i}\\mathbf{\\hat{\\pi}}_{i}^{\\mathrm{ I}}\\right)\\odot\\phi^{\\prime}\\left(\\mathbf{\\hat{\\pi}}_{i}^{\\mathrm{E}}\\right) \\right]\\left(\\mathbf{\\hat{\\pi}}_{i-1}^{\\mathrm{E}}\\right)^{T}\\quad. \\tag{11}\\]\n\nHere the parameter \\(\\theta_{i}\\) takes the role of a postsynaptic plasticity threshold, common to many phenomenological plasticity models [6; 7; 8], while \\(\\delta_{i}\\) adds an inhibitory current dependence to this threshold. In practice, we obtain \\(\\theta_{i}\\) and \\(\\delta_{i}\\) through a first-order Taylor expansion of the inverse activation function around a given linearization point \\(\\tilde{r}\\) (see Appendix B). In the next sections, we will see that, depending on the inhibitory activation function and the linearization parameters, the model reconciles aspects of phenomenological plasticity with an effective error-modulation mechanism as demanded by normative theories of gradient-based learning.\n\n## 4 Learning with dis-inhibitory control accounts for key plasticity experiments\n\nMost experiments on synaptic plasticity are performed _in vitro_ under highly controlled conditions, in which pairs of connected neurons are isolated. This enables researchers to investigate the plasticity at single synapses in the absence of interfering activity from the local microcircuit or long-range synaptic afferent connections. Such experimental conditions would likely interfere with any putative error-modulation of synaptic plasticity since long-range synaptic afferents are either severed during sample preparation or do not transmit plausible activity levels. To account for such possible experimental confounding factors and to compare experimentally observed Hebbian plasticity, we probe our error-modulated learning rule in three different settings: full microcircuit with \"closed-loop feedback\", intact \"microcircuit without feedback\", and isolated neurons resembling in-vivo experimental conditions with \"direct control of inhibition\" (Fig. 3). In each setting, we vary the excitatory input to the excitatory neuron and calculate the resulting weight change as a function of the postsynaptic firing rate using Eq. (11). For a qualitative comparison to previously suggested error-driven plasticity rules, we evaluated learning rules using dendritic error coding [22; 24; 30] or burst-dependent plasticity [23; 31] in comparable settings (Supplementary Fig. S1; see Appendix A).\n\nDis-inhibition controls the sign of plasticity in the intact microcircuit.We first considered a simplified version of the intact microcircuit with top-down dis-inhibitory feedback, for which our learning rule was derived. In this circuit, top-down control drives neuronal activity towards a target value \\(r^{\\mathrm{tar}}\\). We computed the weight updates dictated by our learning rule for two different targets as a function of the neuronal firing rate. In this setting, the learning rule exhibits two stable fixed points separated by an unstable one (Fig. 3a). Importantly, a stable fixed point exists at the the target firing rate, i.e. \\(r^{\\mathrm{E}}=r^{\\mathrm{tar}}\\) (Fig. 3a). At this fixed point, the sign of plasticity is determined by the top-down controller through the error decoded by the learning rule (11). This behavior is in line with the behavior of learning rules with explicit error-modulation in this simplified setting (cf. Fig. 1b, Supplementary Fig. S1).\n\nMoreover, in contrast to purely error-modulated plasticity rules, our rule exhibits an LTD region flanked by a stable fixed point in neuronal firing rates at zero and an unstable fixed point at intermediate firing rates. The existence of this LTD regime is a direct consequence of the local approximation used in its derivation (cf. Eq. (11)) and depends on the chosen parameters of the linearization (Supplementary Fig. S2; see Appendix B). For proper error-modulated learning in our framework, we have to ensure that each neuron's activity does not exclusively stay in this region over time and across different inputs. In neurobiology, this activity regime could, for instance, be attained through homeostatic plasticity [51]. Thus our learning rule exhibits error-modulated plasticity at the upper fixed point when embedded in an intact microcircuit with top-down feedback.\n\nPlasticity in an isolated microcircuit is self-stabilizing.To examine how our learning rule behaves in the absence of top-down control signals, we investigated plasticity dynamics in an isolated microcircuit without control feedback, while local circuit connectivity between excitatory and inhibitory neurons was left intact. In this setting, the inhibitory membrane potential does not encode an error signal that can be decoded by the learning rule. While explicitly error-modulated learning rules would not exhibit any synaptic weight change in this setting, our Hebbian learning rule predicts weight changes due to the imperfect approximation of the inverse function. Fig. 3b depicts the resulting plasticity dynamics for two different linear approximations. Embedded in a local microcircuit, the plasticity rule still exhibits both an LTD and LTP regime and a stable fixed point that depends on the learning rule parameters. Notably, the plasticity rule embedded in an isolated, but intact microcircuit is self-stabilizing through recurrent inhibition. Interestingly, the necessity of such a stable fixed point at higher activity levels for stable learning has been postulated previously in theoretical work [49, 52]. As before, the presence and location of the stable fixed point depend on the choice of plasticity parameters (Supplementary Fig. S2).\n\nPlasticity induction changes under direct control of inhibition.Experiments on excitatory plasticity _in vitro_ are commonly performed under conditions designed to minimize the interference of inhibitory activity, for example by applying GABA antagonists [53]. To study plasticity induction in our model under such simulated experimental conditions, we blocked recurrent connections from excitatory to inhibitory neurons, so that inhibitory activity is independent of excitatory activity. Additionally, we controlled inhibitory activity, as could be achieved, for instance, through current injection or optogenetic manipulations in experiments. In the absence of any inhibitory activity, i.e. \\(r^{\\mathrm{I}}=0\\), our learning rule reduces to the form \\(\\Delta w\\propto r_{\\mathrm{pre}}\\left(r_{\\mathrm{post}}-\\theta\\right)\\phi^{ \\prime}\\left(r_{\\mathrm{post}}\\right)\\) and loses its stable fixed point (Fig. 3c). Thus, in the absence of inhibition, the weight update prescribed by our plasticity model resembles Hebbian plasticity rules commonly observed under experimental conditions (Fig. 3d). However, our model predicts that direct control over inhibitory inputs to the excitatory neuron should shift the postsynaptic plasticity threshold to larger values.\n\nIn summary, dis-inhibitory control accounts for key plasticity experiments while also supporting error-driven learning in top-down controlled microcircuits. Additionally, its self-stabilizing capabilities provide a possible explanation as to why _in vitro_ experiments have failed to uncover a\n\nFigure 3: A Hebbian learning rule for error-modulated learning through dis-inhibitory control resembles plasticity observed in single-neuron electrophysiology experiments. **(a)** Weight change \\(\\Delta W\\) of a a single synapse as a function of postsynaptic firing rate \\(r_{\\mathrm{E}}\\). Different colored lines indicate two different postsynaptic firing rate targets \\(r^{\\mathrm{tar}}\\) indicated by colored arrows. The top-down feedback onto the interneuron is proportional to the error \\(r^{\\mathrm{tar}}-r^{\\mathrm{E}}\\). In the intact microcircuit with closed-loop feedback, our learning rule naturally leads to error-modulated learning. **(b)** Same as (a), but with top-down connections ablated. The two shades of green represent two different linear approximations of the inverse inhibitory activation function (see inset; inverse activation function in black). The plasticity rule resembles a multi-stable Hebbian plasticity rule [49]. **(c)** Same as before, but for an isolated neuron without microcircuit. Different shades of blue correspond to different amounts of inhibitory current. Different levels of injected inhibitory current lead to different values for the plasticity threshold, but the stable fixed point disappears in the absence of recurrent inhibition. **(d)** Experimentally observed plasticity with activity-dependent LTD and LTP redrawn from [50]. The data qualitatively resembles our learning rule in the open-loop setting (cf. panel (c)).\n\nstabilizing mechanism for excitatory synaptic plasticity. Next, we test whether our learning rule allows hierarchical networks to solve nonlinear function approximation problems.\n\n## 5 Dis-inhibitory control orchestrates learning in multi-layer networks\n\nTo explore our model's ability to train multi-layer networks, we first designed a simple continuous-time low-dimensional student-teacher learning task (Fig. 4a; see Appendix C for details). In this task, a randomly initialized teacher network with fixed parameters \\(f(\\mathbf{x}(t))\\) is given a set of 20 sine wave inputs with randomly chosen amplitudes, frequencies, and phases \\(\\mathbf{x}(t)\\). An architecturally identical student network but with different initial weights receives the same input \\(\\mathbf{x}(t)\\) and is tasked to reproduce the teacher's output, i.e. \\(r_{L}^{\\text{tar}}(t)=f(\\mathbf{x}(t))\\), evaluated through an MSE loss function. A dis-inhibitory feedback controller as described in the previous section is continually driving the student network's activity towards lower loss in real time. We trained the student network with either the exact non-linear inhibitory threshold rule Eq. (10) or the Hebbian learning rule with a linear inhibitory threshold Eq. (11). Neuronal and weight dynamics were simulated in continuous time using an explicit \\(5^{\\text{th}}\\) order Runge-Kutta method. To make sure that the feedback controller is aligned with the changing Jacobian during learning, the feedback weights were plastic and continuously evolving towards the average network Jacobian (see Appendix C).\n\nBefore learning, the student network did not follow the target closely in the open-loop setting, i.e., when the control signal was turned off. However, as soon as dis-inhibitory control was activated, the output activity closely followed the target (Fig. 4b). We then trained the student network for a total of 300 seconds of continuous sine wave inputs using either Eq. (10) or Eq. (11). After learning, the student network output closely followed the target in the open-loop setting, accompanied by a substantial reduction of the open-loop MSE loss (\\(c(t)=0\\)) and the surrogate loss \\(\\mathcal{H}\\) over the course of training (Fig. 4c). Finally, we observed that the linear threshold learning rule resulted in comparable performance to the exact inverse learning rule. Thus, an inhibitory modulated Hebbian learning rule is capable of solving a nonlinear learning task when credit is relayed through local dis-inhibitory microcircuits acting as feedback controller.\n\n### Training multi-layer networks on classification tasks through dis-inhibitory control\n\nHaving confirmed that our learning rule is capable of error-driven learning through minimization of control on a simple continuous-time learning task, we wondered whether we could train a multi-layer perceptron (MLP) on standard image classification tasks. To that end, we implemented a MLP with excitatory-inhibitory microcircuit units. Networks were comprised of either one or three hidden layers\n\nFigure 4: Online learning using dis-inhibitory control of Hebbian plasticity in a student-teacher task. **(a)** A teacher network (left) of size 30-20-2 implements a nonlinear mapping from an array of input sine waves \\(\\mathbf{x}(t)\\) (bottom) to an output target \\(f(\\mathbf{x}(t))\\). A student network (right) of the same size learns to approximate the teacher function by minimizing the control signal \\(\\mathbf{Q}_{i}\\mathbf{c}(t)\\) of a dis-inhibitory feedback controller at each layer. **(b)** One example teacher output neuron (black) and the corresponding student neuron (teal) before and after learning. The yellow bar indicates when feedback control is active. **(c)** MSE loss \\(\\mathcal{L}\\) and least control loss \\(\\mathcal{H}\\) over time for student networks trained with the exact update rule derived in Eq. (10) and the linear threshold rule (Eq. (11)).\n\nand used a parameterized soft rectifier activation function for all units. Numerical integration was performed using a fifth-order Runge-Kutta method with an adaptive step size to allow the network to reach an equilibrium state for each input (see Appendix C for details). Networks were trained either with the exact inverse rule in Eq. (10), or the linear threshold rule (Eq. (11)). To make training more robust and less dependent on initialization, the linear approximation of the inverse function was obtained through first-order Taylor expansion around a neuron-specific parameter \\(\\mathbf{\\tilde{r}}_{i}\\) which tracked the average inhibitory firing rate across each batch (see Appendix C). With these settings we trained the networks on MNIST [54] and Fashion-MNIST [55]. For comparison, we also trained conventional MLPs with the same neuron numbers but strictly feed-forward connectivity using BP.\n\nWe observed that networks trained using dis-inhibitory control with the exact inverse learning rule performed almost on par with standard BP on both datasets (Table 1). Using the linear threshold rule led to a slight drop in accuracy in all cases. It is possible that this gap could be narrowed further by choosing different linearization parameters \\(\\mathbf{\\tilde{r}}_{i}\\) since the performance of the exact inverse rule suggests that improving the approximation of the error translates to higher accuracy.\n\nIn above simulations, the Jacobian used for calculating the feedback signals is input-dependent and feedback weights are thus different for each stimulus. In neurobiology, feedback would presumably be relayed via synaptic connections that do not change this rapidly. To test whether successful learning is possible with more slowly varying feedback weights we repeated the above simulations with feedback weights that slowly tracked the average Jacobian (see Appendix C). This change did not compromise learning, although it resulted in a small but noticeable drop in accuracy compared to the ideal data-dependent Jacobian (Table 1). In summary, the combination of dis-inhibitory control with a Hebbian learning rule with an inhibition-dependent threshold allows training MLPs on vision datasets such as MNIST and Fashion-MNIST to accuracy values close to networks trained with BP.\n\n## 6 Discussion\n\nIn this article, we introduced a Hebbian learning rule with an inhibition-dependent threshold, which, through dis-inhibitory microcircuit dynamics, allows top-down feedback to control the sign of plasticity. Notably, the learning rule captures essential aspects of classic phenomenological Hebbian plasticity models under simulated experimental conditions disrupting recurrent inhibition in the local microcircuit. In contrast to standard Hebbian plasticity models, our model is stable when recurrent inhibition is intact without requiring additional homeostatic or compensatory mechanisms. Finally, we show how dis-inhibitory control is sufficient to train MLPs on vision tasks with performance levels close to classical BP.\n\n**Dis-inhibitory control reconciles error-based learning with classic Hebbian plasticity models.** The learning rule we put forward in this article has a postsynaptic plasticity threshold \\(\\theta\\) (cf. Eq. (11)), a neuron-specific parameter potentially subject to its own temporal dynamics. This threshold makes it reminiscent of the BCM rule [36]. Moreover, our model extends classic plasticity models by an explicit dependence on inhibition, which adds a dynamic, rapidly evolving component to the plasticity threshold. We derived this dual-threshold mechanism within a normative minimization-of-control objective. Its functional role is to decode a credit signal, relayed through dis-inhibitory afferents, from changes in the inhibitory current. Notably, the inhibitory threshold in our model confers an additional helpful property in that it induces a rapid compensatory plasticity mechanism that prevents\n\n\\begin{table}\n\\begin{tabular}{l l c c c c} \\hline \\hline  & & \\multicolumn{2}{c}{MNIST} & \\multicolumn{2}{c}{Fashion-MNIST} \\\\ \\cline{3-6} Number of hidden layers & 1 & 3 & 1 & 3 \\\\ \\hline \\hline Backprop & & \\(98.1\\pm 0.2\\) & \\(98.3\\pm 0.1\\) & \\(89.3\\pm 0.3\\) & \\(89.4\\pm 0.2\\) \\\\ \\hline \\multirow{3}{*}{Dis-inhibitory control} & Exact inverse & \\(97.7\\pm 0.2\\) & \\(98.0\\pm 0.2\\) & \\(89.1\\pm 0.2\\) & \\(89.1\\pm 0.2\\) \\\\  & Linear threshold & \\(97.1\\pm 0.1\\) & \\(96.5\\pm 0.1\\) & \\(87.6\\pm 0.4\\) & \\(86.8\\pm 0.2\\) \\\\ \\cline{2-6}  & Exact inverse (Avg \\(\\mathbf{J}\\)) & \\(97.8\\pm 0.1\\) & \\(97.0\\pm 1.5\\) & \\(88.7\\pm 1.8\\) & \\(88.2\\pm 0.5\\) \\\\  & Linear threshold (Avg \\(\\mathbf{J}\\)) & \\(96.5\\pm 0.1\\) & \\(96.0\\pm 0.3\\) & \\(86.4\\pm 0.3\\) & \\(84.6\\pm 0.3\\) \\\\ \\hline \\hline \\end{tabular}\n\\end{table}\nTable 1: Test accuracy in % for networks trained with BP or dis-inhibitory feedback control. Reported values are mean \\(\\pm\\)stdev (\\(n=10\\)). For validation accuracy see Appendix C.\n\npathological runaway LTP, usually associated with Hebbian plasticity, through recurrent inhibition and in the absence of top-down control signals or any additional homeostatic mechanisms. Theoretical studies have argued that such a stabilizing mechanism for high firing rates should exist to counteract the runaway potentiation of Hebbian plasticity [49, 52].\n\nExperimentally testable predictions.Our model puts forth several testable predictions. First, we anticipate a direct modulation of plasticity induction through inhibitory currents in conventional excitatory long-term plasticity induction protocols. We predict that postsynaptic plasticity thresholds [32, 33] should be influenced by inhibitory current injection, even when the postsynaptic activity is kept constant in experiments. Most plasticity experiments proceed in the presence of GABA antagonists or sodium channel blockers, which may obscure the direct impact of inhibition on the plasticity threshold. Nevertheless, abundant experimental evidence supports the idea that inhibition influences classic induction protocols at excitatory synapses [56, 57, 58, 59] and that GABAergic afferents can switch the sign of plasticity [60, 61]. Second, our model suggests that blocking dis-inhibitory circuits during an error-based learning paradigm should block or influence learning. Consistent with this hypothesis, dis-inhibitory microcircuits have been implicated with the gating of plasticity and behaviorally relevant learning in the Amygdala [62, 63], Hippocampus [64], and sensory cortices [65, 66, 67, 68, 69] (for a review, see [70]). Conversely, activating the same circuitry should affect or trigger learning during specific tasks.\n\nLimitations.Several limitations should be considered when interpreting this study's results. First, while our learning rule minimizes the loss, it does not necessarily follow the negative gradient. This difference could lead to sub-optimal learning dynamics and we will explore its impact in future work.\n\nMoreover, our circuit model requires specific one-to-one connectivity between excitatory and inhibitory interneurons that is inconsistent with circuit motifs observed in the brain. In biological microcircuits, excitatory neurons usually exceed the number of inhibitory neurons [71], and inhibitory interneurons typically provide inputs to many local excitatory cells and vice versa. Nevertheless, it may be possible to consider our model's inhibitory threshold (cf. Eq. (11)) as a local inhibitory current derived from multiple presynaptic targets. In this scenario, the challenge for excitatory neurons is to estimate their respective contribution to the inhibitory current they receive. In future work, we will explore the possibility of learning an estimate of the expected inhibitory current by implementing inhibitory plasticity to achieve an excitatory-inhibitory balanced state [72] and its co-dependence with excitatory plasticity [73]. Inhibitory plasticity and its interactions with excitatory plasticity is supported by experiments [74, 75, 76] and has been the focus of recent computational models [77, 78]. However, neither of these studies explored the functional relevance of co-dependent plasticity for credit assignment and circuit-level learning.\n\nFinally, the present model does not adhere to Dale's law in synaptic connections between hidden layers or the feedback pathway. Instead, we integrated the local population of dis-inhibitory interneurons into the dynamics of a top-down controller that modifies inhibitory activity bidirectionally. Biologically, this could be achieved through high baseline firing rates of dis-inhibitory interneurons or top-down feedback connections that target inhibitory interneurons directly [42]. In future work, we will explore detailed cortical architectures for dis-inhibitory feedback controllers that allow bi-directional modulation.\n\nIn summary, we made the first step to reconcile realistic circuit models and phenomenological plasticity rules with normative theories relying on error-modulated plasticity to solve the credit assignment problem. Specifically, we showed how top-down feedback signals targeting specific interneurons could efficiently modulate neuronal activity _and_ plasticity. Our results highlight the potential of learning algorithms beyond BP, showcase their ability to incorporate diverse plasticity phenomena observed in neurobiology, and open the door for exciting future research.\n\n## Acknowledgments\n\nWe thank all members of the Zenke Group for valuable comments and discussions. This project was supported by the Swiss National Science Foundation [grant number PCEFP3_202981] and the Novartis Research Foundation.\n\n## References\n\n* Hebb [1949] D O Hebb. _The organization of behavior; a neuropsychological theory_. The organization of behavior; a neuropsychological theory. Wiley, Oxford, England, 1949.\n* Malenka and Bear [2004] Robert C Malenka and Mark F Bear. LTP and LTD: an embarrassment of riches. _Neuron_, 44(1):5-21, 2004. doi: 10.1016/j.neuron.2004.09.012.\n* Feldman [2012] Daniel E Feldman. The spike-timing dependence of plasticity. _Neuron_, 75(4):556-571, 2012. doi: 10.1016/j.neuron.2012.08.001.\n* Gerstner and Kistler [2002] Wulfram Gerstner and Werner M Kistler. Mathematical formulations of hebbian learning. _Biological cybernetics_, 87(5-6):404-415, 2002. doi: 10.1007/s00422-002-0353-y.\n* Shouval et al. [2002] Harel Z. Shouval, Mark F. Bear, and Leon N. Cooper. A Unified Model of NMDA Receptor-Dependent Bidirectional Synaptic Plasticity. _Proc Natl Acad Sci U S A_, 99(16):10831-10836, 2002. doi: 10.1073/pnas.152343099.\n* Pfister and Gerstner [2006] Jean-Pascal Pfister and Wulfram Gerstner. Triplets of Spikes in a Model of Spike Timing-Dependent Plasticity. _J Neurosci_, 26(38):9673-9682, 2006. doi: 10.1523/JNEUROSCI.1425-06.2006.\n* Clopath et al. [2010] Claudia Clopath, Lars Busing, Eleni Vasilaki, and Wulfram Gerstner. Connectivity reflects coding: a model of voltage-based STDP with homeostasis. _Nat Neurosci_, 13(3):344-352, 2010. doi: 10.1038/nn.2479.\n* Graupner and Brunel [2012] Michael Graupner and Nicolas Brunel. Calcium-based plasticity model explains sensitivity of synaptic changes to spike pattern, rate, and dendritic location. _Proc Natl Acad Sci U S A_, 109(10):3991-3996, 2012. doi: 10.1073/pnas.1109359109.\n* Fremaux and Gerstner [2016] Nicolas Fremaux and Wulfram Gerstner. Neuromodulated Spike-Timing-Dependent plasticity, and theory of Three-Factor learning rules. _Frontiers in neural circuits_, 9:85-85, 2016. doi: 10.3389/fncir.2015.00085.\n* Kusmierz et al. [2017] Lukasz Kusmierz, Takuya Isomura, and Taro Toyoizumi. Learning with three factors: modulating hebbian plasticity with errors. _Current opinion in neurobiology_, 46:170-177, 2017. doi: 10.1016/j.conb.2017.08.020.\n* Pawlak et al. [2010] Verena Pawlak, Jeffery R Wickens, Alfredo Kirkwood, and Jason N D Kerr. Timing is not Everything: Neuromodulation Opens the STDP Gate. _Front Synaptic Neurosci_, 2:146, 2010. doi: 10.3389/fnsyn.2010.00146.\n* Brzosko et al. [2019] Zuzanna Brzosko, Susanna B Mierau, and Ole Paulsen. Neuromodulation of Spike-Timing-Dependent plasticity: Past, present, and future. _Neuron_, 103(4):563-581, 2019. doi: 10.1016/j.neuron.2019.05.041.\n* Werfel et al. [2005] Justin Werfel, Xiaohui Xie, and H Sebastian Seung. Learning curves for stochastic gradient descent in linear feedforward networks. _Neural computation_, 17(12):2699-2718, 2005. doi: 10.1162/089976605774320539.\n* Lillicrap et al. [2020] Timothy P Lillicrap, Adam Santoro, Luke Marris, Colin J Akerman, and Geoffrey Hinton. Backpropagation and the brain. _Nature reviews. Neuroscience_, 2020. doi: 10.1038/s41583-020-0277-3.\n* Rumelhart et al. [1986] David E Rumelhart, Geoffrey E Hinton, and Ronald J Williams. Learning representations by back-propagating errors. _Nature_, 323(6088):533-536, 1986. doi: 10.1038/323533a0.\n* Crick [1989] F Crick. The recent excitement about neural networks. _Nature_, 337(6203):129-132, 1989. doi: 10.1038/337129a0.\n* Roelfsema and Holtmaat [2018] Pieter R Roelfsema and Anthony Holtmaat. Control of synaptic plasticity in deep cortical networks. _Nature reviews. Neuroscience_, 19(3):166-180, 2018. doi: 10.1038/nrn.2018.6.\n\n* Richards et al. [2019] Blake A Richards, Timothy P Lillicrap, Philippe Beaudoin, Yoshua Bengio, Rafal Bogacz, Amelia Christensen, Claudia Clopath, Rui Ponte Costa, Archy de Berker, Surya Ganguli, Colleen J Gillon, Danijar Hafner, Adam Kepecs, Nikolaus Kriegeskorte, Peter Latham, Grace W Lindsay, Kenneth D Miller, Richard Naud, Christopher C Pack, Panayiota Poizzi, Pieter Roelfsema, Joao Sacramento, Andrew Saxe, Benjamin Scellier, Anna C Schapiro, Walter Senn, Greg Wayne, Daniel Yamins, Friedemann Zenke, Joel Zylberberg, Denis Therien, and Konrad P Kording. A deep learning framework for neuroscience. _Nature neuroscience_, 22(11):1761-1770, 2019. doi: 10.1038/s41593-019-0520-2.\n* Whittington and Bogacz [2017] James C R Whittington and Rafal Bogacz. An approximation of the error backpropagation algorithm in a predictive coding network with local hebbian synaptic plasticity. _Neural computation_, 29(5):1229-1262, 2017. doi: 10.1162/NECO_a_00949.\n* Whittington and Bogacz [2019] James C R Whittington and Rafal Bogacz. Theories of error Back-Propagation in the brain. _Trends in cognitive sciences_, 23(3):235-250, 2019. doi: 10.1016/j.tics.2018.12.005.\n* Guerguiev et al. [2017] Jordan Guerguiev, Timothy P Lillicrap, and Blake A Richards. Towards deep learning with segregated dendrites. _eLife_, 6, 2017. doi: 10.7554/eLife.22901.\n* Sacramento et al. [2018] Joao Sacramento, Rui Ponte Costa, Yoshua Bengio, and Walter Senn. Dendritic cortical microcircuits approximate the backpropagation algorithm. In S Bengio, H Wallach, H Larochelle, K Grauman, N Cesa-Bianchi, and R Garnett, editors, _Advances in Neural Information Processing Systems 31_, pages 8721-8732. Curran Associates, Inc., 2018.\n* Payeur et al. [2021] Alexandre Payeur, Jordan Guerguiev, Friedemann Zenke, Blake A Richards, and Richard Naud. Burst-dependent synaptic plasticity can coordinate learning in hierarchical circuits. _Nature neuroscience_, 2021. doi: 10.1038/s41593-021-00857-x.\n* Meulemans et al. [2021] Alexander Meulemans, Matilde Tristany Farinha, Javier Garcia Ordonez, Pau Vilimelis Aceituno, Joao Sacramento, and Benjamin F. Grewe. Credit assignment in neural networks through deep feedback control. In M. Ranzato, A. Beygelzimer, Y. Dauphin, P.S. Liang, and J. Wortman Vaughan, editors, _Advances in Neural Information Processing Systems_, volume 34, pages 4674-4687. Curran Associates, Inc., 2021.\n* Song et al. [2022] Yuhang Song, Beren Millidge, Tommaso Salvatori, Thomas Lukasiewicz, Zhenghua Xu, and Rafal Bogacz. Inferring neural activity before plasticity: A foundation for learning beyond backpropagation. _bioRxiv_, 2022. doi: 10.1101/2022.05.17.492325.\n* Richards and Lillicrap [2018] Blake A. Richards and Timothy P. Lillicrap. Can neocortical feedback alter the sign of plasticity? _Nat Rev Neurosci_, 2018. doi: 10.1038/s41583-018-0049-5.\n* Pfeffer et al. [2013] Carsten K. Pfeffer, Mingshan Xue, Miao He, Z. Josh Huang, and Massimo Scanziani. Inhibition of inhibition in visual cortex: the logic of connections between molecularly distinct interneurons. _Nature Neuroscience_, 16(8):1068-1076, 2013. doi: 10.1038/nn.3446.\n* Kepecs and Fishell [2014] Adam Kepecs and Gordon Fishell. Interneuron cell types are fit to function. _Nature_, 505(7483):318-326, 2014. doi: 10.1038/nature12983.\n* Scellier and Bengio [2017] Benjamin Scellier and Yoshua Bengio. Equilibrium propagation: Bridging the gap between energy-based models and backpropagation. _Frontiers in computational neuroscience_, 11:24, 2017. doi: 10.3389/fncom.2017.00024.\n* Urbanczik and Senn [2014] Robert Urbanczik and Walter Senn. Learning by the dendritic prediction of somatic spiking. _Neuron_, 81(3):521-528, 2014. doi: 10.1016/j.neuron.2013.11.030.\n* Greedy et al. [2022] Will Greedy, Heng Wei Zhu, Joseph Pemberton, Jack Mellor, and Rui Ponte Costa. Single-phase deep learning in cortico-cortical networks. In S. Koyejo, S. Mohamed, A. Agarwal, D. Belgrave, K. Cho, and A. Oh, editors, _Advances in Neural Information Processing Systems_, volume 35, pages 24213-24225. Curran Associates, Inc., 2022.\n* Sjostrom et al. [2001] P J Sjostrom, G G Turrigiano, and S B Nelson. Rate, timing, and cooperativity jointly determine cortical synaptic plasticity. _Neuron_, 32(6):1149-1164, 2001. doi: 10.1016/s0896-6273(01)00542-6.\n\n* Artola et al. [1990] A Artola, S Brocher, and W Singer. Different voltage-dependent thresholds for inducing long-term depression and long-term potentiation in slices of rat visual cortex. _Nature_, 347(6288):69-72, 1990. doi: 10.1038/347069a0.\n* Lim et al. [2015] Sukbin Lim, Jillian L McKee, Luke Woloszyn, Yali Amit, David J Freedman, David L Sheinberg, and Nicolas Brunel. Inferring learning rules from distributions of firing rates in cortical neurons. _Nature neuroscience_, 18(12):1804-1810, 2015. doi: 10.1038/nn.4158.\n* Oja [1982] E Oja. A simplified neuron model as a principal component analyzer. _Journal of mathematical biology_, 15(3):267-273, 1982. doi: 10.1007/bf00275687.\n* Bienenstock et al. [1982] E L Bienenstock, L N Cooper, and P W Munro. Theory for the development of neuron selectivity: orientation specificity and binocular interaction in visual cortex. _The Journal of neuroscience_, 2(1):32-48, 1982. doi: 10.1523/JNEUROSCI.02-01-00032.1982.\n* Richards and Lillicrap [2019] Blake A Richards and Timothy P Lillicrap. Dendritic solutions to the credit assignment problem. _Current opinion in neurobiology_, 54:28-36, 2019. doi: 10.1016/j.conb.2018.08.003.\n* Laborieux and Zenke [2022] Axel Laborieux and Friedemann Zenke. Holomorphic equilibrium propagation computes exact gradients through finite size oscillations. In S. Koyejo, S. Mohamed, A. Agarwal, D. Belgrave, K. Cho, and A. Oh, editors, _Advances in Neural Information Processing Systems_, volume 35, pages 12950-12963. Curran Associates, Inc., 2022.\n* Rao and Ballard [1999] R P Rao and D H Ballard. Predictive coding in the visual cortex: a functional interpretation of some extra-classical receptive-field effects. _Nature neuroscience_, 2(1):79-87, 1999. doi: 10.1038/4580.\n* Larkum et al. [1999] M E Larkum, J J Zhu, and B Sakmann. A new cellular mechanism for coupling inputs arriving at different cortical layers. _Nature_, 398(6725):338-341, 1999. doi: 10.1038/18686.\n* Coussens et al. [1997] C M Coussens, D S Kerr, and W C Abraham. Glucocorticoid receptor activation lowers the threshold for NMDA-receptor-dependent homosynaptic long-term depression in the hippocampus through activation of voltage-dependent calcium channels. _Journal of neurophysiology_, 78(1):1-9, 1997. doi: 10.1152/jn.1997.78.1.1.\n* Shen et al. [2022] Shan Shen, Xiaolong Jiang, Federico Scala, Jiakun Fu, Paul Fahey, Dmitry Kobak, Zhenghuan Tan, Na Zhou, Jacob Reimer, Fabian Sinz, and Andreas S Tolias. Distinct organization of two cortico-cortical feedback pathways. _Nature communications_, 13(1):6389, 2022. doi: 10.1038/s41467-022-33883-9.\n* Keller et al. [2020] Andreas J Keller, Morgane M Roth, and Massimo Scanziani. Feedback generates a second receptive field in neurons of the visual cortex. _Nature_, 582(7813):545-549, 2020. doi: 10.1038/s41586-020-2319-4.\n* Zagha et al. [2013] Edward Zagha, Amanda E Casale, Robert N S Sachdev, Matthew J McGinley, and David A McCormick. Motor cortex feedback influences sensory processing by modulating network state. _Neuron_, 79(3):567-578, 2013. doi: 10.1016/j.neuron.2013.06.008.\n* Gilra and Gerstner [2017] Aditya Gilra and Wulfram Gerstner. Predicting non-linear dynamics by stable local learning in a recurrent spiking neural network. _eLife_, 6:e28295, 2017. doi: 10.7554/eLife.28295.\n* Alemi et al. [2018] Alireza Alemi, Christian Machens, Sophie Deneve, and Jean-Jacques Slotine. Learning Nonlinear Dynamics in Efficient, Balanced Spiking Networks Using Local Plasticity Rules. _Proceedings of the AAAI Conference on Artificial Intelligence_, 32(1), 2018. doi: 10.1609/aaai.v32i1.11320.\n* Meulemans et al. [2022] Alexander Meulemans, Matilde Tristany Farinha, Maria R. Cervera, Joao Sacramento, and Benjamin F. Grewe. Minimizing control for credit assignment with strong feedback. In Kamalika Chaudhuri, Stefanie Jegelka, Le Song, Csaba Szepesvari, Gang Niu, and Sivan Sabato, editors, _Proceedings of the 39th International Conference on Machine Learning_, volume 162 of _Proceedings of Machine Learning Research_, pages 15458-15483. PMLR, 2022.\n\n* [48] Pau Vilimelis Aceituno, Matilde Tristany Farinha, Reinhard Loidl, and Benjamin F Grewe. Learning cortical hierarchies with temporal hebbian updates. _bioRxiv_, 2023. doi: 10.1101/2023.01.02.522459.\n* Zenke et al. [2015] Friedemann Zenke, Everton J Agnes, and Wulfram Gerstner. Diverse synaptic plasticity mechanisms orchestrated to form and retrieve memories in spiking neural networks. _Nature communications_, 6:6922, 2015. doi: 10.1038/ncomms7922.\n* Kirkwood et al. [1996] A Kirkwood, M C Rioult, and M F Bear. Experience-dependent modification of synaptic plasticity in visual cortex. _Nature_, 381(6582):526-528, 1996. doi: 10.1038/381526a0.\n* Turrigiano [2012] Gina Turrigiano. Homeostatic synaptic plasticity: local and global mechanisms for stabilizing neuronal function. _Cold Spring Harbor perspectives in biology_, 4(1):a005736, 2012. doi: 10.1101/cshperspect.a005736.\n* Zenke and Gerstner [2017] Friedemann Zenke and Wulfram Gerstner. Hebbian plasticity requires compensatory processes on multiple timescales. _Philosophical transactions of the Royal Society of London. Series B, Biological sciences_, 372(1715), 2017. doi: 10.1098/rstb.2016.0259.\n* Kruijssen and Wierenga [2019] Dennis L H Kruijssen and Corette J Wierenga. Single synapse LTP: A matter of context? _Frontiers in cellular neuroscience_, 13:496, 2019. doi: 10.3389/fncel.2019.00496.\n* Deng [2012] Li Deng. The mnist database of handwritten digit images for machine learning research. _IEEE Signal Processing Magazine_, 29(6):141-142, 2012.\n* Xiao et al. [2017] Han Xiao, Kashif Rasul, and Roland Vollgraf. Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. _arXiv_, 2017. doi: [https://doi.org/10.48550/arXiv.1708.07747](https://doi.org/10.48550/arXiv.1708.07747).\n* Higley [2014] Michael J Higley. Localized GABAergic inhibition of dendritic ca(2+) signalling. _Nature reviews. Neuroscience_, 15(9):567-572, 2014. doi: 10.1038/nrn3803.\n* Hasuo and Akasu [2001] H Hasuo and T Akasu. Activation of inhibitory pathways suppresses the induction of long-term potentiation in neurons of the rat lateral septal nucleus. _Neuroscience_, 105(2):343-352, 2001. doi: 10.1016/s0306-4522(01)00195-6.\n* Douglas et al. [1982] R M Douglas, G V Goddard, and M Rives. Inhibitory modulation of long-term potentiation: evidence for a postsynaptic locus of control. _Brain research_, 240(2):259-272, 1982. doi: 10.1016/0006-8993(82)90221-9.\n* Kotak et al. [2017] Vibhakar C Kotak, Ana Mirallave, Todd M Mowery, and Dan H Sanes. GABAergic inhibition gates excitatory LTP in perirhinal cortex. _Hippocampus_, 27(12):1217-1223, 2017. doi: 10.1002/hipo.22799.\n* Hayama et al. [2013] Tatsuya Hayama, Jun Noguchi, Satoshi Watanabe, Noriko Takahashi, Akiko Hayashi-Takagi, Graham C R Ellis-Davies, Masanori Matsuzaki, and Haruo Kasai. GABA promotes the competitive selection of dendritic spines by controlling local ca2+ signaling. _Nature neuroscience_, 16(10):1409-1416, 2013. doi: 10.1038/nn.3496.\n* Paille et al. [2013] Vincent Paille, Elodie Fino, Kai Du, Teresa Morera-Herreras, Sylvie Perez, Jeanette Hellgren Kotaleski, and Laurent Venance. GABAergic circuits control spike-timing-dependent plasticity. _The Journal of neuroscience_, 33(22):9353-9363, 2013. doi: 10.1523/JNEUROSCI.5796-12.2013.\n* Wolff et al. [2014] Steffen B E Wolff, Jan Grundemann, Philip Tovote, Sabine Krabbe, Gilad A Jacobson, Christian Muller, Cyril Herry, Ingrid Ehrlich, Rainer W Friedrich, Johannes J Letzkus, and Andreas Luthi. Amygdala interneuron subtypes control fear learning through disinhibition. _Nature_, 509(7501):453-458, 2014. doi: 10.1038/nature13258.\n* Krabbe et al. [2019] Sabine Krabbe, Enrica Paradiso, Simon d'Aquin, Yael Bitterman, Julien Courtin, Chun Xu, Keisuke Yonehara, Milica Markovic, Christian Muller, Tobias Eichlisberger, Jan Grundemann, Francesco Ferraguti, and Andreas Luthi. Adaptive disinhibitory gating by VIP interneurons associative learning. _Nature neuroscience_, 22(11):1834-1843, 2019. doi: 10.1038/s41593-019-0508-y.\n\n* Donato et al. [2013] Flavio Donato, Santiago Belluco Rompani, and Pico Caroni. Parvalbumin-expressing basketball network plasticity induced by experience regulates adult learning. _Nature_, 504(7479):272-276, 2013. doi: 10.1038/nature12866.\n* Froemke et al. [2007] Robert C Froemke, Michael M Merzenich, and Christoph E Schreiner. A synaptic memory trace for cortical receptive field plasticity. _Nature_, 450(7168):425-429, 2007. doi: 10.1038/nature06289.\n* Letzkus et al. [2011] Johannes J Letzkus, Steffen B E Wolff, Elisabeth M M Meyer, Philip Tovote, Julien Courtin, Cyril Herry, and Andreas Luthi. A disinhibitory microcircuit for associative fear learning in the auditory cortex. _Nature_, 480(7377):331-335, 2011. doi: 10.1038/nature10674.\n* Williams and Holtmaat [2019] Leena E Williams and Anthony Holtmaat. Higher-Order thalamocortical inputs gate synaptic Long-Term potentiation via disinhibition. _Neuron_, 101(1):91-102.e4, 2019. doi: 10.1016/j.neuron.2018.10.049.\n* Pages et al. [2021] Stephane Pages, Nicolas Chenouard, Ronan Chereau, Vladimir Kouskoff, Frederic Gambino, and Anthony Holtmaat. An increase in dendritic plateau potentials is associated with experience-dependent cortical map reorganization. _Proceedings of the National Academy of Sciences of the United States of America_, 118(9), 2021. doi: 10.1073/pnas.2024920118.\n* Canto-Bustos et al. [2022] Martha Canto-Bustos, F Kathryn Friason, Constanza Bassi, and Anne-Marie M Oswald. Disinhibitory circuitry gates associative synaptic plasticity in olfactory cortex. _The Journal of neuroscience_, 42(14):2942-2950, 2022. doi: 10.1523/JNEUROSCI.1369-21.2021.\n* Letzkus et al. [2015] Johannes J Letzkus, Steffen B E Wolff, and Andreas Luthi. Disinhibition, a circuit mechanism for associative learning and memory. _Neuron_, 88(2):264-276, 2015. doi: 10.1016/j.neuron.2015.09.024.\n* Sahara et al. [2012] Setsuko Sahara, Yuchio Yanagawa, Dennis D M O'Leary, and Charles F Stevens. The fraction of cortical GABAergic neurons is constant from near the start of cortical neurogenesis to adulthood. _The Journal of neuroscience_, 32(14):4755-4761, 2012. doi: 10.1523/JNEUROSCI.6412-11.2012.\n* Vogels et al. [2011] T P Vogels, H Sprekeler, F Zenke, C Clopath, and W Gerstner. Inhibitory plasticity balances excitation and inhibition in sensory pathways and memory networks. _Science_, 334(6062):1569-1573, 2011. doi: 10.1126/science.1211095.\n* Hennequin et al. [2017] Guillaume Hennequin, Everton J Agnes, and Tim P Vogels. Inhibitory plasticity: Balance, control, and codependence. _Annual review of neuroscience_, 40:557-579, 2017. doi: 10.1146/annurev-neuro-072116-031005.\n* Wang and Maffei [2014] Lang Wang and Arianna Maffei. Inhibitory plasticity dictates the sign of plasticity at excitatory synapses. _The Journal of neuroscience_, 34(4):1083-1093, 2014. doi: 10.1523/JNEUROSCI.4711-13.2014.\n* D'amour and Froemke [2015] James A D'amour and Robert C Froemke. Inhibitory and excitatory spike-timing-dependent plasticity in the auditory cortex. _Neuron_, 86(2):514-528, 2015. doi: 10.1016/j.neuron.2015.03.014.\n* Mapelli et al. [2016] Jonathan Mapelli, Daniela Gandolfi, Antonietta Vilella, Michele Zoli, and Albertino Bigiani. Heterosynaptic GABAergic plasticity bidirectionally driven by the activity of pre- and postsynaptic NMDA receptors. _Proceedings of the National Academy of Sciences of the United States of America_, 113(35):9898-9903, 2016. doi: 10.1073/pnas.1601194113.\n* Agnes and Vogels [2022] Everton J Agnes and Tim P Vogels. Codependent excitatory and inhibitory plasticity accounts for quick, stable and long-lasting memories in biological networks. _bioRxiv_, 2022. doi: 10.1101/2021.04.01.437962.\n* Miehl and Gjorgjieva [2022] Christoph Miehl and Julijana Gjorgjieva. Stability and learning in excitatory synapses by nonlinear inhibitory plasticity. _PLoS computational biology_, 18(12):e1010682, 2022. doi: 10.1371/journal.pcbi.1010682.\n\n## Appendix A Comparison with explicitly error-modulated plasticity rules\n\nNormative theories of functional learning in neuronal circuits of the brain commonly derive biologically plausible solutions for the credit assignment problem that approximate BP. Like BP, these models result in explicitly error-modulated update rules for synaptic weights of the general form\n\n\\[\\Delta w_{ij}\\propto\\text{pre}_{j}\\times g(\\text{post})_{i}\\times e_{i} \\tag{12}\\]\n\nwhere \\(\\text{pre}_{j}\\) is presynaptic neuronal activity, \\(g(\\text{post})_{i}\\) is a nonlinear function \\(g(.)\\) of postsynaptic neuronal activity and \\(e_{i}\\) is a neuron-specific error signal that determines the sign of plasticity. In this section, we showcase how error-modulated plasticity rules drive neuronal activity to a stable fixed point and discuss their predictions for _in vitro_ electrophysiology experiments (Supplementary Fig. S1).\n\nDendritic error rules.Dendritic error rules employ neuron models with multiple segregated compartments to encode a neuron-specific error signal \\(e_{i}\\) in the difference between neuronal activity of different compartments [1, 6]. Here, we implement the model described by Sacramento et al. [1], in which the error is encoded in the difference between a somatic voltage \\(u_{\\text{S}}\\) and a basal dendritic voltage \\(u_{\\text{B}}\\). For simplicity, we consider a single pyramidal neuron receiving presynaptic input \\(x\\) weighted by a feed-forward weight \\(w\\). Top-down feedback provides a current to an apical dendritic compartment \\(u_{\\text{A}}\\) that is canceled out by recurrent inhibition, leaving an error signal in the apical dendrite that in turn influences the somatic voltage (Supplementary Fig. S1a). The error signal usedfor the bottom-up weight update is then decoded from the difference between the output of the neuron, \\(r=\\phi(u_{\\text{S}})\\), and the output that would be observed if the apical dendrite potential (i.e. the top-down error) was zero. Specifically, the weight change at a synapse with presynaptic input \\(x\\) is given by\n\n\\[\\Delta w^{\\text{Dendr.\\ Error}}=\\eta\\left(\\phi\\left(u_{\\text{S}}\\right)-\\phi \\left(\\hat{u}_{\\text{B}}\\right)\\right)x \\tag{13}\\]\n\nwhere \\(\\hat{u}_{b}\\) is a function of the dendritic potential that takes into account dendritic attenuation parameters of the model, \\(\\phi(\\cdot)\\) is a nonlinear activation function and \\(\\eta\\) is a learning rate (see Sacramento et al. [1] for a detailed description of the model dynamics).\n\nWhen this microcircuit is intact, top-down feedback is proportional to a target firing rate \\(r^{\\text{tar}}\\) and the neuron-specific error \\(e_{i}\\) determining the sign of plasticity is proportional to the output error \\(r^{\\text{tar}}-r\\). Consequently, the synaptic weight will increase when the neuronal firing rate is below the target rate and vice versa (Supplementary Fig. S1a).\n\nIn contrast, when the apical dendrite does not receive top-down inputs, for example when top-down afferents are severed in an _in vitro_ slice preparation, the plasticity rule in Eq. (13) cannot decode an error. In this setting, recurrent inhibition would still try to cancel out an expected top-down input to the apical compartment, leading to a negative apical potential and in turn to LTD (Supplementary Fig. S1b). Note that in other implementations of dendritic error rules, that do not employ a recurrent inhibitory circuit [6, 7], \\(\\Delta w=0\\) in this setting.\n\nIf the local inhibitory microcircuit is also impaired, for example by blocking recurrent connections from excitatory neurons to inhibitory neurons, the apical voltage is independent of presynaptic stimulation. In this isolated neuron setting, which arguably resembles most experimental conditions for _in vitro_ electrophysiology, dendritic error rules would predict no weight change, except if current is injected into the apical compartment experimentally, in which case the weight change would be proportional to the injected current (Supplementary Fig. S1c)\n\nDeep Feedback Control.The DFC framework put forward by Meulemans et al. [2, 3] also makes use of a dendritic error rule, but considers a simpler microcircuit without recurrent inhibition. The weight change in DFC is proportional to the difference between a neuron's bottom-up inputs, encoded in a basal dendritic potential \\(u_{\\text{B}}\\), and its somatic potential \\(u_{\\text{S}}\\). The somatic potential integrates both bottom-up inputs and a top-down control signal that moves the neuronal firing rate towards the target \\(r^{\\text{tar}}\\). The DFC update rule is defined as\n\n\\[\\Delta w^{\\text{DFC}} = \\eta\\left(u_{\\text{S}}-u_{\\text{B}}\\right)x \\tag{14}\\] \\[= \\eta\\left(u_{\\text{S}}-wx\\right)x\\quad. \\tag{15}\\]\n\nWhen the top-down control loop is intact, the DFC learning rule resembles the learning dynamics of dendritic error rules and creates a stable fixed point at \\(r=r^{\\text{tar}}\\) (Supplementary Fig. S1d). As was the case in dendritic error rules, when the top-down control signal is impaired, the sign and magnitude of the weight change are proportional to externally injected currents (Supplementary Fig. S1e).\n\nBurst-dependent plasticity.Payeur et al. [4] suggested an alternative mechanism to leverage segregated neuronal compartments for error-driven plasticity. As in dendritic error rules, burst-dependent plasticity relies on top-down feedback connections to an apical dendritic compartment. However, in contrast to the previously described learning rules, burst-dependent plasticity does not rely on compute \\(e_{i}\\) using differences in neuronal activity between compartments. Instead, burst-dependent plasticity exploits the unique spiking behavior of cortical layer 5 pyramidal neurons [8], in which inputs to the apical dendrite determine whether bottom-up inputs to the neuron result in a single output spike (\"event\") or several output spikes fired in quick succession (\"burst\"). This approach allows the neuron to multiplex two streams of information in its firing rate: bottom-up forward signals are encoded in the event rate \\(r_{\\text{event}}\\) while top-down errors are encoded in the burst probability \\(p\\). To illustrate the dynamics arising from burst-dependent plasticity, we consider the rate-based version of the model proposed by Greedy et al. [5], for which the weight change is defined as\n\n\\[\\Delta w^{\\text{Burst-dependent}}=\\eta\\:r_{\\text{event}}\\left(p-p_{b}\\right)x \\tag{16}\\]\n\nwhere \\(p_{b}\\) is a parameter denoting the baseline burst probability. The sign of plasticity is thus determined by deviations from the baseline burst probability (see Greedy et al. [5] for details on model dynamics).\n\nIn the intact, closed-loop microcircuit, top-down feedback communicates an error signal to the apical compartment that in- or decreases the burst probability depending on whether the neuronal output \\(r_{\\text{event}}\\) is smaller or larger than the target rate \\(r^{\\text{tar}}\\). Similar to other error-modulated plasticity rules, this creates a stable fixed point for plasticity at \\(r_{\\text{event}}=r^{\\text{tar}}\\) (Supplementary Fig. S1f).\n\nBecause the baseline burst probability \\(p_{b}\\) acts like a postsynaptic plasticity threshold, burst-dependent plasticity can resemble some aspects of phenomenological Hebbian plasticity rules. Specifically, when synaptic inputs to the neuron follow a Poisson distribution over time, its output will also be Poisson distributed and the burst probability is not solely determined by top-down apical inputs. To illustrate this scenario, we consider an isolated neuron without top-down feedback to its apical dendrite. Given Poisson-distributed inputs \\(x\\), we can assume that the burst probability of the postsynaptic neuron is determined by the input Poisson rate, i.e. \\(p=f(x)\\). Plotting the weight change as a function of the input Poisson rate while keeping the baseline burst probability \\(p_{b}\\) fixed results in a learning rule that resembles phenomenological Hebbian plasticity with a postsynaptic threshold (Supplementary Fig. S1g).\n\n## Appendix B Analysis of the learning rule for dis-inhibitory control\n\nLet's first recall the learning rule expressed in Eq. (9), where local error is encoded in the difference between the excitatory firing rate and inhibitory membrane potential. To enhance clarity, we reformulate the weight change for a singular synapse, \\(w_{ij}\\), established between presynaptic excitatory neuron \\(j\\) and postsynaptic excitatory neuron \\(i\\):\n\n\\[\\Delta w_{ij}=\\phi^{\\prime}\\left(u_{i}^{\\text{E}}\\right)\\left(r_{i}^{\\text{E} }-u_{i}^{\\text{I}}\\right)r_{j}^{\\text{E}}. \\tag{17}\\]\n\nwhere we assume all quantities are evaluated at equilibrium state. Here, \\(r_{j}^{E}\\) and \\(r_{i}^{E}\\) represent the pre- and post-synaptic firing rates respectively. \\(u_{i}^{E}\\) symbolizes the postsynaptic membrane potential, whereas \\(u_{i}^{I}\\) denotes the membrane potential of the inhibitory neuron linked to the postsynaptic cell. This learning rule is non-local since the postsynaptic neuron doesn't possess direct access to the membrane potential of an inhibitory neuron \\(u_{i}^{I}\\).\n\nTo navigate around this limitation, we introduced a linear approximation of the inverse inhibitory activation function in Eq. (11), yielding a local learning rule of the generic form:\n\n\\[\\Delta w_{ij}=\\phi^{\\prime}\\left(u_{i}^{\\text{E}}\\right)\\left(r_{i}^{\\text{E} }-\\theta-\\delta r_{i}^{\\text{I}}\\right)r_{j}^{\\text{E}}. \\tag{18}\\]\n\nThe simplified microcircuit model we studied here assumes a particular form of one-to-one connectivity in which each excitatory neuron only receives input from one inhibitory neuron and vice-versa. Conceptually, \\(r_{i}^{I}\\) can thus be interpreted as the inhibitory current received by the postsynaptic neuron.\n\n### Relationship between \\(\\mathcal{L}\\) and \\(\\mathcal{H}\\)\n\nFollowing [3], it is straight forward to show that a reduction in \\(\\mathcal{H}\\) leads to a reduction in \\(\\mathcal{L}\\) by observing that, if \\(\\mathcal{H}\\) is minimized to zero, \\(\\mathcal{L}\\) is also minimized to zero. This is simple to show by observing that \\(\\mathcal{H}=0\\) only occurs when \\(\\mathcal{L}=0\\) (cf. Eq. (4)), i.e. when the uncontrolled network output equals the target. Formally, for any given loss that evaluates to zero if \\(\\mathbf{r}_{L}=\\mathbf{\\tilde{r}}_{L}^{\\text{tar}}\\), i.e. \\(\\mathcal{L}\\left(\\mathbf{x},\\mathbf{x}\\right)=0\\), we have\n\n\\[\\mathcal{H}=\\sum_{n}\\left\\|Q\\mathbf{\\tilde{\\epsilon}}\\left(n\\right)\\right\\|_{ 2}^{2}=0\\Longleftrightarrow\\sum_{n}\\mathcal{L}\\left(\\mathbf{\\tilde{r}}_{L} \\left(n\\right),\\mathbf{r}_{L}^{\\text{true}}\\left(n\\right)\\right)=0 \\tag{19}\\]\n\nwhere \\(\\mathbf{\\tilde{r}}_{L}\\) denotes the output of the network in the absence of a controller. However, in contrast to the networks considered in [3], the dis-inhibitory controller considered in this work is not guaranteed to minimize the surrogate loss \\(\\mathcal{H}\\), and thus \\(\\mathcal{L}\\) to zero.\n\n### Linear approximation of the inverse function\n\nThe learning dynamics described by the learning rule in Eq. (11) depend implicitly on the chosen nonlinear activation function \\(r=\\phi(u)\\), which dictates the error induced by the linear approximation. For all simulations in this work, we use a soft rectifying nonlinearity as neuronal activation function:\n\n\\[\\phi(u)=\\beta\\log(1+\\exp(u-\\gamma)) \\tag{20}\\]which is parameterized through its scale \\(\\beta\\) and shift \\(\\gamma\\). Additionally, the learning dynamics are contingent on the choice of linearization parameters \\(\\theta\\) and \\(\\delta\\) used to approximate the inverse activation function,\n\n\\[\\phi^{-1}(r)=\\gamma+\\log\\left(\\exp\\left(\\frac{r}{\\beta}\\right)-1\\right)\\approx \\theta+\\delta r. \\tag{21}\\]\n\nFig. S2 depicts functionally and phenomenologically different learning dynamics in the same three settings we already investigated in Section 4, that can be obtained by changing the linearization parameters.\n\nFor the training of the deep neural networks discussed in Section 5, we adopted a practical approach and linearize the inverse function using a first-order Taylor expansion. This method effectively reduces the linear approximation to a single hyperparameter, the linearization point \\(\\tilde{r}\\). Therefore, we approximate the inverse activation function as follows:\n\n\\[\\phi^{-1}(r)=f(r)\\approx f(\\tilde{r})+f^{\\prime}(\\tilde{r})(r-\\tilde{r}), \\tag{22}\\]\n\nwhere \\(f^{\\prime}(\\tilde{r})\\) denotes the derivative of the inverse activation function evaluated at the linearization point \\(\\tilde{r}\\). From this linearization, we can extract the parameters \\(\\theta\\) and \\(\\delta\\) in relation to \\(\\tilde{r}\\):\n\n\\[\\theta=f(\\tilde{r})-\\tilde{r}f^{\\prime}(\\tilde{r})\\qquad\\delta=f^{\\prime}( \\tilde{r}). \\tag{23}\\]\n\nDenoting \\(\\tilde{u}=f(\\tilde{r})=\\phi^{-1}(\\tilde{r})\\), we can use the inverse function theorem to formulate the relationship between the linearization point \\(\\tilde{r}\\) and the general linearization parameters, \\(\\theta\\) and \\(\\delta\\), in a more interpretable fashion:\n\n\\[\\theta=\\tilde{u}-\\frac{\\tilde{r}}{\\phi^{\\prime}(\\tilde{u})}\\qquad\\delta=\\frac {1}{\\phi^{\\prime}(\\tilde{u})}. \\tag{24}\\]\n\n## Appendix C Simulation details\n\nAll numerical simulations were implemented using Python 3.8.10 and were executed on NVIDIA Quadro RTX 5000 GPUs. The software stack includes Jax [9], Flax [10], and Diffrax [11]. Datasets were obtained from the TensorFlow datasets library [12]. The code to reproduce our results is publicly available at [https://github.com/fmi-basel/disinhibitory-control](https://github.com/fmi-basel/disinhibitory-control).\n\n### Single-synapse experiments\n\nFor the investigation of learning dynamics at a single synapse (Section 4), we simulated a single microcircuit unit. The microcircuit unit was driven by increasing the strength of afferent input into the excitatory neuron. To capture the input-output curve of neurons in the absence of any background inputs, we parameterized the neuronal activation function Eq. (20) with \\(\\beta=1.0\\) and \\(\\gamma=3.0\\) for both excitatory and inhibitory neurons. For each input strength, the microcircuit was simulated for \\(600\\) ms to ensure that neuronal dynamics settled to an equilibrium. Synaptic weight updates where computed at the equilibrium state. Numerical integration was performed using the forward Euler method with a time step of \\(1\\) ms.\n\n### Student-teacher task\n\nInputs.For the student-teacher learning task (Section 5), we generated 30 sine wave inputs with random frequency, amplitude and phase shift. Frequencies were randomly drawn from a uniform distribution \\(\\mathcal{U}(0.1\\text{Hz},2.0\\text{Hz})\\). Similarly, sine amplitudes were randomly drawn from the distribution \\(\\mathcal{U}(0.1,1.0)\\) and phase shifts were drawn from the distribution \\(\\mathcal{U}(0,\\pi)\\). Sine waves were continuously generated and used as input to both the student and teacher network during training. Evaluation for panel B in Fig. 4 before and after training was performed on a 5-second window of the training data.\n\nFeedback weights \\(\\mathbf{Q}_{i}\\) were initialized proportional to the normalized network Jacobian at the beginning of training:\n\n\\[-\\mathbf{Q}_{i}(0)=\\alpha\\frac{\\mathbf{J}_{i}(0)}{\\|\\mathbf{J}_{i}(0)\\|_{F}} \\tag{25}\\]\n\nwhere we normalized the Jacobian by dividing it by its Frobenius norm. The parameters \\(\\alpha\\) controls the norm of the effective Jacobian, and thus the magnitude of the feedback weights. We found that normalizing the Jacobian is not necessary for good training performance, but prevents exorbitantly large values in the feedback weights and can sometimes speed up numerical integration of the neuronal dynamics and aid the stability of the dynamical system.\n\nAs the forward weights \\(\\mathbf{W}_{i}\\) change over the course of learning, the network Jacobian of the student network relevant for computing suitable feedback weights changes as well. To account for the learning-induced changes in the network Jacobian, we make the feedback weights \\(\\mathbf{Q}_{i}\\) part of the network dynamics and continuously nudge them towards the normalized network Jacobian with a slow time constant \\(\\tau_{Q}\\):\n\n\\[\\tau_{Q}\\frac{d\\mathbf{Q}_{i}}{dt}=-\\mathbf{Q}_{i}(t)+\\alpha\\frac{-\\mathbf{J}_{i} (t)}{\\|-\\mathbf{J}_{i}(t)\\|_{F}} \\tag{26}\\]\n\nAs a result, feedback weights are dynamically adjusted to retain good control performance. We chose \\(\\tau_{Q}\\) to be faster than the plasticity time constant \\(\\tau_{W}\\) and much slower than changes in the input \\(x(t)\\). As a result, the feedback weights reflect the average network Jacobian over time, but adapt to changes in the neuronal dynamics caused by feed-forward plasticity. Meulemans et al. [2, 3] have demonstrated that feedback weights reflecting the average Jacobian can also be learned using a simple, fully local anti-Hebbian learning rule based on noise correlations between neuronal activity and top-down control.\n\nTraining.Both student and teacher networks were randomly initialized using Kaiming initialization and hyperparameters relating to numerical integration and neuronal dynamics were kept identical for student and teacher networks (see Table S1). Student networks were trained continuously for a total of 5 minutes, split into 60 periods each lasting 5 seconds. For each 5-second window, we first computed the output of the teacher network to obtain a target for the top-down controller. We then computed the output of the student network in an open-loop setting without top-down control to calculate the MSE loss \\(\\mathcal{L}\\). Finally, we simulated the student network with active top-down control and continuously monitored the least-control loss \\(\\mathcal{H}\\). Numerical integration was performed using Tsitouras' 5th order explicit Runge Kutta method [13] with an adaptive step size.\n\n### Computer vision benchmarks\n\nData preprocessing and network architecture.MNIST and Fashion-MNIST images were rescaled to values between 0 and 1 and flattened before being used as inputs to the networks. Each hidden layer was composed of 256 excitatory-inhibitory microcircuit units for networks trained with disinhibitory control or 256 neurons for networks trained with BP.\n\nClassification with Softmax and cross-entropy loss.Conventionally, classification tasks are solved using a Softmax layer as output of the network. The Softmax layer is trained to match a one-hot encoded label \\(\\mathbf{r}_{L}^{\\text{tar}}\\). As pointed out by Meulemans et al. [3], because the Softmax layer output only matches the one-hot vector when it has an infinite input, this can lead to problems when feedback control is strong. Thus, we use soft targets where the value of \\(\\mathbf{r}_{L}^{\\text{tar}}\\) is \\(0.99\\) for the correct target and \\(\\nicefrac{{0.01}}{{n_{L}-1}}\\) for all other entries. Following [3], we use a linear output layer and absorb the Softmax operation into the cross-entropy loss function:\n\n\\[\\mathcal{L}^{\\text{classification}}=-\\sum_{n}\\mathbf{r}_{L}^{\\text{tar}}(n) \\log\\left(\\text{Softmax}\\left(\\mathbf{r}_{L}(n)\\right)\\right) \\tag{27}\\]\n\nOur dis-inhibitory controller and the associated learning rules require a nonlinear activation function and recurrent inhibition, which could interfere with the Softmax in the loss function. Thus, we implemented the readout layer for classification tasks as a linear layer without excitatory-inhibitory units:\n\n\\[\\tau_{E}\\frac{d}{dt}\\mathbf{r}_{L}=-\\mathbf{r}_{L}(t)+\\mathbf{W}_{L}\\mathbf{r}_{L- 1}^{E}(t)+\\mathbf{Q}_{L}\\mathbf{c}(t) \\tag{28}\\]\n\nwith \\(\\mathbf{Q}_{L}=I\\), and performed weight updates according to the deep feedback control learning rule [3].\n\nHyperparameter optimization.Optimal values for the controller parameters \\(k_{\\mathrm{p}}\\) and \\(k_{\\mathrm{i}}\\) were obtained using the Optuna package [14]. Optimization was performed using a Tree-structured Parzen Estimator [15] with 25 starts. Specifically, hyperparameters were chosen to minimize the validation loss of a 3-layer network after 20 training epochs using the exact inverse learning rule on the MNIST dataset.\n\nTraining.Network weights were initialized using Xavier initialization and trained for 50 epochs with a batchsize of 100. For every input sample, we first let the network settle to an equilibrium in the absence of feedback control. This uncontrolled equilibrium state was used to calculate the feedback weights \\(\\mathbf{Q}\\) and used as initial state for the second convergence with top-down control. Numerical integration was performed using Tsitouras' 5th order explicit Runge-Kutta method [13] with an adaptive step size for a maximum duration of \\(2\\) seconds or until equilibrium was reached, defined by an absolute tolerance value of \\(10^{-6}\\). Synaptic weight updates were performed at the controlled equilibrium state and averaged across each minibatch. For all experiments, we used the ADAM optimizer with with learning rate \\(0.001\\).\n\nFeedback weights.Feedback weights were calculated proportional to the normalized network Jacobian:\n\n\\[-\\mathbf{Q}_{i}=\\alpha\\frac{\\mathbf{\\tilde{J}}_{i}}{\\|\\mathbf{\\tilde{J}}_{i}\\| _{F}} \\tag{29}\\]\n\nwhere \\(\\mathbf{\\tilde{J}}_{i}\\) is the Jacobian obtained at the uncontrolled equilibrium state (cf. Eq. (7)). Because this calculation results in input-dependent feedback weights, which are biologically implausible, we also consider the case in which feedback weights reflect the average Jacobian. As demonstrated in Meulemans et al. [2, 3], such feedback connections could be learned locally making use of noise correlations between neuronal compartments receiving top-down feedback and the strength of the control signal. Here, we instead take a practical approach and take the average of the Jacobian across each minibatch:\n\n\\[-\\mathbf{Q}_{i}^{\\mathrm{(Avg\\,\\mathbf{J})}} = \\alpha\\frac{\\mathbf{\\tilde{J}}_{i}^{\\mathrm{Avg}}}{\\|\\mathbf{ \\tilde{J}}_{i}^{\\mathrm{Avg}}\\|_{F}} \\tag{30}\\] \\[\\mathbf{\\tilde{J}}_{i}^{\\mathrm{Avg}} = \\frac{1}{N}\\sum_{n=1}^{N}\\mathbf{\\tilde{J}}_{i} \\tag{31}\\]\n\n## References\n\n* [1] Joao Sacramento, Rui Ponte Costa, Yoshua Bengio, and Walter Senn. Dendritic cortical microcircuits approximate the backpropagation algorithm. In S Bengio, H Wallach, H Larochelle, K Grauman, N Cesa-Bianchi, and R Garnett, editors, _Advances in Neural Information Processing Systems 31_, pages 8721-8732. Curran Associates, Inc., 2018.\n* [2] Alexander Meulemans, Matilde Tristany Farinha, Javier Garcia Ordonez, Pau Vilimelis Aceituno, Joao Sacramento, and Benjamin F. Grewe. Credit assignment in neural networks through deep feedback control. In M. Ranzato, A. Beygelzimer, Y. Dauphin, P.S. Liang, and J. Wortman Vaughan, editors, _Advances in Neural Information Processing Systems_, volume 34, pages 4674-4687. Curran Associates, Inc., 2021.\n* [3] Alexander Meulemans, Matilde Tristany Farinha, Maria R. Cervera, Joao Sacramento, and Benjamin F. Grewe. Minimizing control for credit assignment with strong feedback. In Kamalika Chaudhuri, Stefanie Jegelka, Le Song, Csaba Szepesvari, Gang Niu, and Sivan Sabato, editors, _Proceedings of the 39th International Conference on Machine Learning_, volume 162 of _Proceedings of Machine Learning Research_, pages 15458-15483. PMLR, 2022.\n* [4] Alexandre Payeur, Jordan Guerguiev, Friedemann Zenke, Blake A Richards, and Richard Naud. Burst-dependent synaptic plasticity can coordinate learning in hierarchical circuits. _Nature neuroscience_, 2021. doi: 10.1038/s41593-021-00857-x.\n* [5] Will Greedy, Heng Wei Zhu, Joseph Pemberton, Jack Mellor, and Rui Ponte Costa. Single-phase deep learning in cortico-cortical networks. In S. Koyejo, S. Mohamed, A. Agarwal, D. Belgrave, K. Cho, and A. Oh, editors, _Advances in Neural Information Processing Systems_, volume 35, pages 24213-24225. Curran Associates, Inc., 2022.\n* [6] Robert Urbanczik and Walter Senn. Learning by the dendritic prediction of somatic spiking. _Neuron_, 81(3):521-528, 2014. doi: 10.1016/j.neuron.2013.11.030.\n* [7] Jordan Guerguiev, Timothy P Lillicrap, and Blake A Richards. Towards deep learning with segregated dendrites. _eLife_, 6, 2017. doi: 10.7554/eLife.22901.\n* [8] M E Larkum, J J Zhu, and B Sakmann. A new cellular mechanism for coupling inputs arriving at different cortical layers. _Nature_, 398(6725):338-341, 1999. doi: 10.1038/18686.\n* [9] James Bradbury, Roy Frostig, Peter Hawkins, Matthew James Johnson, Chris Leary, Dougal Maclaurin, George Necula, Adam Paszke, Jake VanderPlas, Skye Wanderman-Milne, and Qiao Zhang. JAX: composable transformations of Python+NumPy programs, 2018. URL [http://github.com/google/jax](http://github.com/google/jax).\n* [10] Jonathan Heek, Anselm Levskaya, Avital Oliver, Marvin Ritter, Bertrand Rondepierre, Andreas Steiner, and Marc van Zee. Flax: A neural network library and ecosystem for JAX, 2023. URL [http://github.com/google/flax](http://github.com/google/flax).\n* [11] Patrick Kidger. _On Neural Differential Equations_. PhD thesis, University of Oxford, 2021.\n* [12] TensorFlow Datasets, a collection of ready-to-use datasets. [https://www.tensorflow.org/datasets](https://www.tensorflow.org/datasets).\n* [13] Ch Tsitouras. Runge-Kutta pairs of order 5(4) satisfying only the first column simplifying assumption. _Computers & mathematics with applications_, 62(2):770-775, 2011. doi: 10.1016/j.camwa.2011.06.002.\n* [14] Takuya Akiba, Shotaro Sano, Toshihiko Yanase, Takeru Ohta, and Masanori Koyama. Optuna: A next-generation hyperparameter optimization framework. _arXiv_, 2019.\n* [15] James Bergstra, Remi Bardenet, Yoshua Bengio, and Balazs Kegl. Algorithms for Hyper-Parameter optimization. In J Shawe-Taylor, R Zemel, P Bartlett, F Pereira, and K Q Weinberger, editors, _Advances in Neural Information Processing Systems_, volume 24. Curran Associates, Inc., 2011.", "response": "## Review\n\n### Summary\n\nThe paper proposes a model of neural plasticity that integrates recurrent inhibition into deep neural networks (DNNs) to enhance biological plausibility and performance. It utilizes adaptive control theory to derive plasticity rules that can closely mimic experimentally observed plasticity phenomena while allowing DNNs to achieve comparable performance to backpropagation on tasks such as MNIST. The model includes feedback-driven inhibitory neurons that modulate excitatory connections, providing predictions that align with known plasticity data. Although the reviewers acknowledged the potential significance of the contributions, they raised concerns regarding the novelty and biological realism of the model, leading to a mixed reception that ultimately favors acceptance after further clarification from the authors.\n\n### Strengths\n\n- The microcircuit is biologically motivated.\n- The insights into how experimental conditions influence plasticity results are valuable.\n- The model represents one of the most biologically plausible models of backpropagation to date.\n- All sections are clearly articulated, making the work accessible to a broad audience.\n- The paper presents well-grounded theoretical derivations supported by experimental results.\n\n### Weaknesses\n\n- The motivation and novelty of the work are not fully articulated.\n- The results may not represent a substantial advancement over prior models.\n- The experimental evaluation could be broader and more comprehensive.\n- There are lingering issues of biological plausibility, particularly with the one-to-one mapping of excitatory to inhibitory neurons.\n- Some descriptions are confusing and contain typos that detract from clarity.\n\n### Questions\n\n- Why is there no weight matrix for the recurrent connections?\n- How does the model ensure stability, particularly regarding the rapid compensatory mechanism described?\n- What are the key differences between this model and others like Payeur et al. 2021?\n- Can the authors clarify the distinction between error and feedback signals in their model?\n- How does the model's feedback mechanism compare to existing models, particularly in relation to the role of inhibitory neurons?\n\n### Soundness\n\n**Score:** 3\n\n**Description:** 3 = good; the model is technically solid and presents a realistic plasticity mechanism, though with some concerns regarding biological plausibility and the complexity of its implementation.\n\n### Presentation\n\n**Score:** 3\n\n**Description:** 3 = good; the paper is well-written and organized, though some sections contain typos and unclear descriptions that could be improved.\n\n### Contribution\n\n**Score:** 3\n\n**Description:** 3 = good; while the contributions are meaningful, they are incremental compared to existing work, and the extent of novelty could be better articulated.\n\n### Rating\n\n**Score:** 7\n\n**Description:** 7 = accept, but needs minor improvements; the paper has potential impact but requires clarification on several points to strengthen its case.\n\n### Paper Decision\n\n**Decision:** Accept (poster)\n\n**Reasons:** The paper presents a novel integration of biological principles into neural networks, demonstrating both theoretical and experimental merit. Although it raises questions regarding its contributions relative to existing literature and exhibits some clarity issues, the reviewers generally found the work technically sound and significant in its implications for understanding neural plasticity. The decision to accept is supported by the paper's overall quality and the authors' satisfactory responses to reviewer concerns.\n"}