Title: EMaQ: Expected-Max Q-Learning Operator for Simple Yet Effective Offline and Online RL. Abstract: Off-policy reinforcement learning (RL) holds the promise of sample-efficient learning of decision-making policies by leveraging past experience. However, in the offline RL setting -- where a fixed collection of interactions are provided and no further interactions are allowed -- it has been shown that standard off-policy RL methods can significantly underperform. Recently proposed methods often aim to address this shortcoming by constraining learned policies to remain close to the given dataset of interactions. In this work, we closely investigate an important simplification of BCQ~\citep{fujimoto2018off} -- a prior approach for offline RL -- which removes a heuristic design choice and naturally restrict extracted policies to remain \emph{exactly} within the support of a given behavior policy. Importantly, in contrast to their original theoretical considerations, we derive this simplified algorithm through the introduction of a novel backup operator, Expected-Max Q-Learning (EMaQ), which is more closely related to the resulting practical algorithm. Specifically, in addition to the distribution support, EMaQ explicitly considers the number of samples and the proposal distribution, allowing us to derive new sub-optimality bounds which can serve as a novel measure of complexity for offline RL problems. In the offline RL setting -- the main focus of this work -- EMaQ matches and outperforms prior state-of-the-art in the D4RL benchmarks~\citep{fu2020d4rl}. In the online RL setting, we demonstrate that EMaQ is competitive with Soft Actor Critic (SAC). The key contributions of our empirical findings are demonstrating the importance of careful generative model design for estimating behavior policies, and an intuitive notion of complexity for offline RL problems. With its simple interpretation and fewer moving parts, such as no explicit function approximator representing the policy, EMaQ serves as a strong yet easy to implement baseline for future work.