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app/src/index.html

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-    <title>Scaling FineWeb to 1000+ languages: Step 1: finding signal in 100s of evaluation tasks</title>
+    <title>Scaling test-time compute for open models: How we implemented DeepMind’s compute-optimal recipe to solve hard math problems like OpenAI’s o1</title>
     <link rel="stylesheet" href="style.css">
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@@ -139,7 +139,9 @@ def get_canonical_form(expression: str) -&gt; str:
             
 <p id="15d1384e-bcac-804e-a99c-fe5e83313a3d" class="">This approach was significantly faster than checking each pair of solutions independently for equality.</p></div></details><p id="15b1384e-bcac-80f7-83e8-e1d6b360faa4" class="">Here’s how majority voting performs when applied to the generations from Llama 3.2 1B Instruct:</p><figure id="15b1384e-bcac-8072-9987-d80031b97793" class="image"><a href="Scaling%20test-time%20compute%20with%20open%20models%201531384ebcac800b9d73fca3503eb783/methods-maj.png"><img style="width:707.9891357421875px" src="https://huggingface.co/datasets/HuggingFaceH4/blogpost-images/resolve/main/methods-maj.png"/></a></figure><p id="15b1384e-bcac-8020-8688-fe1713e92c2b" class="">The results show that majority voting yields a significant improvement over the greedy decoding baseline, but its gains start to plateau after approximately <style>@import url('https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.9/katex.min.css')</style><span data-token-index="0" contenteditable="false" class="notion-text-equation-token" style="user-select:all;-webkit-user-select:all;-moz-user-select:all"><span></span><span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>N</mi><mo>=</mo><mn>64</mn></mrow><annotation encoding="application/x-tex">N=64</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">N</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">64</span></span></span></span></span><span></span></span> generations. This limitation arises because majority voting struggles with problems that require nuanced reasoning or tasks where errors are consistent across generations. If you’re also wondering why the majority voting accuracy is worse than the 0-shot CoT baseline for N=1 and 2, that’s because we sample at T=0.8, which makes it less likely we produce the correct answer among a handful of candidates.</p><p id="15b1384e-bcac-8075-8fef-f26f0b8e5559" class="">Building on the limitations of majority voting, let’s see how incorporating a reward model can enhance performance.</p>
 
-<h2 id="1591384e-bcac-8098-9db5-f76c9ce00e7a" class="">Beyond majority: Best-of-N</h2><p id="15b1384e-bcac-8019-9b5c-d11bae74628d" class="">Best-of-N is a simple, but effective extension to majority voting that uses a reward model to determine the most plausible answer. This method comes in two main variants:</p><ul id="15b1384e-bcac-80b4-aae4-d5e98e29debf" class="bulleted-list"><li style="list-style-type:disc"><strong>Vanilla Best-of-N:</strong> Generate <style>@import url('https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.9/katex.min.css')</style><span data-token-index="0" contenteditable="false" class="notion-text-equation-token" style="user-select:all;-webkit-user-select:all;-moz-user-select:all"><span></span><span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>N</mi></mrow><annotation encoding="application/x-tex">N</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">N</span></span></span></span></span><span></span></span> independent responses and select the one with the <em>highest  RM reward</em> as the final answer. This ensures that the most confident individual response is chosen, but it doesn’t account for consistency across answers.</li></ul><ul id="15b1384e-bcac-8035-a394-fbd954af1984" class="bulleted-list"><li style="list-style-type:disc"><strong>Weighted Best-of-N:</strong> Aggregate scores across all identical responses and select the answer with the <em>highest total reward</em>. This approach prioritises high-quality answers by boosting their scores through repeated occurrences. Mathematically, the weighting across answers <style>@import url('https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.9/katex.min.css')</style><span data-token-index="0" contenteditable="false" class="notion-text-equation-token" style="user-select:all;-webkit-user-select:all;-moz-user-select:all"><span></span><span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>a</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">a_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span><span></span></span> is performed as follows:<figure id="15d1384e-bcac-80e5-8d68-fe7bad033482" class="equation"><style>@import url('https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.9/katex.min.css')</style><div class="equation-container"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>a</mi><mrow><mi mathvariant="normal">w</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">g</mi><mi mathvariant="normal">h</mi><mi mathvariant="normal">t</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">d</mi></mrow></msub><mo>=</mo><mi>arg</mi><mo>⁡</mo><munder><mrow><mi>max</mi><mo>⁡</mo></mrow><mi>a</mi></munder><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>N</mi></munderover><mi mathvariant="double-struck">I</mi><mo stretchy="false">(</mo><msub><mi>a</mi><mi>i</mi></msub><mo>=</mo><mi>a</mi><mo stretchy="false">)</mo><mo>⋅</mo><mrow><mi mathvariant="normal">R</mi><mi mathvariant="normal">M</mi></mrow><mo stretchy="false">(</mo><mi>p</mi><mo separator="true">,</mo><msub><mi>s</mi><mi>i</mi></msub><mo stretchy="false">)</mo><mtext> </mtext><mo separator="true">,</mo></mrow><annotation encoding="application/x-tex">a_\mathrm{weighted} = \arg\max_{a} \sum_{i=1}^{N} \mathbb{I}(a_i = a) \cdot \mathrm{RM}(p, s_i) \,,</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7167em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathrm mtight">weighted</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:3.106em;vertical-align:-1.2777em;"></span><span class="mop">ar<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.4306em;"><span style="top:-2.4em;margin-left:0em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">a</span></span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span><span class="mop">max</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.7em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8283em;"><span style="top:-1.8723em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.10903em;">N</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2777em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathbb">I</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">a</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathrm">RM</span></span><span class="mopen">(</span><span class="mord mathnormal">p</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mpunct">,</span></span></span></span></span></div></figure><p id="15d1384e-bcac-8083-8f2a-d5701df84dcd" class="">where <style>@import url('https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.9/katex.min.css')</style><span data-token-index="0" contenteditable="false" class="notion-text-equation-token" style="user-select:all;-webkit-user-select:all;-moz-user-select:all"><span></span><span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mrow><mi mathvariant="normal">R</mi><mi mathvariant="normal">M</mi></mrow><mo stretchy="false">(</mo><mi>p</mi><mo separator="true">,</mo><msub><mi>s</mi><mi>i</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathrm{RM}(p, s_i)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathrm">RM</span></span><span class="mopen">(</span><span class="mord mathnormal">p</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span><span></span></span> is the reward model score of the <style>@import url('https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.9/katex.min.css')</style><span data-token-index="0" contenteditable="false" class="notion-text-equation-token" style="user-select:all;-webkit-user-select:all;-moz-user-select:all"><span></span><span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span></span><span></span></span>-th solution solution <style>@import url('https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.9/katex.min.css')</style><span data-token-index="0" contenteditable="false" class="notion-text-equation-token" style="user-select:all;-webkit-user-select:all;-moz-user-select:all"><span></span><span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>s</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">s_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span><span></span></span> to problem <style>@import url('https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.9/katex.min.css')</style><span data-token-index="0" contenteditable="false" class="notion-text-equation-token" style="user-select:all;-webkit-user-select:all;-moz-user-select:all"><span></span><span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">p</span></span></span></span></span><span></span></span>.</p></li></ul><p id="15d1384e-bcac-8012-8282-c0ed1215a611" class="">Typically, one usually uses an outcome reward model (ORM) to get a single, solution-level score. But to allow for fair comparison with the other search strategies discussed later, we will use the same PRM to score the solutions from Best-of-N. As illustrated below, PRMs produce a <em>cumulative</em> <em>sequence of step-level scores</em> per solution, so we need to perform a reduction over the steps to obtain a single solution-level score: </p><figure id="15d1384e-bcac-80d6-815f-c7d87fe313a6" class="image"><a href="https://huggingface.co/datasets/HuggingFaceH4/blogpost-images/resolve/main/prm-reductions.png"><img style="width:700px" src="https://huggingface.co/datasets/HuggingFaceH4/blogpost-images/resolve/main/prm-reductions.png"/></a></figure><p id="15d1384e-bcac-80e7-8d1a-e0aab286f9f4" class="">In the literature, the most common reductions are the following:</p><ul id="15b1384e-bcac-80e4-92b4-e2bc90a9130a" class="bulleted-list"><li style="list-style-type:disc"><strong>Min: </strong>use the minimum score across all steps.</li></ul><ul id="15b1384e-bcac-8073-b4dc-fbfcfc0567bc" class="bulleted-list"><li style="list-style-type:disc"><strong>Prod: </strong>use the product of step-level scores.</li></ul><ul id="15b1384e-bcac-80ed-8cc5-fa6e2ce330fb" class="bulleted-list"><li style="list-style-type:disc"><strong>Last: </strong>use the final score in the steps. This score contains the cumulative information from all prior steps, so treats the PRM effectively as an ORM that is able to score partial solutions.</li></ul><p id="15b1384e-bcac-80ad-96d1-d313ae3e1954" class="">We experimented with each reduction and found—like DeepMind—that <em><strong>“last” performs best for our choice of task and PRM</strong></em>. We use this aggregation throughout all of our experiments and you can expand the detail below to see how we implemented it, along with the weighting procedure discussed above.</p><details><summary style="font-weight:600;font-size:1.25em;line-height:1.3;margin:0">Implementation detail</summary></details><p id="15d1384e-bcac-809a-8aa8-c52ca7301b52" class="">Here’s the results one gets from applying both variants of Best-of-N:</p><figure id="15b1384e-bcac-808d-857e-d492683a4a91" class="image"><a href="https://huggingface.co/datasets/HuggingFaceH4/blogpost-images/resolve/main/methods-maj-bon.png"><img style="width:707.9891357421875px" src="https://huggingface.co/datasets/HuggingFaceH4/blogpost-images/resolve/main/methods-maj-bon.png"/></a></figure><p id="15b1384e-bcac-8001-9320-ff788bab0c52" class="">The results reveal a clear advantage: <strong>weighted Best-of-N</strong> consistently outperforms vanilla Best-of-N, especially with larger generation budgets. Its ability to aggregate scores across identical responses ensures that even less frequent but higher-quality answers are effectively prioritized.</p><p id="15b1384e-bcac-808a-b3ff-ee08c05a20af" class="">However, despite these improvements, we’re still falling short of the performance achieved by the Llama 8B model and the Best-of-N approach is starting to plateau at <style>@import url('https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.9/katex.min.css')</style><span data-token-index="0" contenteditable="false" class="notion-text-equation-token" style="user-select:all;-webkit-user-select:all;-moz-user-select:all"><span></span><span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>N</mi><mo>=</mo><mn>256</mn></mrow><annotation encoding="application/x-tex">N=256</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">N</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">256</span></span></span></span></span><span></span></span> generations. Can we push the boundaries further by supervising the search process step-by-step? Let’s find out 🚀!</p>
+<h2 id="1591384e-bcac-8098-9db5-f76c9ce00e7a" class="">Beyond majority: Best-of-N</h2><p id="15b1384e-bcac-8019-9b5c-d11bae74628d" class="">Best-of-N is a simple, but effective extension to majority voting that uses a reward model to determine the most plausible answer. This method comes in two main variants:</p><ul id="15b1384e-bcac-80b4-aae4-d5e98e29debf" class="bulleted-list"><li style="list-style-type:disc"><strong>Vanilla Best-of-N:</strong> Generate <style>@import url('https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.9/katex.min.css')</style><span data-token-index="0" contenteditable="false" class="notion-text-equation-token" style="user-select:all;-webkit-user-select:all;-moz-user-select:all"><span></span><span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>N</mi></mrow><annotation encoding="application/x-tex">N</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">N</span></span></span></span></span><span></span></span> independent responses and select the one with the <em>highest  RM reward</em> as the final answer. This ensures that the most confident individual response is chosen, but it doesn’t account for consistency across answers.</li></ul><ul id="15b1384e-bcac-8035-a394-fbd954af1984" class="bulleted-list"><li style="list-style-type:disc"><strong>Weighted Best-of-N:</strong> Aggregate scores across all identical responses and select the answer with the <em>highest total reward</em>. This approach prioritises high-quality answers by boosting their scores through repeated occurrences. Mathematically, the weighting across answers <style>@import url('https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.9/katex.min.css')</style><span data-token-index="0" contenteditable="false" class="notion-text-equation-token" style="user-select:all;-webkit-user-select:all;-moz-user-select:all"><span></span><span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>a</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">a_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span><span></span></span> is performed as follows:<figure id="15d1384e-bcac-80e5-8d68-fe7bad033482" class="equation"><style>@import url('https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.9/katex.min.css')</style><div class="equation-container"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>a</mi><mrow><mi mathvariant="normal">w</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">g</mi><mi mathvariant="normal">h</mi><mi mathvariant="normal">t</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">d</mi></mrow></msub><mo>=</mo><mi>arg</mi><mo>⁡</mo><munder><mrow><mi>max</mi><mo>⁡</mo></mrow><mi>a</mi></munder><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>N</mi></munderover><mi mathvariant="double-struck">I</mi><mo stretchy="false">(</mo><msub><mi>a</mi><mi>i</mi></msub><mo>=</mo><mi>a</mi><mo stretchy="false">)</mo><mo>⋅</mo><mrow><mi mathvariant="normal">R</mi><mi mathvariant="normal">M</mi></mrow><mo stretchy="false">(</mo><mi>p</mi><mo separator="true">,</mo><msub><mi>s</mi><mi>i</mi></msub><mo stretchy="false">)</mo><mtext> </mtext><mo separator="true">,</mo></mrow><annotation encoding="application/x-tex">a_\mathrm{weighted} = \arg\max_{a} \sum_{i=1}^{N} \mathbb{I}(a_i = a) \cdot \mathrm{RM}(p, s_i) \,,</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7167em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathrm mtight">weighted</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:3.106em;vertical-align:-1.2777em;"></span><span class="mop">ar<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.4306em;"><span style="top:-2.4em;margin-left:0em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">a</span></span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span><span class="mop">max</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.7em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8283em;"><span style="top:-1.8723em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.10903em;">N</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2777em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathbb">I</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">a</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathrm">RM</span></span><span class="mopen">(</span><span class="mord mathnormal">p</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mpunct">,</span></span></span></span></span></div></figure><p id="15d1384e-bcac-8083-8f2a-d5701df84dcd" class="">where <style>@import url('https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.9/katex.min.css')</style><span data-token-index="0" contenteditable="false" class="notion-text-equation-token" style="user-select:all;-webkit-user-select:all;-moz-user-select:all"><span></span><span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mrow><mi mathvariant="normal">R</mi><mi mathvariant="normal">M</mi></mrow><mo stretchy="false">(</mo><mi>p</mi><mo separator="true">,</mo><msub><mi>s</mi><mi>i</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathrm{RM}(p, s_i)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathrm">RM</span></span><span class="mopen">(</span><span class="mord mathnormal">p</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span><span></span></span> is the reward model score of the <style>@import url('https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.9/katex.min.css')</style><span data-token-index="0" contenteditable="false" class="notion-text-equation-token" style="user-select:all;-webkit-user-select:all;-moz-user-select:all"><span></span><span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span></span><span></span></span>-th solution solution <style>@import url('https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.9/katex.min.css')</style><span data-token-index="0" contenteditable="false" class="notion-text-equation-token" style="user-select:all;-webkit-user-select:all;-moz-user-select:all"><span></span><span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>s</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">s_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span><span></span></span> to problem <style>@import url('https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.9/katex.min.css')</style><span data-token-index="0" contenteditable="false" class="notion-text-equation-token" style="user-select:all;-webkit-user-select:all;-moz-user-select:all"><span></span><span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">p</span></span></span></span></span><span></span></span>.</p></li></ul><p id="15d1384e-bcac-8012-8282-c0ed1215a611" class="">Typically, one usually uses an outcome reward model (ORM) to get a single, solution-level score. But to allow for fair comparison with the other search strategies discussed later, we will use the same PRM to score the solutions from Best-of-N. As illustrated below, PRMs produce a <em>cumulative</em> <em>sequence of step-level scores</em> per solution, so we need to perform a reduction over the steps to obtain a single solution-level score: </p><figure id="15d1384e-bcac-80d6-815f-c7d87fe313a6" class="image"><a href="https://huggingface.co/datasets/HuggingFaceH4/blogpost-images/resolve/main/prm-reductions.png"><img style="width:700px" src="https://huggingface.co/datasets/HuggingFaceH4/blogpost-images/resolve/main/prm-reductions.png"/></a></figure><p id="15d1384e-bcac-80e7-8d1a-e0aab286f9f4" class="">In the literature, the most common reductions are the following:</p><ul id="15b1384e-bcac-80e4-92b4-e2bc90a9130a" class="bulleted-list"><li style="list-style-type:disc"><strong>Min: </strong>use the minimum score across all steps.</li></ul><ul id="15b1384e-bcac-8073-b4dc-fbfcfc0567bc" class="bulleted-list"><li style="list-style-type:disc"><strong>Prod: </strong>use the product of step-level scores.</li></ul><ul id="15b1384e-bcac-80ed-8cc5-fa6e2ce330fb" class="bulleted-list"><li style="list-style-type:disc"><strong>Last: </strong>use the final score in the steps. This score contains the cumulative information from all prior steps, so treats the PRM effectively as an ORM that is able to score partial solutions.</li></ul><p id="15b1384e-bcac-80ad-96d1-d313ae3e1954" class="">We experimented with each reduction and found—like DeepMind—that <em><strong>“last” performs best for our choice of task and PRM</strong></em>. We use this aggregation throughout all of our experiments and you can expand the detail below to see how we implemented it, along with the weighting procedure discussed above.</p>
+
+<p id="15d1384e-bcac-809a-8aa8-c52ca7301b52" class="">Here’s the results one gets from applying both variants of Best-of-N:</p><figure id="15b1384e-bcac-808d-857e-d492683a4a91" class="image"><a href="https://huggingface.co/datasets/HuggingFaceH4/blogpost-images/resolve/main/methods-maj-bon.png"><img style="width:707.9891357421875px" src="https://huggingface.co/datasets/HuggingFaceH4/blogpost-images/resolve/main/methods-maj-bon.png"/></a></figure><p id="15b1384e-bcac-8001-9320-ff788bab0c52" class="">The results reveal a clear advantage: <strong>weighted Best-of-N</strong> consistently outperforms vanilla Best-of-N, especially with larger generation budgets. Its ability to aggregate scores across identical responses ensures that even less frequent but higher-quality answers are effectively prioritized.</p><p id="15b1384e-bcac-808a-b3ff-ee08c05a20af" class="">However, despite these improvements, we’re still falling short of the performance achieved by the Llama 8B model and the Best-of-N approach is starting to plateau at <style>@import url('https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.9/katex.min.css')</style><span data-token-index="0" contenteditable="false" class="notion-text-equation-token" style="user-select:all;-webkit-user-select:all;-moz-user-select:all"><span></span><span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>N</mi><mo>=</mo><mn>256</mn></mrow><annotation encoding="application/x-tex">N=256</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">N</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">256</span></span></span></span></span><span></span></span> generations. Can we push the boundaries further by supervising the search process step-by-step? Let’s find out 🚀!</p>
 
 			<h2 id="1591384e-bcac-8065-a02c-cd760ebd6cd1" class="">Beam search with process reward models</h2><p id="15a1384e-bcac-80e1-9e0e-c01f5f373805" class="">Beam search is a structured search method that systematically explores the solution space, making it a powerful tool for improving model outputs at test-time. When combined with a PRM, beam search can optimize both the generation and evaluation of intermediate steps in problem-solving. The way it works is as follows:</p><ol type="1" id="15d1384e-bcac-8007-8d79-cdaa74e4c8c0" class="numbered-list" start="1"><li>Generate multiple candidate solutions <em>iteratively</em> by maintaining a fixed number of &quot;beams&quot; or active paths <style>@import url('https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.9/katex.min.css')</style><span data-token-index="0" contenteditable="false" class="notion-text-equation-token" style="user-select:all;-webkit-user-select:all;-moz-user-select:all"><span></span><span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>N</mi></mrow><annotation encoding="application/x-tex">N</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">N</span></span></span></span></span><span></span></span>.</li></ol><ol type="1" id="15d1384e-bcac-8020-bf69-e67fd962062b" class="numbered-list" start="2"><li>In the first iteration, sample <style>@import url('https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.9/katex.min.css')</style><span data-token-index="0" contenteditable="false" class="notion-text-equation-token" style="user-select:all;-webkit-user-select:all;-moz-user-select:all"><span></span><span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>N</mi></mrow><annotation encoding="application/x-tex">N</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">N</span></span></span></span></span><span></span></span> independent steps from the LLM with temperature <style>@import url('https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.9/katex.min.css')</style><span data-token-index="0" contenteditable="false" class="notion-text-equation-token" style="user-select:all;-webkit-user-select:all;-moz-user-select:all"><span></span><span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">T</span></span></span></span></span><span></span></span> to introduce diversity in the responses. These steps are usually defined by a stopping criterion like terminating on a new line <code>\n</code> or double new line <code>\n\n</code>.</li></ol><ol type="1" id="15d1384e-bcac-80c2-aeaa-f6d73682eb8c" class="numbered-list" start="3"><li>Score each step with the PRM and select the top <style>@import url('https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.9/katex.min.css')</style><span data-token-index="0" contenteditable="false" class="notion-text-equation-token" style="user-select:all;-webkit-user-select:all;-moz-user-select:all"><span></span><span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>N</mi><mi mathvariant="normal">/</mi><mi>M</mi></mrow><annotation encoding="application/x-tex">N/M</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">N</span><span class="mord">/</span><span class="mord mathnormal" style="margin-right:0.10903em;">M</span></span></span></span></span><span></span></span> steps as candidates for the next round of generation. Here <style>@import url('https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.9/katex.min.css')</style><span data-token-index="0" contenteditable="false" class="notion-text-equation-token" style="user-select:all;-webkit-user-select:all;-moz-user-select:all"><span></span><span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">M</span></span></span></span></span><span></span></span> denotes the “beam width” of a given active path. As in Best-of-N, we used the “last” reduction to score the partial solutions at each iteration.</li></ol><ol type="1" id="15d1384e-bcac-8022-966b-e1dae6845cc1" class="numbered-list" start="4"><li>Expand the steps selected in step (3) by sampling <style>@import url('https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.9/katex.min.css')</style><span data-token-index="0" contenteditable="false" class="notion-text-equation-token" style="user-select:all;-webkit-user-select:all;-moz-user-select:all"><span></span><span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">M</span></span></span></span></span><span></span></span> next steps in the solution.</li></ol><ol type="1" id="15d1384e-bcac-8023-b6b6-f470e22ac78a" class="numbered-list" start="5"><li>Repeat steps (3) and (4) until the EOS token is reached or the maximum search depth is exceeded.</li></ol><p id="15a1384e-bcac-8003-a9d9-da7f3a4dc321" class="">By allowing the PRM to evaluate the correctness of intermediate steps, beam search can identify and prioritize promising paths early in the process. This step-by-step evaluation is particularly beneficial for complex reasoning tasks like mathematics, where verifying partial solutions can significantly improve final outcomes.</p><details><summary style="font-weight:600;font-size:1.25em;line-height:1.3;margin:0">Implementation detail</summary><div class="indented"><p id="15b1384e-bcac-8065-a739-d24b699106be" class="">When we implemented beam search with process supervision, we encountered two major footguns with the Llama 3 chat template that are worth mentioning:</p><ul id="15d1384e-bcac-803c-84b3-d881bc2ca3b5" class="bulleted-list"><li style="list-style-type:disc">By default, the chat template trims trailing new lines from every assistant turn. As a result, if one uses <code>\n</code> or <code>\n\n</code> to terminate a step, these tokens are lost on subsequent steps and force the model to produce peculiar outputs.</li></ul><ul id="15d1384e-bcac-808f-97f1-fb7d27565e36" class="bulleted-list"><li style="list-style-type:disc">The chat template is prefixed with Llama’s BOS token. When the formatted string is fed to vLLM a <em>second</em> BOS token is added which completely ruins performance, even though the generations look mostly coherent 🤯</li></ul><p id="15d1384e-bcac-8041-9164-ecc3d9497886" class="">The solution is to overwrite the Llama 3 chat template to prevent trimming and exclude the BOS token prefix. </p><p id="15a1384e-bcac-8090-b5fc-eb36a6588e60" class="">
 </p></div></details><p id="15d1384e-bcac-80e9-8e65-e1b58080b94c" class="">In our experiments, we followed DeepMind’s hyperparameter choices and ran beam search with the following:</p><ul id="15d1384e-bcac-8098-8574-e16392fc6123" class="bulleted-list"><li style="list-style-type:disc"><style>@import url('https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.9/katex.min.css')</style><span data-token-index="0" contenteditable="false" class="notion-text-equation-token" style="user-select:all;-webkit-user-select:all;-moz-user-select:all"><span></span><span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>N</mi></mrow><annotation encoding="application/x-tex">N</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">N</span></span></span></span></span><span></span></span> beams in compute scalings of 4, 16, 64, 256</li></ul><ul id="15d1384e-bcac-8067-b37c-e9692e34678c" class="bulleted-list"><li style="list-style-type:disc">Fixed beam width <style>@import url('https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.9/katex.min.css')</style><span data-token-index="0" contenteditable="false" class="notion-text-equation-token" style="user-select:all;-webkit-user-select:all;-moz-user-select:all"><span></span><span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>M</mi><mo>=</mo><mn>4</mn></mrow><annotation encoding="application/x-tex">M=4</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">M</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">4</span></span></span></span></span><span></span></span></li></ul><ul id="15d1384e-bcac-8093-a928-c16e31e29e3f" class="bulleted-list"><li style="list-style-type:disc">Sampling with temperature <style>@import url('https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.9/katex.min.css')</style><span data-token-index="0" contenteditable="false" class="notion-text-equation-token" style="user-select:all;-webkit-user-select:all;-moz-user-select:all"><span></span><span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>T</mi><mo>=</mo><mn>0.8</mn></mrow><annotation encoding="application/x-tex">T=0.8</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">T</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0.8</span></span></span></span></span><span></span></span></li></ul><ul id="15d1384e-bcac-802a-8416-e332ca20237f" class="bulleted-list"><li style="list-style-type:disc">Up to 40 iterations, i.e. a tree of maximum depth with 40 steps.</li></ul><p id="15d1384e-bcac-8051-abe5-dc84c42a1b5f" class="">As shown below, the results are striking: with a test-time budget of <style>@import url('https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.9/katex.min.css')</style><span data-token-index="0" contenteditable="false" class="notion-text-equation-token" style="user-select:all;-webkit-user-select:all;-moz-user-select:all"><span></span><span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>N</mi><mo>=</mo><mn>4</mn></mrow><annotation encoding="application/x-tex">N=4</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">N</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">4</span></span></span></span></span><span></span></span>, beam search achieves the same accuracy as Best-of-N for <style>@import url('https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.9/katex.min.css')</style><span data-token-index="0" contenteditable="false" class="notion-text-equation-token" style="user-select:all;-webkit-user-select:all;-moz-user-select:all"><span></span><span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>N</mi><mo>=</mo><mn>16</mn></mrow><annotation encoding="application/x-tex">N=16</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">N</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">16</span></span></span></span></span><span></span></span>, i.e. it is 4x more compute efficient! Moreover, beam search matches the performance of Llama 3.1 8B with just <style>@import url('https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.9/katex.min.css')</style><span data-token-index="0" contenteditable="false" class="notion-text-equation-token" style="user-select:all;-webkit-user-select:all;-moz-user-select:all"><span></span><span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>N</mi><mo>=</mo><mn>32</mn></mrow><annotation encoding="application/x-tex">N=32</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">N</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">32</span></span></span></span></span><span></span></span> solutions per problem. The average performance on MATH by computer science PhD students is around 40%,  so reaching nearly 55% isn’t too bad for a 1B model 💪!</p><figure id="15b1384e-bcac-80e9-97fa-fe50d1811f5b" class="image"><a href="https://huggingface.co/datasets/HuggingFaceH4/blogpost-images/resolve/main/methods-maj-bon-beam.png"><img style="width:707.9891357421875px" src="https://huggingface.co/datasets/HuggingFaceH4/blogpost-images/resolve/main/methods-maj-bon-beam.png"/></a></figure><h3 id="15a1384e-bcac-800c-baee-fb99b242ef87" class="">Which problems does beam search solve best?</h3><p id="15d1384e-bcac-80e3-938a-c3f09db2e9ff" class="">Although in aggregate it is clear that beam search is a better search strategy than Best-of-N or majority voting, the DeepMind paper showed that <em><strong>each strategy has tradeoffs that depend on the problem difficulty</strong></em> and test-time compute budget. </p><p id="15d1384e-bcac-8015-a8f0-c2323b9e535f" class="">To see which problems are best suited for which strategy, DeepMind computed a distribution over estimated problem difficulty, and then binned the results into quintiles. In other words, each problem is assigned one of 5 levels, where level 1 indicates easier problems and level 5 indicates the hardest ones. To estimate problem difficulty, DeepMind generated 2048 candidate solutions with standard sampling per problem and then proposed the following heuristics:</p><ul id="15d1384e-bcac-8080-9152-caeaa288073c" class="bulleted-list"><li style="list-style-type:disc"><strong>Oracle: </strong>use the ground truth labels to estimate the <style>@import url('https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.9/katex.min.css')</style><span data-token-index="0" contenteditable="false" class="notion-text-equation-token" style="user-select:all;-webkit-user-select:all;-moz-user-select:all"><span></span><span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi>a</mi><mi>s</mi><mi>s</mi><mi mathvariant="normal">@</mi><mn>1</mn></mrow><annotation encoding="application/x-tex">pass@1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">p</span><span class="mord mathnormal">a</span><span class="mord mathnormal">ss</span><span class="mord">@1</span></span></span></span></span><span></span></span> score per problem. Bin the distribution of <style>@import url('https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.9/katex.min.css')</style><span data-token-index="0" contenteditable="false" class="notion-text-equation-token" style="user-select:all;-webkit-user-select:all;-moz-user-select:all"><span></span><span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi>a</mi><mi>s</mi><mi>s</mi><mi mathvariant="normal">@</mi><mn>1</mn></mrow><annotation encoding="application/x-tex">pass@1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">p</span><span class="mord mathnormal">a</span><span class="mord mathnormal">ss</span><span class="mord">@1</span></span></span></span></span><span></span></span> scores to determine the quintiles.</li></ul><ul id="15d1384e-bcac-80f9-8778-d4045c6faa7d" class="bulleted-list"><li style="list-style-type:disc"><strong>Model: </strong>use the distribution of average PRM scores per problem to determine the quintiles. The intuition here is that harder problems will have lower scores.</li></ul><p id="15d1384e-bcac-80a3-af7c-f3497126ab1e" class="">Here’s the breakdown of the various methods according to the pass@1 scores and across four test-time compute budgets of <style>@import url('https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.9/katex.min.css')</style><span data-token-index="0" contenteditable="false" class="notion-text-equation-token" style="user-select:all;-webkit-user-select:all;-moz-user-select:all"><span></span><span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>N</mi><mo>=</mo><mo stretchy="false">[</mo><mn>4</mn><mo separator="true">,</mo><mn>16</mn><mo separator="true">,</mo><mn>64</mn><mo separator="true">,</mo><mn>256</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">N = [4,16,64, 256]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">N</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord">4</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">16</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">64</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">256</span><span class="mclose">]</span></span></span></span></span><span></span></span>:</p><figure id="15b1384e-bcac-80ad-9cf3-cf5bcbd3f53b" class="image"><a href="https://huggingface.co/datasets/HuggingFaceH4/blogpost-images/resolve/main/levels-maj-bon-beam.png"><img style="width:707.9891357421875px" src="https://huggingface.co/datasets/HuggingFaceH4/blogpost-images/resolve/main/levels-maj-bon-beam.png"/></a></figure><p id="15d1384e-bcac-80c3-93b3-fa4c071ac807" class="">In this plot, each bar denotes a test-time compute budget, and within each bar we show the relative accuracy of each method. For example, in the group of four bars on difficulty level 2 we see that:</p><ul id="15d1384e-bcac-8091-b3fb-cad0ab99b2c1" class="bulleted-list"><li style="list-style-type:disc">Majority voting is the worst performer for all compute budgets, except for <style>@import url('https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.9/katex.min.css')</style><span data-token-index="0" contenteditable="false" class="notion-text-equation-token" style="user-select:all;-webkit-user-select:all;-moz-user-select:all"><span></span><span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>N</mi><mo>=</mo><mn>256</mn></mrow><annotation encoding="application/x-tex">N=256</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">N</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">256</span></span></span></span></span><span></span></span>, where beam search is worst.</li></ul><ul id="15d1384e-bcac-8076-b88c-c7f55fa0cdbc" class="bulleted-list"><li style="list-style-type:disc">Beam search is best for <style>@import url('https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.9/katex.min.css')</style><span data-token-index="0" contenteditable="false" class="notion-text-equation-token" style="user-select:all;-webkit-user-select:all;-moz-user-select:all"><span></span><span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>N</mi><mo>=</mo><mo stretchy="false">[</mo><mn>4</mn><mo separator="true">,</mo><mn>16</mn><mo separator="true">,</mo><mn>64</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">N=[4,16,64]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">N</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord">4</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">16</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">64</span><span class="mclose">]</span></span></span></span></span><span></span></span>, but Best-of-N is best for <style>@import url('https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.9/katex.min.css')</style><span data-token-index="0" contenteditable="false" class="notion-text-equation-token" style="user-select:all;-webkit-user-select:all;-moz-user-select:all"><span></span><span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>N</mi><mo>=</mo><mn>256</mn></mrow><annotation encoding="application/x-tex">N=256</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">N</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">256</span></span></span></span></span><span></span></span>.</li></ul><p id="15a1384e-bcac-80d4-af98-eaebf5fcf84e" class="">Although we see that beam search gives consistent gains in the medium and hard problems (levels 3-5), it tends to do worse than Best-of-N (and even majority voting!) on the simpler problems and especially at large compute budgets. </p><p id="15a1384e-bcac-805b-9949-f0cdc44c9e3c" class="">We realized from looking at the resulting trees produced by beam search, that if a single step is assigned high reward, then the whole tree collapses to that trace and thus diversity is impacted. This prompted us to explore an extension to beam search that maximises diversity - let’s take a look!</p>
@@ -150,7 +152,7 @@ def get_canonical_form(expression: str) -&gt; str:
 
 <h2 id="1591384e-bcac-809a-96d2-e928398d159a" class="">Scaling up to larger models</h2><p id="15a1384e-bcac-8078-86d7-f48c2146444e" class="">We also explored scaling up the compute-optimal recipe to Llama 3.2 3B Instruct to see at what point the benefits of the PRM fade in comparison to the policy’s own capacity. To our surprise, compute-optimal scaling works remarkably well, with the 3B model surpassing the performance of Llama 3.1 70B Instruct (22x it's size!):</p><figure id="15b1384e-bcac-80b3-bc58-d20ba41d3950" class="image"><a href="https://huggingface.co/datasets/HuggingFaceH4/blogpost-images/resolve/main/methods-opt-3b.png"><img style="width:707.9891357421875px" src="https://huggingface.co/datasets/HuggingFaceH4/blogpost-images/resolve/main/methods-opt-3b.png"/></a></figure>
 
-<h2 id="15a1384e-bcac-809c-b5e7-eb92dadaebb4" class="">Where to go from here?</h2><p id="15b1384e-bcac-8052-91d7-d6e1f6f66e09" class="">This exploration of test-time compute scaling has revealed both the potential and the challenges of leveraging search-based methods. As we look ahead, several exciting directions emerge:</p><ol type="1" id="15b1384e-bcac-8040-91d7-e4236e1530ef" class="numbered-list" start="1"><li><strong>The Power of Strong Verifiers:</strong><p id="15b1384e-bcac-8032-85d2-e0e0283f7e4a" class="">Strong verifiers play a critical role in enhancing performance. However, their current limitations are apparent, as highlighted in benchmarks like <em>ProcessBench</em>. Improving the robustness and generalization of verifiers will be crucial for advancing these methods.</p></li></ol><ol type="1" id="15b1384e-bcac-80da-a940-e4ad39ff493d" class="numbered-list" start="2"><li><strong>The Challenge of Self-Verification:</strong><p id="15b1384e-bcac-8077-a5bb-c75fb34b60eb" class="">The ultimate goal—or &quot;holy grail&quot;—is achieving self-verification, where models can validate their own outputs autonomously. This approach appears to be what models like o1 are doing, but remains difficult to implement in practice. Unlike standard supervised fine-tuning (SFT), self-verification demands more nuanced strategies. The recent DeepMind paper on self-verification and <em>Score</em> sheds light on this challenge and offers a pathway for future research.</p></li></ol><ol type="1" id="15b1384e-bcac-8038-88f9-dd21d5cf272c" class="numbered-list" start="3"><li><strong>Integrating “Thoughts” into the Process:</strong><p id="15b1384e-bcac-801d-923f-d1ea46844321" class="">Incorporating explicit intermediate steps or “thoughts” during generation could further enhance reasoning and decision-making. By integrating structured reasoning into the search process, we may unlock better performance on complex tasks.</p></li></ol><ol type="1" id="15b1384e-bcac-802b-aa65-d84fb31ed476" class="numbered-list" start="4"><li><strong>Search as a Data Generation Tool:</strong><p id="15b1384e-bcac-802a-8488-ef74d5fb0a55" class="">This method can also serve as a powerful data generation process, creating high-quality training datasets. For example, fine-tuning models like Llama 1B on correct traces produced by search could yield significant gains. This on-policy approach resembles techniques like ReST or V-StaR but with the added benefits of search, offering a promising direction for iterative improvement.</p></li></ol><ol type="1" id="15b1384e-bcac-804a-8a0e-dcaf06767339" class="numbered-list" start="5"><li><strong>A Call for More PRMs:</strong><p id="15b1384e-bcac-803c-9524-f563cd123121" class="">Open process reward models (PRMs) are relatively rare, limiting their broader application. Developing and sharing more PRMs for different domains is a critical area where the community can contribute significantly.</p><ul id="15c1384e-bcac-80af-97dd-fcabe67a54a8" class="bulleted-list"><li style="list-style-type:disc">Link to TRL!</li></ul></li></ol><ol type="1" id="15b1384e-bcac-805f-b97c-d3383a2c9682" class="numbered-list" start="6"><li><strong>Expanding Beyond Verifiable Domains:</strong><p id="15b1384e-bcac-8073-9fea-c717f2d50df5" class="">While current methods excel in domains like math and code, where solutions are inherently verifiable, extending these techniques to other areas remains a major challenge. How can we adapt these strategies for less structured or subjective tasks? This is a vital question for future exploration.</p></li></ol><h2 id="15b1384e-bcac-8093-82c6-d9c951dc0bab" class="">Acknowledgements</h2><p id="15c1384e-bcac-80d5-8ad4-d7f6874404c5" class="">We are grateful to Aviral Kumar for many discussions about test-time compute scaling and for sharing implementation details from their work. We thank Chun Te Lee for designing the lovely banner and Thomas Wolf, Leandro von Werra, Colin Raffel, and Quentin Gallouédec for many helpful suggestions to improve the blog post. We also thank Hugo Larcher and Mathieu Morlon for continually optimising the Hugging Face Science Cluster to make the GPUs go brrr 🔥!</p>
+<h2 id="15a1384e-bcac-809c-b5e7-eb92dadaebb4" class="">Where to go from here?</h2><p id="15b1384e-bcac-8052-91d7-d6e1f6f66e09" class="">This exploration of test-time compute scaling has revealed both the potential and the challenges of leveraging search-based methods. As we look ahead, several exciting directions emerge:</p><ol type="1" id="15b1384e-bcac-8040-91d7-e4236e1530ef" class="numbered-list" start="1"><li><strong>The Power of Strong Verifiers:</strong><p id="15b1384e-bcac-8032-85d2-e0e0283f7e4a" class="">Strong verifiers play a critical role in enhancing performance. However, their current limitations are apparent, as highlighted in benchmarks like <em>ProcessBench</em>. Improving the robustness and generalization of verifiers will be crucial for advancing these methods.</p></li></ol><ol type="1" id="15b1384e-bcac-80da-a940-e4ad39ff493d" class="numbered-list" start="2"><li><strong>The Challenge of Self-Verification:</strong><p id="15b1384e-bcac-8077-a5bb-c75fb34b60eb" class="">The ultimate goal—or &quot;holy grail&quot;—is achieving self-verification, where models can validate their own outputs autonomously. This approach appears to be what models like o1 are doing, but remains difficult to implement in practice. Unlike standard supervised fine-tuning (SFT), self-verification demands more nuanced strategies. The recent DeepMind paper on self-verification and <em>Score</em> sheds light on this challenge and offers a pathway for future research.</p></li></ol><ol type="1" id="15b1384e-bcac-8038-88f9-dd21d5cf272c" class="numbered-list" start="3"><li><strong>Integrating “Thoughts” into the Process:</strong><p id="15b1384e-bcac-801d-923f-d1ea46844321" class="">Incorporating explicit intermediate steps or “thoughts” during generation could further enhance reasoning and decision-making. By integrating structured reasoning into the search process, we may unlock better performance on complex tasks.</p></li></ol><ol type="1" id="15b1384e-bcac-802b-aa65-d84fb31ed476" class="numbered-list" start="4"><li><strong>Search as a Data Generation Tool:</strong><p id="15b1384e-bcac-802a-8488-ef74d5fb0a55" class="">This method can also serve as a powerful data generation process, creating high-quality training datasets. For example, fine-tuning models like Llama 1B on correct traces produced by search could yield significant gains. This on-policy approach resembles techniques like ReST or V-StaR but with the added benefits of search, offering a promising direction for iterative improvement.</p></li></ol><ol type="1" id="15b1384e-bcac-804a-8a0e-dcaf06767339" class="numbered-list" start="5"><li><strong>A Call for More PRMs:</strong><p id="15b1384e-bcac-803c-9524-f563cd123121" class="">Open process reward models (PRMs) are relatively rare, limiting their broader application. Developing and sharing more PRMs for different domains is a critical area where the community can contribute significantly.</p></li></ol><ol type="1" id="15b1384e-bcac-805f-b97c-d3383a2c9682" class="numbered-list" start="6"><li><strong>Expanding Beyond Verifiable Domains:</strong><p id="15b1384e-bcac-8073-9fea-c717f2d50df5" class="">While current methods excel in domains like math and code, where solutions are inherently verifiable, extending these techniques to other areas remains a major challenge. How can we adapt these strategies for less structured or subjective tasks? This is a vital question for future exploration.</p></li></ol><h2 id="15b1384e-bcac-8093-82c6-d9c951dc0bab" class="">Acknowledgements</h2><p id="15c1384e-bcac-80d5-8ad4-d7f6874404c5" class="">We are grateful to Charlie Snell and Aviral Kumar for many discussions about test-time compute scaling and for sharing implementation details from their work. We thank Chun Te Lee for designing the lovely banner and Thomas Wolf, Leandro von Werra, Colin Raffel, and Quentin Gallouédec for many helpful suggestions to improve the blog post. We also thank Hugo Larcher and Mathieu Morlon for continually optimising the Hugging Face Science Cluster to make the GPUs go brrr 🔥!</p>
 
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