Papers › Math-Shepherd: Verify and Reinforce LLMs Step-by-step without Human Annotations
Math-Shepherd: Verify and Reinforce LLMs Step-by-step without Human Annotations
Peiyi Wang, Lei LI, Zhihong Shao, R. X. Xu, Damai Dai, Yifei Li, Deli Chen, Y. Wu, Zhifang Sui
In this paper, we present an innovative process-oriented math process reward model called \textbf{Math-Shepherd}, which assigns a reward score to each step of math problem solutions. The training of Math-Shepherd is achieved using automatically constructed process-wise supervision data, breaking the bottleneck of heavy reliance on manual annotation in existing work. We explore the effectiveness of Math-Shepherd in two scenarios: 1) \textit{Verification}: Math-Shepherd is utilized for reranking multiple outputs generated by Large Language Models (LLMs); 2) \textit{Reinforcement Learning}: Math-Shepherd is employed to reinforce LLMs with step-by-step Proximal Policy Optimization (PPO). With Math-Shepherd, a series of open-source LLMs demonstrates exceptional performance. For instance, the step-by-step PPO with Math-Shepherd significantly improves the accuracy of Mistral-7B (77.9\%→84.1\% on GSM8K and 28.6\%→33.0\% on MATH). The accuracy can be further enhanced to 89.1\% and 43.5\% on GSM8K and MATH with the verification of Math-Shepherd, respectively. We believe that automatic process supervision holds significant potential for the future evolution of LLMs.
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Code
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Tasks
Results from the paper archive 2025-07-28
| Task | Dataset | Model | Metric | Value | Rank at snapshot | Leaderboard | Report |
|---|---|---|---|---|---|---|---|
| Arithmetic Reasoning | GSM8K | Shepherd+Mistral-7B (SFT on MetaMATH + PRM RL+ PRM rerank, k=256) | Accuracy | 89.1 | #24 of 164 | Archive leaderboard | report |
| Arithmetic Reasoning | GSM8K | Shepherd+Mistral-7B (SFT on MetaMATH + PRM RL+ PRM rerank, k=256) | Parameters (Billion) | 7 | #24 of 164 | Archive leaderboard | report |
| Arithmetic Reasoning | GSM8K | Shepherd + Mistral-7B (SFT on MetaMATH + PRM RL) | Accuracy | 84.1 | #52 of 164 | Archive leaderboard | report |
| Arithmetic Reasoning | GSM8K | Shepherd + Mistral-7B (SFT on MetaMATH + PRM RL) | Parameters (Billion) | 7 | #52 of 164 | Archive leaderboard | report |
| Math Word Problem Solving | MATH | Shepherd + DeepSeek-67B (SFT on MetaMATH + PRM rerank, k=256) | Accuracy | 48.1 | #55 of 135 | Archive leaderboard | report |
| Math Word Problem Solving | MATH | Shepherd + DeepSeek-67B (SFT on MetaMATH + PRM rerank, k=256) | Parameters (Billions) | 67 | #55 of 135 | Archive leaderboard | report |
| Math Word Problem Solving | MATH | Shepherd+Mistral-7B (SFT on MetaMATH + PRM RL+ PRM rerank, k=256) | Accuracy | 43.5 | #71 of 135 | Archive leaderboard | report |
| Math Word Problem Solving | MATH | Shepherd+Mistral-7B (SFT on MetaMATH + PRM RL+ PRM rerank, k=256) | Parameters (Billions) | 7 | #71 of 135 | Archive leaderboard | report |
| Math Word Problem Solving | MATH | Shepherd + Mistral-7B (SFT on MetaMATH + PRM RL) | Accuracy | 33.0 | #84 of 135 | Archive leaderboard | report |
| Math Word Problem Solving | MATH | Shepherd + Mistral-7B (SFT on MetaMATH + PRM RL) | Parameters (Billions) | 7 | #84 of 135 | Archive leaderboard | report |
Ranks are positions in the archive's leaderboards as they stood at the 2025-07-28 snapshot. Results published since then are not among these rows, so a rank here is not a current standing.
Methods
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