# DeepSeekMath: Pushing the Limits of Mathematical Reasoning in Open Language Models

Mathematical reasoning poses a significant challenge for language models due to its complex and structured nature. In this paper, we introduce DeepSeekMath 7B, which continues pre-training DeepSeek-Coder-Base-v1.5 7B with 120B math-related tokens sourced from Common Crawl, together with natural language and code data. DeepSeekMath 7B has achieved an impressive score of 51.7% on the competition-level MATH benchmark without relying on external toolkits and voting techniques, approaching the performance level of Gemini-Ultra and GPT-4. Self-consistency over 64 samples from DeepSeekMath 7B achieves 60.9% on MATH. The mathematical reasoning capability of DeepSeekMath is attributed to two key factors: First, we harness the significant potential of publicly available web data through a meticulously engineered data selection pipeline. Second, we introduce Group Relative Policy Optimization (GRPO), a variant of Proximal Policy Optimization (PPO), that enhances mathematical reasoning abilities while concurrently optimizing the memory usage of PPO.

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## Results from the Paper

Ranked #9 on Math Word Problem Solving on MATH (using extra training data)

Task | Dataset | Model | Metric Name | Metric Value | Global Rank | Uses Extra Training Data |
Benchmark |
---|---|---|---|---|---|---|---|

Arithmetic Reasoning | GSM8K | DeepSeekMATH-RL-7B | Accuracy | 88.2 | # 23 | ||

Parameters (Billion) | 7 | # 4 | |||||

Math Word Problem Solving | MATH | DeepSeekMATH-RL-7B (w/ code, greedy decoding) | Accuracy | 58.8 | # 9 | ||

Parameters (Billions) | 7 | # 47 | |||||

Math Word Problem Solving | MATH | DeepSeekMATH-RL-7B (greedy decoding) | Accuracy | 51.7 | # 21 | ||

Parameters (Billions) | 7 | # 47 |