Deepseekmath: Pushing the limits of mathematical reasoning in open language models

Last updated at 2025-08-06Posted at 2025-08-06

Deepseekmath: Pushing the limits of mathematical reasoning in open language models

Z. Shao, P. Wang, Q. Zhu, R. Xu, J. Song, X. Bi, H. Zhang, M. Zhang, Y. K. Li, Y. Wu, and D. Guo. Deepseekmath: Pushing the limits of mathematical reasoning in open language models, 2024. URL https://arxiv.org/abs/2402.03300.

References

R. Anil, S. Borgeaud, Y. Wu, J. Alayrac, J. Yu, R. Soricut, J. Schalkwyk, A. M. Dai, A. Hauth, K. Millican, D. Silver, S. Petrov, M. Johnson, I. Antonoglou, J. Schrittwieser, A. Glaese, J. Chen, E. Pitler, T. P. Lillicrap, A. Lazaridou, O. Firat, J. Molloy, M. Isard, P. R. Barham, T. Hennigan, B. Lee, F. Viola, M. Reynolds, Y. Xu, R. Doherty, E. Collins, C. Meyer, E. Rutherford, E. Moreira, K. Ayoub, M. Goel, G. Tucker, E. Piqueras, M. Krikun, I. Barr, N. Savinov, I. Danihelka, B. Roelofs, A. White, A. Andreassen, T. von Glehn, L. Yagati, M. Kazemi, L. Gonzalez, M. Khalman, J. Sygnowski, and et al. Gemini: A family of highly capable multimodal models. CoRR, abs/2312.11805, 2023. doi: 10.48550/ARXIV.2312.11805. URL https://doi.org/10.48550/arXiv.2312.11805.
J. Austin, A. Odena, M. Nye, M. Bosma, H. Michalewski, D. Dohan, E. Jiang, C. Cai, M. Terry, Q. Le, et al. Program synthesis with large language models. arXiv preprint arXiv:2108.07732, 2021.
Z. Azerbayev, H. Schoelkopf, K. Paster, M. D. Santos, S. McAleer, A. Q. Jiang, J. Deng, S. Biderman, and S. Welleck. Llemma: An open language model for mathematics. arXiv preprint arXiv:2310.10631, 2023.
J. Bai, S. Bai, Y. Chu, Z. Cui, K. Dang, X. Deng, Y. Fan, W. Ge, Y. Han, F. Huang, et al. Qwen technical report. arXiv preprint arXiv:2309.16609, 2023. C. Burns, P. Izmailov, J. H. Kirchner, B. Baker, L. Gao, L. Aschenbrenner, Y. Chen, A. Ecoffet, M. Joglekar, J. Leike, et al. Weak-to-strong generalization: Eliciting strong capabilities with weak supervision. arXiv preprint arXiv:2312.09390, 2023.
ChatGLM3 Team. Chatglm3 series: Open bilingual chat llms, 2023. URL https://github.com/THUDM/ChatGLM3. M. Chen, J. Tworek, H. Jun, Q. Yuan, H. P. de Oliveira 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. CoRR, abs/2107.03374, 2021. URL https://arxiv.org/abs/2107.03374.
W. Chen, X. Ma, X. Wang, and W. W. Cohen. Program of thoughts prompting: Disentangling computation from reasoning for numerical reasoning tasks. CoRR, abs/2211.12588, 2022. doi: 10.48550/ARXIV.2211.12588. URL https://doi.org/10.48550/arXiv.2211.12588.
K. Cobbe, V. Kosaraju, M. Bavarian, M. Chen, H. Jun, L. Kaiser, M. Plappert, J. Tworek, J. Hilton, R. Nakano, et al. Training verifiers to solve math word problems. arXiv preprint arXiv:2110.14168, 2021.
T. Computer. Redpajama: an open dataset for training large language models, Oct. 2023. URL https://github.com/togethercomputer/RedPajama-Data.
DeepSeek-AI. Deepseek LLM: scaling open-source language models with longtermism. CoRR,abs/2401.02954, 2024. doi: 10.48550/ARXIV.2401.02954. URL https://doi.org/10.48550/arXiv.2401.02954.
Z. Du, Y. Qian, X. Liu, M. Ding, J. Qiu, Z. Yang, and J. Tang. Glm: General language model pretraining with autoregressive blank infilling. In Proceedings of the 60th Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers), pages 320–335, 2022.
L. Gao, A. Madaan, S. Zhou, U. Alon, P. Liu, Y. Yang, J. Callan, and G. Neubig. PAL: program-aided language models. In A. Krause, E. Brunskill, K. Cho, B. Engelhardt, S. Sabato, and J. Scarlett, editors, International Conference on Machine Learning, ICML 2023, 23-29 July 2023, Honolulu, Hawaii, USA, volume 202 of Proceedings of Machine Learning Research, pages 10764–10799. PMLR, 2023. URL https://proceedings.mlr.press/v202/gao23f.html.
Z. Gou, Z. Shao, Y. Gong, Y. Shen, Y. Yang, M. Huang, N. Duan, and W. Chen. Tora: A tool-integrated reasoning agent for mathematical problem solving. CoRR, abs/2309.17452, 2023. doi: 10.48550/ARXIV.2309.17452. URL https://doi.org/10.48550/arXiv.2309.1745 2.
D. Guo, Q. Zhu, D. Yang, Z. Xie, K. Dong, W. Zhang, G. Chen, X. Bi, Y. Wu, Y. K. Li, F. Luo, Y. Xiong, and W. Liang. Deepseek-coder: When the large language model meets programming – the rise of code intelligence, 2024.
D. Hendrycks, C. Burns, S. Basart, A. Zou, M. Mazeika, D. Song, and J. Steinhardt. Measuring massive multitask language understanding. arXiv preprint arXiv:2009.03300, 2020.
D. Hendrycks, C. Burns, S. Kadavath, A. Arora, S. Basart, E. Tang, D. Song, and J. Steinhardt. Measuring mathematical problem solving with the math dataset. arXiv preprint arXiv:2103.03874, 2021.
High-flyer. Hai-llm: 高效且轻量的大模型训练工具, 2023. URL https://www.high-flyer.cn/en/blog/hai-llm.
Inflection AI. Inflection-2, 2023. URL https://inflection.ai/inflection-2.
A. Q. Jiang, S. Welleck, J. P. Zhou, W. Li, J. Liu, M. Jamnik, T. Lacroix, Y. Wu, and G. Lample. Draft, sketch, and prove: Guiding formal theorem provers with informal proofs. arXiv preprint arXiv:2210.12283, 2022.
A. Q. Jiang, A. Sablayrolles, A. Mensch, C. Bamford, D. S. Chaplot, D. d. l. Casas, F. Bressand, G. Lengyel, G. Lample, L. Saulnier, et al. Mistral 7b. arXiv preprint arXiv:2310.06825, 2023.
A. Joulin, E. Grave, P. Bojanowski, M. Douze, H. Jégou, and T. Mikolov. Fasttext. zip: Compressing text classification models. arXiv preprint arXiv:1612.03651, 2016.
W. Kwon, Z. Li, S. Zhuang, Y. Sheng, L. Zheng, C. H. Yu, J. E. Gonzalez, H. Zhang, and I. Stoica. Efficient memory management for large language model serving with pagedattention. In Proceedings of the ACM SIGOPS 29th Symposium on Operating Systems Principles, 2023.
Y. Leviathan, M. Kalman, and Y. Matias. Fast inference from transformers via speculative decoding. 2023. In International Conference on Machine Learning, pages 19274–19286. PMLR,
A. Lewkowycz, A. Andreassen, D. Dohan, E. Dyer, H. Michalewski, V. Ramasesh, A. Slone, C. Anil, I. Schlag, T. Gutman-Solo, et al. Solving quantitative reasoning problems with language models. Advances in Neural Information Processing Systems, 35:3843–3857, 2022a.
A. Lewkowycz, A. Andreassen, D. Dohan, E. Dyer, H. Michalewski, V. V. Ramasesh, A. Slone, C. Anil, I. Schlag, T. Gutman-Solo, Y. Wu, B. Neyshabur, G. Gur-Ari, and V. Misra. Solving quantitative reasoning problems with language models. In S. Koyejo, S. Mohamed, A. Agarwal, D. Belgrave, K. Cho, and A. Oh, editors, Advances in Neural Information Processing Systems 35: Annual Conference on Neural Information Processing Systems 2022, NeurIPS 2022, New Orleans, LA, USA, November 28 - December 9, 2022, 2022b. URL http://papers.nips.cc/paper_files/paper/2022/hash/18abbeef8cfe9203fdf9053c9c4fe191-Abstract-Conference.html.
H. Lightman, V. Kosaraju, Y. Burda, H. Edwards, B. Baker, T. Lee, J. Leike, J. Schulman, I. Sutskever, and K. Cobbe. Let’s verify step by step. arXiv preprint arXiv:2305.20050, 2023.
I. Loshchilov and F. Hutter. arXiv:1711.05101, 2017. Decoupled weight decay regularization. arXiv preprint
H. Luo, Q. Sun, C. Xu, P. Zhao, J. Lou, C. Tao, X. Geng, Q. Lin, S. Chen, and D. Zhang. Wizardmath: Empowering mathematical reasoning for large language models via reinforced evol-instruct. arXiv preprint arXiv:2308.09583, 2023.
S. Mishra, M. Finlayson, P. Lu, L. Tang, S. Welleck, C. Baral, T. Rajpurohit, O. Tafjord, A. Sabharwal, P. Clark, and A. Kalyan. LILA: A unified benchmark for mathematical reasoning. In Y. Goldberg, Z. Kozareva, and Y. Zhang, editors, Proceedings of the 2022 Conference on Empirical Methods in Natural Language Processing, EMNLP 2022, Abu Dhabi, United Arab Emirates, December 7-11, 2022, pages 5807–5832. Association for Computational Linguistics,2022. doi: 10.18653/V1/2022.EMNLP-MAIN.392. URL https://doi.org/10.18653/v1/2022.emnlp-main.392.
X. Nguyen, W. Zhang, X. Li, M. M. Aljunied, Q. Tan, L. Cheng, G. Chen, Y. Deng, S. Yang, C. Liu, H. Zhang, and L. Bing. Seallms - large language models for southeast asia. CoRR, abs/2312.00738, 2023. doi: 10.48550/ARXIV.2312.00738. URL https://doi.org/10.48550/arXiv.2312.00738.
OpenAI. GPT4 technical report. arXiv preprint arXiv:2303.08774, 2023.
L. Ouyang, J. Wu, X. Jiang, D. Almeida, C. Wainwright, P. Mishkin, C. Zhang, S. Agarwal, K. Slama, A. Ray, et al. Training language models to follow instructions with human feedback. Advances in Neural Information Processing Systems, 35:27730–27744, 2022.
K. Paster, M. D. Santos, Z. Azerbayev, and J. Ba. Openwebmath: An open dataset of high-quality mathematical web text. CoRR, abs/2310.06786, 2023. doi: 10.48550/ARXIV.2310.06786. URL https://doi.org/10.48550/arXiv.2310.06786.
L. C. Paulson. Three years of experience with sledgehammer, a practical link between automatic and interactive theorem provers. In R. A. Schmidt, S. Schulz, and B. Konev, editors, Proceedings of the 2nd Workshop on Practical Aspects of Automated Reasoning, PAAR-2010, Edinburgh, Scotland, UK, July 14, 2010, volume 9 of EPiC Series in Computing, pages 1–10. EasyChair, 2010. doi: 10.29007/TNFD. URL https://doi.org/10.29007/tnfd.
S. Polu and I. Sutskever. Generative language modeling for automated theorem proving. CoRR, abs/2009.03393, 2020. URL https://arxiv.org/abs/2009.03393.
R. Rafailov, A. Sharma, E. Mitchell, S. Ermon, C. D. Manning, and C. Finn. Direct preference optimization: Your language model is secretly a reward model. 2023.
J. Schulman. Approximating kl divergence, 2020. URL http://joschu.net/blog/kl-approx.html.
J. Schulman, P. Moritz, S. Levine, M. Jordan, and P. Abbeel. High-dimensional continuous control using generalized advantage estimation. arXiv preprint arXiv:1506.02438, 2015.
J. Schulman, F. Wolski, P. Dhariwal, A. Radford, and O. Klimov. Proximal policy optimization algorithms. arXiv preprint arXiv:1707.06347, 2017.
F. Shi, M. Suzgun, M. Freitag, X. Wang, S. Srivats, S. Vosoughi, H. W. Chung, Y. Tay, S. Ruder, D. Zhou, D. Das, and J. Wei. Language models are multilingual chain-of-thought reasoners. In The Eleventh International Conference on Learning Representations, ICLR 2023, Kigali, Rwanda, May 1-5, 2023. OpenReview.net, 2023. URL https://openreview.net/pdf?id=fR3wGCk-IXp.
F. Song, B. Yu, M. Li, H. Yu, F. Huang, Y. Li, and H. Wang. Preference ranking optimization for human alignment. arXiv preprint arXiv:2306.17492, 2023.
M. Suzgun, N. Scales, N. Schärli, S. Gehrmann, Y. Tay, H. W. Chung, A. Chowdhery, Q. V. Le, E. H. Chi, D. Zhou, et al. Challenging big-bench tasks and whether chain-of-thought can solve them. arXiv preprint arXiv:2210.09261, 2022.
T. Tao. Embracing change and resetting expectations, 2023. URL https://unlocked.microsoft.com/ai-anthology/terence-tao/.
H. Touvron, L. Martin, K. Stone, P. Albert, A. Almahairi, Y. Babaei, N. Bashlykov, S. Batra, P. Bhargava, S. Bhosale, D. Bikel, L. Blecher, C. Canton-Ferrer, M. Chen, G. Cucurull, D. Esiobu, J. Fernandes, J. Fu, W. Fu, B. Fuller, C. Gao, V. Goswami, N. Goyal, A. Hartshorn, S. Hosseini, R. Hou, H. Inan, M. Kardas, V. Kerkez, M. Khabsa, I. Kloumann, A. Korenev, P. S. Koura, M. Lachaux, T. Lavril, J. Lee, D. Liskovich, Y. Lu, Y. Mao, X. Martinet, T. Mihaylov, P. Mishra, I. Molybog, Y. Nie, A. Poulton, J. Reizenstein, R. Rungta, K. Saladi, A. Schelten, R. Silva, E. M. Smith, R. Subramanian, X. E. Tan, B. Tang, R. Taylor, A. Williams, J. X. Kuan, P. Xu, Z. Yan, I. Zarov, Y. Zhang, A. Fan, M. Kambadur, S. Narang, A. Rodriguez, R. Stojnic, S. Edunov, and T. Scialom. Llama 2: Open foundation and fine-tuned chat models. CoRR, abs/2307.09288,2023. doi: 10.48550/arXiv.2307.09288. URL https://doi.org/10.48550/arXiv.2307.09288.
T. H. Trinh, Y. Wu, Q. V. Le, H. He, and T. Luong. Solving olympiad geometry without human demonstrations. Nature, 625(7995):476–482, 2024.
P. Wang, L. Li, L. Chen, F. Song, B. Lin, Y. Cao, T. Liu, and Z. Sui. Making large language models better reasoners with alignment. arXiv preprint arXiv:2309.02144, 2023a.
P. Wang, L. Li, Z. Shao, R. Xu, D. Dai, Y. Li, D. Chen, Y. Wu, and Z. Sui. Math-shepherd: Verify and reinforce llms step-by-step without human annotations. CoRR, abs/2312.08935, 2023b.
Z. Wang, R. Xia, and P. Liu. Generative AI for math: Part I - mathpile: A billion-token-scale pretraining corpus for math. CoRR, abs/2312.17120, 2023c. doi: 10.48550/ARXIV.2312.17120. URL https://doi.org/10.48550/arXiv.2312.17120.
J. Wei, X. Wang, D. Schuurmans, M. Bosma, B. Ichter, F. Xia, E. H. Chi, Q. V. Le, and D. Zhou. Chain-of-thought prompting elicits reasoning in large language models. In NeurIPS, 2022. URL http://papers.nips.cc/paper_files/paper/2022/hash/9d5609613524ecf4f15af0f7b31abca4-Abstract-Conference.html.
T. Wei, J. Luan, W. Liu, S. Dong, and B. Wang. Cmath: Can your language model pass chinese elementary school math test?, 2023.
M. Wenzel, L. C. Paulson, and T. Nipkow. The isabelle framework. In O. A. Mohamed, C. A. Muñoz, and S. Tahar, editors, Theorem Proving in Higher Order Logics, 21st International Conference, TPHOLs 2008, Montreal, Canada, August 18-21, 2008. Proceedings, volume 5170 of Lecture Notes in Computer Science, pages 33–38. Springer, 2008. doi: 10.1007/978-3-540-71067-7_7. URL https://doi.org/10.1007/978-3-540-71067-7_7.
H. Xia, T. Ge, P. Wang, S.-Q. Chen, F. Wei, and Z. Sui. Speculative decoding: Exploiting speculative execution for accelerating seq2seq generation. In H. Bouamor, J. Pino, and K. Bali, editors, Findings of the Association for Computational Linguistics: EMNLP 2023, pages 3909–3925, Singapore, Dec. 2023. Association for Computational Linguistics. doi: 10.18653/v1/2023.findings-emnlp.257. URL https://aclanthology.org/2023.findings-emnlp.257.
H. Xia, Z. Yang, Q. Dong, P. Wang, Y. Li, T. Ge, T. Liu, W. Li, and Z. Sui. Unlocking efficiency in large language model inference: A comprehensive survey of speculative decoding. arXiv preprint arXiv:2401.07851, 2024.
S. Yao, D. Yu, J. Zhao, I. Shafran, T. L. Griffiths, Y. Cao, and K. Narasimhan. Tree of thoughts: Deliberate problem solving with large language models. arXiv preprint arXiv:2305.10601, 2023.
L. Yu, W. Jiang, H. Shi, J. Yu, Z. Liu, Y. Zhang, J. T. Kwok, Z. Li, A. Weller, and W. Liu. Metamath: Bootstrap your own mathematical questions for large language models. CoRR, abs/2309.12284, 2023. doi: 10.48550/ARXIV.2309.12284. URL https://doi.org/10.48550/arXiv.2309.12284.
Z. Yuan, H. Yuan, C. Li, G. Dong, C. Tan, and C. Zhou. Scaling relationship on learning mathematical reasoning with large language models. arXiv preprint arXiv:2308.01825, 2023a.
Z. Yuan, H. Yuan, C. Tan, W. Wang, S. Huang, and F. Huang. Rrhf: Rank responses to align language models with human feedback without tears. arXiv preprint arXiv:2304.05302, 2023b.
X. Yue, X. Qu, G. Zhang, Y. Fu, W. Huang, H. Sun, Y. Su, and W. Chen. Mammoth: Building math generalist models through hybrid instruction tuning. CoRR, abs/2309.05653, 2023. doi: 10.48550/ARXIV.2309.05653. URL https://doi.org/10.48550/arXiv.2309.05653.
K. Zheng, J. M. Han, and S. Polu. Minif2f: a cross-system benchmark for formal olympiad-level mathematics. arXiv preprint arXiv:2109.00110, 2021.
W. Zhong, R. Cui, Y. Guo, Y. Liang, S. Lu, Y. Wang, A. Saied, W. Chen, and N. Duan. AGIEval: A human-centric benchmark for evaluating foundation models. CoRR, abs/2304.06364, 2023. doi: 10.48550/arXiv.2304.06364. URL https://doi.org/10.48550/arXiv.2304.06364.

Deepseekmath: Pushing the limits of mathematical reasoning in open language models

Deepseekmath: Pushing the limits of mathematical reasoning in open language models

References

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