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home / papers / 2023 / ToRA: A Tool-Integrated Reasoning Agent for Mathematical Problem Solving

ToRA: A Tool-Integrated Reasoning Agent for Mathematical Problem Solving

Written for the Research library (claude-opus-5-5) from the paper text, in our own words: the license "arXiv — non-exclusive licence to distribute" does not allow redistributing the text itself. Original — arxiv.org/pdf/2309.17452v4

Summary

Open-source LLMs such as LLaMA-2 still trail closed systems by a wide margin on hard mathematics. Step-by-step natural-language reasoning is good for planning but makes arithmetic and symbolic mistakes. Pure program synthesis calculates precisely but handles the meaning of a problem poorly. ToRA (Tool-integrated Reasoning Agent) combines the two. The model alternates between short textual rationales and Python code that calls tools such as SymPy. It runs the code, reads the result and continues until it writes a boxed final answer. To build training data, the authors prompted GPT-4 to produce such interleaved solutions for GSM8k and MATH training questions and kept only the correct ones. This gave ToRA-Corpus. LLaMA-2 and CodeLLaMA models from 7B to 70B were first fine-tuned by imitation learning. They were then refined with "output space shaping": each model samples its own solutions, invalid ones are repaired by a larger teacher model, and the model is retrained on the combined set. The result shows that tight tool interaction can bring mid-sized open models close to GPT-4 on competition-level math.

Key points

  • ToRA-Corpus has about 16k GPT-4 tool-use trajectories (up to 3 interaction rounds). It covers 98.2% of GSM8k and 83.1% of MATH training questions.
  • On 10 math reasoning benchmarks, ToRA beats earlier open-source models at every size, by 13–19 points absolute on average.
  • ToRA-7B scores 44.6% on MATH, 22 points above WizardMath-70B, the previous best open model.
  • ToRA-Code-34B reaches 50.8% on MATH, the first open model above 50%. That is higher than GPT-4 chain-of-thought (42.5%) and close to GPT-4 with code (51.8%).
  • Code and models are available at github.com/microsoft/ToRA. The work comes from Tsinghua University and Microsoft.

Our summary of the paper, in our own words — not its text. The work belongs to its authors; read it at the links above.

Fields: Mathematics & statistics

Cited by

202 papers of our library cite it (of the 140673 whose bibliographies we read); 11 of them have a page here, 191 not yet.

Cite

The paper itself, not this page. BibTeX from the library's record of the work. DOI 10.48550/arxiv.2309.17452 · arXiv 2309.17452

@misc{gou2023tora,
  title = {{ToRA: A Tool-Integrated Reasoning Agent for Mathematical Problem Solving}},
  author = {Gou, Zhibin and Shao, Zhihong and Gong, Yeyun and Shen, Yelong and Yang, Yujiu and Huang, Minlie and Duan, Nan and Chen, Weizhu},
  year = {2023},
  doi = {10.48550/arxiv.2309.17452},
  eprint = {2309.17452},
  archivePrefix = {arXiv},
  url = {https://doi.org/10.48550/arxiv.2309.17452}
}

References

56 references, 34 of them with a card of ours. ↗ leads to the work, «find» searches for one we have no id for yet.

  1. Rohan Anil, Andrew M Dai, Orhan Firat, Melvin Johnson, Dmitry Lepikhin, Alexandre Passos, Siamak Shakeri, Emanuel Taropa, Paige Bailey, Zhifeng Chen, et al. Palm 2 technical report. arXiv preprint arXiv:2305.10403, 2023. find our card
  2. Sebastian Borgeaud, Arthur Mensch, Jordan Hoffmann, Trevor Cai, Eliza Rutherford, Katie Millican, George Bm Van Den Driessche, Jean-Baptiste Lespiau, Bogdan Damoc, Aidan Clark, et al. Improving language models by retrieving from trillions of tokens. In International conference on machine learning, p… find
  3. Sébastien Bubeck, Varun Chandrasekaran, Ronen Eldan, Johannes Gehrke, Eric Horvitz, Ece Kamar, Peter Lee, Yin Tat Lee, Yuanzhi Li, Scott M. Lundberg, Harsha Nori, Hamid Palangi, Marco Túlio Ribeiro, and Yi Zhang. Sparks of artificial general intelligence: Early experiments with GPT-4. CoRR, abs/2303… find our card
  4. Cristian Buciluǎ, Rich Caruana, and Alexandru Niculescu-Mizil. Model compression. In Proceedings of the 12th ACM SIGKDD international conference on Knowledge discovery and data mining, pp. 535–541, 2006. find
  5. Wenhu Chen, Xueguang Ma, Xinyi Wang, and William W Cohen. Program of thoughts prompting: Disentangling computation from reasoning for numerical reasoning tasks. arXiv preprint arXiv:2211.12588, 2022. find our card
  6. Karl Cobbe, Vineet Kosaraju, Mohammad Bavarian, Mark Chen, Heewoo Jun, Lukasz Kaiser, Matthias Plappert, Jerry Tworek, Jacob Hilton, Reiichiro Nakano, Christopher Hesse, and John Schulman. Training verifiers to solve math word problems, 2021. URL https://arxiv.org/abs/2110.14168. ↗ our card
  7. Tri Dao. FlashAttention-2: Faster attention with better parallelism and work partitioning. 2023. find
  8. Edward A Feigenbaum, Julian Feldman, et al. Computers and thought, volume 7. New York McGraw-Hill, 1963. find
  9. Yao Fu, Hao Peng, Litu Ou, Ashish Sabharwal, and Tushar Khot. Specializing smaller language models towards multi-step reasoning. In Andreas Krause, Emma Brunskill, Kyunghyun Cho, Barbara Engelhardt, Sivan Sabato, and Jonathan Scarlett (eds.), International Conference on Machine Learning, ICML 2023,… find our card
  10. PMLR, 2023. URL https://proceedings.mlr.press/v202/fu23d.html. ↗
  11. Luyu Gao, Aman Madaan, Shuyan Zhou, Uri Alon, Pengfei Liu, Yiming Yang, Jamie Callan, and Graham Neubig. Pal: Program-aided language models. arXiv preprint arXiv:2211.10435, 2022. find
  12. Zhibin Gou, Zhihong Shao, Yeyun Gong, Yelong Shen, Yujiu Yang, Nan Duan, and Weizhu Chen. Critic: Large language models can self-correct with tool-interactive critiquing. arXiv preprint arXiv:2305.11738, 2023. find our card
  13. Dan Hendrycks, Collin Burns, Saurav Kadavath, Akul Arora, Steven Basart, Eric Tang, Dawn Song, and Jacob Steinhardt. Measuring mathematical problem solving with the math dataset. NeurIPS, 2021. find our card
  14. Geoffrey Hinton, Oriol Vinyals, and Jeff Dean. Distilling the knowledge in a neural network. arXiv preprint arXiv:1503.02531, 2015. find our card
  15. Namgyu Ho, Laura Schmid, and Se-Young Yun. Large language models are reasoning teachers. In Proceedings of the 61st Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers), pp. 14852–14882, Toronto, Canada, July 2023. Association for Computational Linguistics. doi: 10… find our card
  16. Mohammad Javad Hosseini, Hannaneh Hajishirzi, Oren Etzioni, and Nate Kushman. Learning to solve arithmetic word problems with verb categorization. In Proceedings of the 2014 Conference on Empirical Methods in Natural Language Processing (EMNLP), pp. 523–533, 2014. find our card
  17. Jiaxin Huang, Shixiang Shane Gu, Le Hou, Yuexin Wu, Xuezhi Wang, Hongkun Yu, and Jiawei Han. Large language models can self-improve. CoRR, abs/2210.11610, 2022. doi: 10.48550/arXiv.2210.11610. URL https://doi.org/10.48550/arXiv.2210.11610. ↗ our card
  18. Weisen Jiang, Han Shi, Longhui Yu, Zhengying Liu, Yu Zhang, Zhenguo Li, and James T Kwok. Backward reasoning in large language models for verification. arXiv preprint arXiv:2308.07758, 2023. find
  19. Rik Koncel-Kedziorski, Subhro Roy, Aida Amini, Nate Kushman, and Hannaneh Hajishirzi. MAWPS: A math word problem repository. In Proceedings of the 2016 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, pp. 1152–1157, San Diego, Ca… find
  20. Ariel N Lee, Cole J Hunter, and Nataniel Ruiz. Platypus: Quick, cheap, and powerful refinement of llms. arXiv preprint arXiv:2308.07317, 2023. find
  21. Yifei Li, Zeqi Lin, Shizhuo Zhang, Qiang Fu, Bei Chen, Jian-Guang Lou, and Weizhu Chen. Making language models better reasoners with step-aware verifier. In Proceedings of the 61st Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers), pp. 5315–5333, 2023. find our card
  22. Tian Liang, Zhiwei He, Wenxiang Jiao, Xing Wang, Yan Wang, Rui Wang, Yujiu Yang, Zhaopeng Tu, and Shuming Shi. Encouraging divergent thinking in large language models through multi-agent debate. arXiv preprint arXiv:2305.19118, 2023. find our card
  23. Hunter Lightman, Vineet Kosaraju, Yura Burda, Harri Edwards, Bowen Baker, Teddy Lee, Jan Leike, John Schulman, Ilya Sutskever, and Karl Cobbe. Let’s verify step by step. arXiv preprint arXiv:2305.20050, 2023. find our card
  24. Pan Lu, Liang Qiu, Kai-Wei Chang, Ying Nian Wu, Song-Chun Zhu, Tanmay Rajpurohit, Peter Clark, and Ashwin Kalyan. Dynamic prompt learning via policy gradient for semi-structured mathematical reasoning. In The Eleventh International Conference on Learning Representations, 2023. URL https://openreview… find our card
  25. Haipeng Luo, Qingfeng Sun, Can Xu, Pu Zhao, Jianguang Lou, Chongyang Tao, Xiubo Geng, Qingwei Lin, Shifeng Chen, and Dongmei Zhang. Wizardmath: Empowering mathematical reasoning for large language models via reinforced evol-instruct. arXiv preprint arXiv:2308.09583, 2023. find our card
  26. Grégoire Mialon, Roberto Dessì, Maria Lomeli, Christoforos Nalmpantis, Ram Pasunuru, Roberta Raileanu, Baptiste Rozière, Timo Schick, Jane Dwivedi-Yu, Asli Celikyilmaz, et al. Augmented language models: a survey. arXiv preprint arXiv:2302.07842, 2023. find our card
  27. Shen-yun Miao, Chao-Chun Liang, and Keh-Yih Su. A diverse corpus for evaluating and developing English math word problem solvers. In Proceedings of the 58th Annual Meeting of the Association for Computational Linguistics, pp. 975–984, Online, July 2020. Association for Computational Linguistics. doi… find
  28. Swaroop Mishra, Matthew Finlayson, Pan Lu, Leonard Tang, Sean Welleck, Chitta Baral, Tanmay Rajpurohit, Oyvind Tafjord, Ashish Sabharwal, Peter Clark, and Ashwin Kalyan. Lila: A unified benchmark for mathematical reasoning. In Proceedings of the 2022 Conference on Empirical Methods in Natural Langua… find our card
  29. Reiichiro Nakano, Jacob Hilton, Suchir Balaji, Jeff Wu, Long Ouyang, Christina Kim, Christopher Hesse, Shantanu Jain, Vineet Kosaraju, William Saunders, et al. Webgpt: Browser-assisted question-answering with human feedback. arXiv preprint arXiv:2112.09332, 2021. find our card
  30. OpenAI. Gpt-4 technical report, 2023. find our card
  31. Bhargavi Paranjape, Scott Lundberg, Sameer Singh, Hannaneh Hajishirzi, Luke Zettlemoyer, and Marco Tulio Ribeiro. Art: Automatic multi-step reasoning and tool-use for large language models. arXiv preprint arXiv:2303.09014, 2023. find
  32. Aaron Parisi, Yao Zhao, and Noah Fiedel. Talm: Tool augmented language models. arXiv preprint arXiv:2205.12255, 2022. find our card
  33. Arkil Patel, Satwik Bhattamishra, and Navin Goyal. Are NLP models really able to solve simple math word problems? In Proceedings of the 2021 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, pp. 2080–2094, Online, June 2021. Assoc… find our card
  34. Guilherme Penedo, Quentin Malartic, Daniel Hesslow, Ruxandra Cojocaru, Alessandro Cappelli, Hamza Alobeidli, Baptiste Pannier, Ebtesam Almazrouei, and Julien Launay. The RefinedWeb dataset for Falcon LLM: outperforming curated corpora with web data, and web data only. arXiv preprint arXiv:2306.01116… find
  35. Baolin Peng, Chunyuan Li, Pengcheng He, Michel Galley, and Jianfeng Gao. Instruction tuning with gpt-4. arXiv preprint arXiv:2304.03277, 2023. find our card
  36. Stanislas Polu and Ilya Sutskever. Generative language modeling for automated theorem proving. arXiv preprint arXiv:2009.03393, 2020. find our card
  37. Samyam Rajbhandari, Olatunji Ruwase, Jeff Rasley, Shaden Smith, and Yuxiong He. Zero-infinity: Breaking the gpu memory wall for extreme scale deep learning. In Proceedings of the International Conference for High Performance Computing, Networking, Storage and Analysis, pp. 1–14, 2021. find
  38. Baptiste Rozière, Jonas Gehring, Fabian Gloeckle, Sten Sootla, Itai Gat, Xiaoqing Ellen Tan, Yossi Adi, Jingyu Liu, Tal Remez, Jérémy Rapin, et al. Code llama: Open foundation models for code. arXiv preprint arXiv:2308.12950, 2023. find our card
  39. Timo Schick, Jane Dwivedi-Yu, Roberto Dessì, Roberta Raileanu, Maria Lomeli, Luke Zettlemoyer, Nicola Cancedda, and Thomas Scialom. Toolformer: Language models can teach themselves to use tools. arXiv preprint arXiv:2302.04761, 2023. find our card
  40. Zhihong Shao, Fei Huang, and Minlie Huang. Chaining simultaneous thoughts for numerical reasoning. In Yoav Goldberg, Zornitsa Kozareva, and Yue Zhang (eds.), Findings of the Association for Computational Linguistics: EMNLP 2022, Abu Dhabi, United Arab Emirates, December 7-11, 2022, pp. 2533–2547. As… find
  41. Zhihong Shao, Yeyun Gong, Yelong Shen, Minlie Huang, Nan Duan, and Weizhu Chen. Synthetic prompting: Generating chain-of-thought demonstrations for large language models. In Andreas Krause, Emma Brunskill, Kyunghyun Cho, Barbara Engelhardt, Sivan Sabato, and Jonathan Scarlett (eds.), International C… find
  42. PMLR, 2023a. URL https://proceedings.mlr.press/v202/shao23a.html. ↗
  43. Zhihong Shao, Yeyun Gong, Yelong Shen, Minlie Huang, Nan Duan, and Weizhu Chen. Enhancing retrieval-augmented large language models with iterative retrieval-generation synergy. CoRR, abs/2305.15294, 2023b. doi: 10.48550/arXiv.2305.15294. URL https://doi.org/10.48550/arXiv.2305.15294. ↗ our card
  44. Rohan Taori, Ishaan Gulrajani, Tianyi Zhang, Yann Dubois, Xuechen Li, Carlos Guestrin, Percy Liang, and Tatsunori B. Hashimoto. Stanford alpaca: An instruction-following llama model. https://github.com/tatsu-lab/stanford\_alpaca, 2023. ↗
  45. Hugo Touvron, Thibaut Lavril, Gautier Izacard, Xavier Martinet, Marie-Anne Lachaux, Timothée Lacroix, Baptiste Rozière, Naman Goyal, Eric Hambro, Faisal Azhar, et al. Llama: Open and efficient foundation language models. arXiv preprint arXiv:2302.13971, 2023a. find
  46. Hugo Touvron, Louis Martin, Kevin Stone, Peter Albert, Amjad Almahairi, Yasmine Babaei, Nikolay Bashlykov, Soumya Batra, Prajjwal Bhargava, Shruti Bhosale, Dan Bikel, Lukas Blecher, Cristian Canton-Ferrer, Moya Chen, Guillem Cucurull, David Esiobu, Jude Fernandes, Jeremy Fu, Wenyin Fu, Brian Fuller,… find
  47. Jason Wei, Xuezhi Wang, Dale Schuurmans, Maarten Bosma, brian ichter, Fei Xia, Ed H. Chi, Quoc V Le, and Denny Zhou. Chain of thought prompting elicits reasoning in large language models. In Alice H. Oh, Alekh Agarwal, Danielle Belgrave, and Kyunghyun Cho (eds.), Advances in Neural Information Proce… find our card
  48. Shunyu Yao, Jeffrey Zhao, Dian Yu, Nan Du, Izhak Shafran, Karthik R Narasimhan, and Yuan Cao. React: Synergizing reasoning and acting in language models. In The Eleventh International Conference on Learning Representations, 2023. URL https://openreview.net/forum?id=WE\_vluYUL-X. ↗ our card
  49. Zheng Yuan, Hongyi Yuan, Chengpeng Li, Guanting Dong, Chuanqi Tan, and Chang Zhou. Scaling relationship on learning mathematical reasoning with large language models. arXiv preprint arXiv:2308.01825, 2023. find our card
  50. Xiang Yue, Xingwei Qu, Ge Zhang, Yao Fu, Wenhao Huang, Huan Sun, Yu Su, and Wenhu Chen. Mammoth: Building math generalist models through hybrid instruction tuning. arXiv preprint arXiv:2309.05653, 2023. find our card
  51. Eric Zelikman, Yuhuai Wu, Jesse Mu, and Noah Goodman. Star: Bootstrapping reasoning with reasoning. Advances in Neural Information Processing Systems, 35:15476–15488, 2022. find our card
  52. Beichen Zhang, Kun Zhou, Xilin Wei, Wayne Xin Zhao, Jing Sha, Shijin Wang, and Ji-Rong Wen. Evaluating and improving tool-augmented computation-intensive math reasoning. arXiv preprint arXiv:2306.02408, 2023. — [arXiv:2306.02408](https://arxiv.org/abs/2306.02408) find
  53. Aojun Zhou, Ke Wang, Zimu Lu, Weikang Shi, Sichun Luo, Zipeng Qin, Shaoqing Lu, Anya Jia, Linqi Song, Mingjie Zhan, et al. Solving challenging math word problems using gpt-4 code interpreter with code-based self-verification. arXiv preprint arXiv:2308.07921, 2023a. find our card
  54. Denny Zhou, Nathanael Schärli, Le Hou, Jason Wei, Nathan Scales, Xuezhi Wang, Dale Schuurmans, Claire Cui, Olivier Bousquet, Quoc V. Le, and Ed H. Chi. Least-to-most prompting enables complex reasoning in large language models. In The Eleventh International Conference on Learning Representations, IC… find our card
  55. OpenReview.net, 2023b. URL https://openreview.net/pdf?id=WZH7099tgfM. ↗
  56. Xinyu Zhu, Junjie Wang, Lin Zhang, Yuxiang Zhang, Yongfeng Huang, Ruyi Gan, Jiaxing Zhang, and Yujiu Yang. Solving math word problems via cooperative reasoning induced language models. In Proceedings of the 61st Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers),… find