Authors:
(1) Haolong Li, Tongji Universiy and work done during internship at ByteDance (furlongli322@gmail.com);
(2) Yu Ma, Seed Foundation, ByteDance (mayu.1231@bytedance.com);
(3) Yinqi Zhang, East China Normal University and work done during internship at ByteDance (zhang.inch@gmail.com);
(4) Chen Ye (Corresponding Author), ESSC Lab, Tongji Universiy (yechen@tongji.edu.cn);
(5) Jie Chen, Seed Foundation, ByteDance and a Project Leader (chenjiexjtu@gmail.com). Authors: Authors: (1) Haolong Li, Tongji Universiy and work done during internship at ByteDance (furlongli322@gmail.com); (2) Yu Ma, Seed Foundation, ByteDance (mayu.1231@bytedance.com); (3) Yinqi Zhang, East China Normal University and work done during internship at ByteDance (zhang.inch@gmail.com); (4) Chen Ye (Corresponding Author), ESSC Lab, Tongji Universiy (yechen@tongji.edu.cn); (5) Jie Chen, Seed Foundation, ByteDance and a Project Leader (chenjiexjtu@gmail.com). Table of Links Abstract and 1 Introduction Abstract and 1 Introduction 2 Problem Definition 2.1 Arithmetical Puzzle Problem 2.1 Arithmetical Puzzle Problem 2.2 Data Synthesizing 2.2 Data Synthesizing 2.3 Dataset 2.3 Dataset 3 Model 3 Model 4 Experiments 4.1 Evaluation 4.1 Evaluation 4.2 Results 4.2 Results 4.3 Case Studies 4.3 Case Studies 5 Conclusion and Acknowledgements 5 Conclusion and Acknowledgements 6 Limitations 6 Limitations 7 Ethics Statement and References 7 Ethics Statement and References A Appendix A Appendix A.1 Hyperparameter Settings A.1 Hyperparameter Settings A.2 Evaluation of the Base Model A.2 Evaluation of the Base Model A.3 Case Study A.3 Case Study A.4 Visualization of the Proposed Puzzle A.4 Visualization of the Proposed Puzzle 5 Conclusion Large language models (LLMs) are intrinsically zero-shot and multi-task learners. However, mathematical reasoning still poses challenges for LLMs, we propose that the reasons can be mainly categorized into three folds: (1) Requirement of multistep derivation; (2) Lack of high-quality data for fine-tuning; (3) Difficulty in extrapolation. In this paper, we design an arithmetical puzzle and make an early attempt to solve these challenges. We develop a 24-point puzzle-like problem which asks for multi-step calculations to arrive at the correct answer. A corresponding data synthesis pipeline is proposed to generate an arbitrary amount of high-quality data, on which a series of LLMs is fine-tuned. In order to verify the extrapolation capability of our models, we have designed two out-of-domain benchmarks and show that our model achieves competitive performance. Furthermore, a data scaling experiment is conducted, and it is concluded that by increasing the amount of training data, both the training loss and in/out-of-domain performance of the fine-tuned model improve accordingly. Acknowledgements We appreciate Peng Sun for providing the initial SFT dataset, and Xintian Han for suggestions about the reward calculation and ablation study. We would also like to thank Liang Xiang and Xun Zhou for the helpful discussions across the project. 6 Limitations In this study, we have explored the mathematical extrapolation of Large Language Models (LLMs) and discovered that, with high-quality synthetic data, LLMs demonstrates certain generalization capabilities in mathematical extrapolation. However, LLMs have not yet fully mastered this capability, and it remains uncertain if this ability can be extended to other complex mathematical tasks. In the future, our research will focus on investigating and enhancing this capability, aiming to empower LLMs to explore unsolved mathematical problems through leveraging our existing knowledge. 7 Ethics Statement In this research, we adhere to strict ethical guidelines and principles. The study has been designed and implemented with respect for the rights, privacy, and well-being of all individuals involved. All of our data is synthesized using our proposed data synthesis algorithm, ensuring compliance with relevant regulations and standards. Our findings and conclusions are reported accurately and objectively, avoiding any misrepresentation or manipulation of data. The entire process and outcomes are free from intellectual property and ethical legal disputes. References Josh Achiam, Steven Adler, Sandhini Agarwal, Lama Ahmad, Ilge Akkaya, Florencia Leoni Aleman, Diogo Almeida, Janko Altenschmidt, Sam Altman, Shyamal Anadkat, et al. 2023. 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