思维链在数学推理中的形态（SHAPE）

摘要

大型语言模型（LLMs）在数学推理基准上表现出色，然而支撑其推理的具有数学意义的技能仍未得到充分探索。我们提出SHAPE，一个通过数学教育中发展出的两个视角来分析思维链（Chain-of-Thought, CoT）轨迹的框架：（1）语义空间：模型对问题不断演化的数学解释（如代数、几何）；（2）启发式策略：模型在这些空间内采取的具体数学行动（如简化问题、倒推求解）。我们首先使用SHAPE分析多种模型的推理模式。研究发现，相较于传统的CoT特征，模型所采用的数学启发式策略能更好地解释最终答案的正确性。此外，模型倾向于将推理努力集中在少数几个语义空间内，而非探索许多彼此不同的空间，从而更有可能得出正确答案——这一模式与人类行为一致。接下来，我们利用SHAPE视角评估后训练是否真正提升了数学能力。我们发现强化学习会在启发式使用中诱发模式寻求（mode-seeking）现象。最后，我们通过促进启发式的多样性对LLMs进行后训练，并证明其在提升准确率方面的有效性。总体而言，SHAPE为解码LLM推理提供了一个有理论基础的诊断框架，并为面向数学推理的LLM后训练开辟了新路径。我们模型的代码可在https://github.com/holi-lab/SHAPE-of-CoT获取。

English

Large language models (LLMs) achieve strong performance on mathematical reasoning benchmarks, yet the mathematically meaningful skills underlying their reasoning remain underexplored. We introduce SHAPE, a framework that analyzes Chain-of-Thought (CoT) trajectories through two lenses developed in mathematics education: (1) semantic spaces: the model's evolving mathematical interpretations of a problem (e.g., algebraic, geometric), and (2) heuristics: the specific mathematical actions taken within those spaces (e.g., simplifying the problem, working backward). We first use SHAPE to analyze the reasoning patterns of various models. Our findings reveal that the mathematical heuristics employed by a model better explain final answer correctness than traditional CoT features. Furthermore, models are likely to reach correct solutions by concentrating their reasoning effort within a few semantic spaces rather than exploring many disparate ones -- a pattern consistent with human behavior. Next, we utilize the SHAPE lens to evaluate whether post-training truly enhances mathematical proficiency. We find that reinforcement learning induces mode-seeking in heuristic usage. Lastly, we post-train LLMs by promoting diverse heuristics and demonstrate its effectiveness in improving accuracy. Overall, SHAPE provides a theoretically-grounded diagnostic framework for decoding LLM reasoning and offers a new path toward post-training LLMs for math reasoning. The code for our model is available at https://github.com/holi-lab/SHAPE-of-CoT