[论文] The Missing Primitive: Diagnosing and Repairing Mathematical Reasoning...

论文概要

研究领域: ML 作者: Shuo Xing, Zilin Dai, Chengyuan Qian, Fangzhou Lin, Wenjing Chen, Ping He, Pan Lu, Alvaro Velasquez, Mohit Bansal, Zhengzhong Tu 发布时间: 2026-10-01 arXiv: 2610.02191

中文摘要

大语言模型（LLM）已在前沿数学问题上展现惊人能力，但它们是否具备解答背后的结构性数学理解，仍不清楚。本文迈出系统研究 LLM 数学理解的第一步：从诊断其不同能力，到利用发现改进后训练。第一，引入「数学原语」（Mathematical Primitive）概念探测结构性数学理解，提出一个新基准，沿四个维度评估数学推理：发现（Discovery）、生成（Generation）、消化（Digestion）与执行（Execution）。第二，系统诊断表明：解题准确率掩盖了迥异的能力画像；原语能释放大量潜在的执行能力；而「发现」是数学推理中的主导瓶颈。后训练分析进一步显示，受限于发现能力的失败尤其容易修复。最后，基于这些发现提出原语优先的自蒸馏框架，选择性地将原语引导的推理迁移进学生模型。大量实验证明，该框架在多种模型规模与挑战性基准上持续提升数学推理能力，稳定优于基线。

原文摘要

While Large Language Models (LLMs) have demonstrated striking capabilities on frontier mathematical problems, it remains unclear whether they possess the structural mathematical understanding underlying their solutions. In this paper, we take a first step toward systematically studying mathematical understanding in LLMs, from diagnosing its distinct capabilities to leveraging these findings to improve post-training. First, we introduce the notion of Mathematical Primitive to probe structural mathematical understanding and propose a novel benchmark that evaluates mathematical reasoning along four distinct dimensions: Discovery, Generation, Digestion, and Execution. Second, our systematic diagnosis shows that solution accuracy masks distinct capability profiles, primitives unlock substantial...

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