Lean 4中的统计学习理论：从零开始的实证过程

摘要

我们首次在经验过程理论基础上实现了统计学习理论（SLT）的完整Lean 4形式化。该端到端的形式化基础设施填补了最新Lean 4 Mathlib库的空白，包含高斯利普希茨集中性的完整推导、达德利熵积分定理在次高斯过程中的首次形式化，以及带有尖锐收敛速率的最小二乘（稀疏）回归应用。项目采用人机协同工作流完成：人类设计证明策略，智能体执行战术性证明构建，最终形成经过人工验证的SLT工具箱。除实现外，形式化过程还揭示并修正了标准SLT教材中隐含的假设与缺失细节，推动了对理论逐行级的精细化理解。此项工作建立了可复用的形式化基础，为机器学习理论的未来发展开辟了道路。代码详见https://github.com/YuanheZ/lean-stat-learning-theory。

English

We present the first comprehensive Lean 4 formalization of statistical learning theory (SLT) grounded in empirical process theory. Our end-to-end formal infrastructure implement the missing contents in latest Lean 4 Mathlib library, including a complete development of Gaussian Lipschitz concentration, the first formalization of Dudley's entropy integral theorem for sub-Gaussian processes, and an application to least-squares (sparse) regression with a sharp rate. The project was carried out using a human-AI collaborative workflow, in which humans design proof strategies and AI agents execute tactical proof construction, leading to the human-verified Lean 4 toolbox for SLT. Beyond implementation, the formalization process exposes and resolves implicit assumptions and missing details in standard SLT textbooks, enforcing a granular, line-by-line understanding of the theory. This work establishes a reusable formal foundation and opens the door for future developments in machine learning theory. The code is available at https://github.com/YuanheZ/lean-stat-learning-theory

Lean 4中的统计学习理论：从零开始的实证过程

Statistical Learning Theory in Lean 4: Empirical Processes from Scratch

摘要

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