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arXiv · 2503.19605

Lean Formalization of Generalization Error Bound by Rademacher Complexity and Dudley's Entropy Integral

Sho Sonoda·Kazumi Kasaura·Yuma Mizuno·Kei Tsukamoto·Naoto Onda
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Abstract

Understanding and certifying the generalization performance of machine learning algorithms -- i.e. obtaining theoretical estimates of the test error from the training error -- is a central theme of statistical learning theory. Among the many complexity measures used to derive such guarantees, Rademacher complexity yields sharp, data-dependent bounds that apply well beyond classical VC-dimension theory. In this study, we formalize the generalization error bound by Rademacher complexity in Lean 4, building on measure-theoretic probability theory available in the Mathlib library. Our development provides a mechanically-checked pipeline from the definitions of empirical and expected Rademacher complexity, through a formal symmetrization argument and a bounded-differences analysis, to high-probability uniform deviation bounds via a formally proved McDiarmid inequality. A key technical contribution is a reusable mechanism for lifting results from countable hypothesis classes (where measurability of suprema is straightforward in Mathlib) to separable topological index sets via a reduction to a countable dense subset. As worked applications of the abstract theorem, we mechanize standard empirical Rademacher bounds for linear predictors under $\ell_2$ and $\ell_1$ regularizations, and we also formalize a Dudley-type entropy integral bound based on covering numbers and a chaining construction.

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Sho Sonoda, Kazumi Kasaura, Yuma Mizuno, Kei Tsukamoto, Naoto Onda. 2025-03-25. Lean Formalization of Generalization Error Bound by Rademacher Complexity and Dudley's Entropy Integral. https://arxiv.org/abs/2503.19605

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