Lean Formalization of Generalization Error Bound by Rademacher Complexity and Dudley's Entropy Integral
Year: 2025
Type: preprint
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 𝓁₂ and 𝓁₁ regularizations, and we also formalize a Dudley-type entropy integral bound based on covering numbers and a chaining construction.
Source: arXiv (Cornell University)
Authors: Sonoda, Sho, Kasaura, Kazumi, Mizuno, Yuma, Tsukamoto, Kei, Onda, Naoto
Language: English
Subfield: Artificial Intelligence
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7168698263