Lean Formalization of Generalization Error Bound by Rademacher Complexity and Dudley's Entropy Integral
arXiv:2503.19605v5 Announce Type: replace-cross 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
The increasing complexity and deployment of AI models necessitate rigorous theoretical understanding and certification of generalization performance, pushing for formal verification in platforms like Lean.
Formalization of generalization error bounds enhances the reliability and trustworthiness of machine learning algorithms, which is critical for their adoption in high-stakes applications.
This work makes theoretical guarantees surrounding AI more robust and formally verifiable, moving towards certifiable AI systems rather than relying solely on empirical performance.
- · AI safety researchers
- · Developers of formal verification tools
- · Industries requiring certified AI (e.g., healthcare, autonomous systems)
- · Academia (theoretical ML)
- · Developers of 'black box' AI models
- · Organizations deploying unvalidated AI in critical systems
Increased research and tooling for formal verification of machine learning models.
Higher standards and regulatory pressure for certifiable AI in specific domains.
A foundational shift in AI development towards 'proof-based' rather than 'performance-only' paradigms in critical applications.
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Read at arXiv cs.CL