Certification-Enhanced Generalization Bounds

2026-10-05

AI Summary

This study addresses the challenge that existing methods struggle to provide tight and verifiable generalization error bounds for modern algorithms. We propose a theoretical framework grounded in formal methods that recasts algorithmic stability as system specifications and computes provable generalization bounds via reachability analysis. Furthermore, we construct novel concentration inequalities to bridge sample-level results with distribution-level analysis, enabling rigorous verification of boundedness without requiring analytic assumptions. By integrating formal verification, reachability analysis, and algorithmic stability theory, our approach yields tighter and computable generalization guarantees compared to traditional methods, particularly in complex scenarios such as large language model fine-tuning.

Abstract

We investigate the use of formal methods to provide tight and sound generalization bounds for learning algorithms. By casting the traditional notion of algorithmic stability as a specification to be verified, we demonstrate that recent advances in reachability analysis can yield provable bounds on the generalization of a given model and algorithm on a sample dataset. As sample-specific algorithmic stability is insufficient to bound the usual distributional notion of generalization, we develop a novel concentration inequality to connect the sample-specific results of formal certification algorithms to the required distributional analysis for bounding the expected generalization gap. The resulting framework enables the analysis of prior generalization bounds to extend far beyond their original restrictive assumptions. Our approach computes sound bounds on the expected generalization gap in a constant number of algorithm runs without making any analytical assumptions on the algorithm; to achieve non-vacuous bounds we only require that the certified reachable parameter set is bounded --- a condition that we do not assume but formally verify. In practice, we demonstrate that our framework provides formal generalization guarantees that are orders of magnitude tighter than alternative sound computational approaches at scales ranging from toy datasets to fine-tuning classification heads on top of modern large language models. While we implement certification-enhanced versions of several well-known stability results, future extensions of our approach will enable tighter bounds and enhanced practical adoption across the spectrum of modern generalization bounds.

Problem

• generalization bounds
• algorithmic stability
• formal verification
• concentration inequality
• expected generalization gap

Innovation

• Formal Methods
• Generalization Bounds
• Algorithmic Stability
• Reachability Analysis
• Concentration Inequality