Certification-Enhanced Generalization Bounds

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.The resulting framework enables the theoretical analysis of prior generalization bounds to extend far beyond their original restrictive analytical assumptions while achieving a provably sound bound on the generalization gap. In practice, we demonstrate that our framework provide formal generalization guarantees that are orders of magnitude tighter than alternative computational approaches at scales ranging from toy datasets to fine-tuning 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.