🤖 AI Summary
Classical learning theory, relying on uniform convergence over hypothesis spaces, struggles to explain the strong generalization performance of over-parameterized deep neural networks. This work proposes a unified framework that, for the first time, incorporates data-dependent worst-case generalization bounds into a single template inequality. The framework systematically integrates extensions of PAC-Bayes theory, geometric and topological characterizations of optimization trajectories—such as fractal dimension and α-weighted persistence sums—and information-theoretic surrogates grounded in algorithmic stability. By enabling direct comparison among diverse generalization bounds, it reveals their intrinsic connections and distinctions, and establishes a non-vacuous, tight family of upper bounds on generalization error that effectively accounts for the empirical generalization behavior of over-parameterized models.
📝 Abstract
Deep neural networks generalize well despite being heavily overparameterized, in apparent contradiction with classical learning theory based on uniform convergence over fixed hypothesis spaces. Uniform bounds over the entire parameter space are vacuous in this regime, and recent work has shown that non-vacuous guarantees can be recovered by restricting attention to the part of parameter space that the algorithm actually visits. This survey paper organizes this line of work around three steps: extending PAC-Bayesian theory to random, data-dependent hypothesis sets (arXiv:2404.17442); refining the complexity term with geometric and topological descriptors of the optimization trajectory, including fractal dimensions, alpha-weighted lifetime sums, and positive magnitude (arXiv:2006.09313, arXiv:2302.02766, arXiv:2407.08723); and replacing the resulting information-theoretic terms by stability assumptions (arXiv:2507.06775). We unify these contributions around a single template inequality and a head-to-head comparison of the resulting bounds.