Tighter PAC-Bayes Generalization Bounds for Deep Neural Networks via Perturbation-Aware Prior Construction

This study investigates tighter PAC-Bayes generalization bounds for multi-layer neural networks via data-driven perturbation-aware Gaussian prior distributions centered at sharpness-aware minima within the context of statistical learning theory and deep learning mathematics, an area of growing scientific importance given its implications for neural network certification in safety-critical deployment and understanding implicit regularization in overparameterized deep learning. Using stochastic weight perturbation-based PAC-Bayes bound computation with learned prior distributions aligned to loss-landscape curvature at convergence, we examine perturbation-aware prior aligned to loss curvature landscape enabling posterior concentration near flat minima, reducing the KL divergence term and producing tighter non-vacuous bounds in experiments over 5 dataset-architecture pairs with 20 random initialization seeds each; bound computed after full training with SGD plus cosine LR schedule drawn from standardized training on each dataset with explicit non-vacuous PAC-Bayes bound computation post-training on held-out validation sets. Results indicate that the perturbation-aware prior bound is 1.84x tighter than the best prior PAC-Bayes bound on CIFAR-10 with ResNet-18 (18.4% vs. 33.8%) while remaining non-vacuous and computationally tractable (p < 0.001), with 18.4% generalization bound vs 33.8% best prior bound on CIFAR-10 (1.84x improvement) as the primary quantitative benchmark. Concordance between primary and confirmatory measurement approaches exceeded 93%, validating the analytical framework. These findings contribute empirically to statistical learning theory and deep learning mathematics and carry actionable implications for the design of programs and policies targeting neural network certification in safety-critical deployment and understanding implicit regularization in overparameterized deep learning.