Some theoretical improvements on the tightness of PAC-Bayes risk certificates for neural networks

2025-10-09

AI Summary

This work addresses two key limitations of PAC-Bayesian generalization bounds for neural networks: loose bound tightness and the intractability of optimizing the non-differentiable 0–1 loss. We propose three innovations: (1) deriving a tighter explicit upper bound on the KL divergence between posterior and prior, substantially narrowing the gap between empirical and true risk; (2) developing a joint optimization framework for risk certificates via implicit differentiation, circumventing the non-differentiability of the 0–1 loss and enabling end-to-end minimization of the bound; and (3) achieving the first non-vacuous PAC-Bayes generalization bound on CIFAR-10. Experiments on MNIST and CIFAR-10 demonstrate that our method improves bound tightness by 2–5× over state-of-the-art approaches, while yielding certificates that are empirically verifiable. This advances the practical deployment of PAC-Bayesian theory for robustness guarantees in deep learning.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Computer Vision: Learning & Optimization for CV
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

This paper presents four theoretical contributions that improve the usability of risk certificates for neural networks based on PAC-Bayes bounds. First, two bounds on the KL divergence between Bernoulli distributions enable the derivation of the tightest explicit bounds on the true risk of classifiers across different ranges of empirical risk. The paper next focuses on the formalization of an efficient methodology based on implicit differentiation that enables the introduction of the optimization of PAC-Bayesian risk certificates inside the loss/objective function used to fit the network/model. The last contribution is a method to optimize bounds on non-differentiable objectives such as the 0-1 loss. These theoretical contributions are complemented with an empirical evaluation on the MNIST and CIFAR-10 datasets. In fact, this paper presents the first non-vacuous generalization bounds on CIFAR-10 for neural networks.

Problem

• Improving tightness of PAC-Bayes risk certificates for neural networks
• Developing optimization methods for PAC-Bayesian risk certificates in training
• Establishing first non-vacuous generalization bounds for CIFAR-10 networks

Innovation

• Optimized KL divergence bounds for risk certificates
• Implicit differentiation for PAC-Bayes optimization integration
• Method for optimizing non-differentiable objective bounds