Gentle Local Robustness implies Generalization

TL;DR

This paper reveals limitations of existing robustness-based generalization bounds for the Bayes optimal classifier and introduces new, tighter bounds that converge to the true error, with empirical validation on deep neural networks.

Contribution

The paper identifies vacuous error bounds for the Bayes classifier and proposes novel, model-dependent bounds that are provably tighter and converge with more data.

Findings

Existing bounds are vacuous for the Bayes optimal classifier.

Proposed bounds are tighter and converge to true error as sample size increases.

Two bounds are often non-vacuous for deep neural networks pretrained on ImageNet.

Abstract

Robustness and generalization ability of machine learning models are of utmost importance in various application domains. There is a wide interest in efficient ways to analyze those properties. One important direction is to analyze connection between those two properties. Prior theories suggest that a robust learning algorithm can produce trained models with a high generalization ability. However, we show in this work that the existing error bounds are vacuous for the Bayes optimal classifier which is the best among all measurable classifiers for a classification problem with overlapping classes. Those bounds cannot converge to the true error of this ideal classifier. This is undesirable, surprizing, and never known before. We then present a class of novel bounds, which are model-dependent and provably tighter than the existing robustness-based ones. Unlike prior ones, our bounds are guaranteed to converge to the true error of the best classifier, as the number of samples increases. We further provide an extensive experiment and find that two of our bounds are often non-vacuous for a large class of deep neural networks, pretrained from ImageNet.