Misclassification excess risk bounds for PAC-Bayesian classification via convexified loss

TL;DR

This paper develops new PAC-Bayesian bounds for classification that directly relate to misclassification risk by leveraging convex surrogate losses and relative bounds, improving understanding of generalization in such settings.

Contribution

It introduces a novel approach to derive misclassification excess risk bounds in PAC-Bayesian classification using convex surrogate losses and expectation-based bounds.

Findings

Provides new theoretical bounds for misclassification risk

Demonstrates approach in multiple applications

Improves understanding of generalization with convex surrogates

Abstract

PAC-Bayesian bounds have proven to be a valuable tool for deriving generalization bounds and for designing new learning algorithms in machine learning. However, it typically focus on providing generalization bounds with respect to a chosen loss function. In classification tasks, due to the non-convex nature of the 0-1 loss, a convex surrogate loss is often used, and thus current PAC-Bayesian bounds are primarily specified for this convex surrogate. This work shifts its focus to providing misclassification excess risk bounds for PAC-Bayesian classification when using a convex surrogate loss. Our key ingredient here is to leverage PAC-Bayesian relative bounds in expectation rather than relying on PAC-Bayesian bounds in probability. We demonstrate our approach in several important applications.