Learning Algorithm Generalization Error Bounds via Auxiliary Distributions

TL;DR

This paper introduces the Auxiliary Distribution Method to derive new, tighter upper bounds on the generalization error in supervised learning, utilizing $oldsymbol{ extit{ extalpha}}$-Jensen-Shannon and Rényi information measures.

Contribution

The work presents a novel auxiliary distribution approach that yields new upper bounds on generalization errors, including bounds involving $ extalpha$-Jensen-Shannon and Rényi divergences, applicable under distribution mismatch scenarios.

Findings

Bounds are specialized for $ extalpha$-Jensen-Shannon and Rényi information.

Bounds are finite and can be tighter than previous bounds.

Method applies to excess risk and distribution mismatch scenarios.

Abstract

Generalization error bounds are essential for comprehending how well machine learning models work. In this work, we suggest a novel method, i.e., the Auxiliary Distribution Method, that leads to new upper bounds on expected generalization errors that are appropriate for supervised learning scenarios. We show that our general upper bounds can be specialized under some conditions to new bounds involving the $\alpha$-Jensen-Shannon, $\alpha$-R\'enyi ($0< \alpha < 1$) information between a random variable modeling the set of training samples and another random variable modeling the set of hypotheses. Our upper bounds based on $\alpha$-Jensen-Shannon information are also finite. Additionally, we demonstrate how our auxiliary distribution method can be used to derive the upper bounds on excess risk of some learning algorithms in the supervised learning context {\blue and the generalization error under the distribution mismatch scenario in supervised learning algorithms, where the distribution mismatch is modeled as $\alpha$-Jensen-Shannon or $\alpha$-R\'enyi divergence between the distribution of test and training data samples distributions.} We also outline the conditions for which our proposed upper bounds might be tighter than other earlier upper bounds.