Generalization Analysis of Machine Learning Algorithms via the Worst-Case Data-Generating Probability Measure

TL;DR

This paper introduces a worst-case data-generating probability measure to analyze the generalization capabilities of machine learning algorithms, providing closed-form expressions and linking it to the Gibbs measure and existing generalization bounds.

Contribution

It defines a worst-case probability measure as a Gibbs measure for characterizing generalization and establishes a novel connection between this measure and the Gibbs algorithm.

Findings

Closed-form expressions for generalization metrics involving the worst-case measure

Recovery of existing results relating Gibbs algorithms to mutual and lautum information

Identification of the Gibbs measure as a fundamental link between model and data spaces

Abstract

In this paper, the worst-case probability measure over the data is introduced as a tool for characterizing the generalization capabilities of machine learning algorithms. More specifically, the worst-case probability measure is a Gibbs probability measure and the unique solution to the maximization of the expected loss under a relative entropy constraint with respect to a reference probability measure. Fundamental generalization metrics, such as the sensitivity of the expected loss, the sensitivity of the empirical risk, and the generalization gap are shown to have closed-form expressions involving the worst-case data-generating probability measure. Existing results for the Gibbs algorithm, such as characterizing the generalization gap as a sum of mutual information and lautum information, up to a constant factor, are recovered. A novel parallel is established between the worst-case data-generating probability measure and the Gibbs algorithm. Specifically, the Gibbs probability measure is identified as a fundamental commonality of the model space and the data space for machine learning algorithms.