Empirical Risk Minimization with Relative Entropy Regularization

TL;DR

This paper explores ERM with relative entropy regularization under a general measure framework, establishing properties of solutions, their generalization guarantees, and connections to information theory.

Contribution

It generalizes ERM-RER to non-probability reference measures, proving solution uniqueness, generalization bounds, and linking sensitivity to lautum information.

Findings

Solution is a unique, absolutely continuous probability measure.

Empirical risk is sub-Gaussian under certain conditions.

Sensitivity analysis relates generalization error to lautum information.

Abstract

The empirical risk minimization (ERM) problem with relative entropy regularization (ERM-RER) is investigated under the assumption that the reference measure is a $\sigma$-finite measure, and not necessarily a probability measure. Under this assumption, which leads to a generalization of the ERM-RER problem allowing a larger degree of flexibility for incorporating prior knowledge, numerous relevant properties are stated. Among these properties, the solution to this problem, if it exists, is shown to be a unique probability measure, mutually absolutely continuous with the reference measure. Such a solution exhibits a probably-approximately-correct guarantee for the ERM problem independently of whether the latter possesses a solution. For a fixed dataset and under a specific condition, the empirical risk is shown to be a sub-Gaussian random variable when the models are sampled from the solution to the ERM-RER problem. The generalization capabilities of the solution to the ERM-RER problem (the Gibbs algorithm) are studied via the sensitivity of the expected empirical risk to deviations from such a solution towards alternative probability measures. Finally, an interesting connection between sensitivity, generalization error, and lautum information is established.