Function-space regularized R\'enyi divergences

TL;DR

This paper introduces a new family of regularized Renyi divergences that incorporate a variational function space, offering lower variance estimators and improved performance in distribution comparison and GAN training.

Contribution

The paper develops a novel class of regularized Renyi divergences with a dual variational representation, addressing variance issues and extending applicability to non-absolutely continuous distributions.

Findings

Lower variance estimators for enyi divergences.

Enhanced GAN training stability and performance.

Ability to compare distributions with low-dimensional support.

Abstract

We propose a new family of regularized R\'enyi divergences parametrized not only by the order $\alpha$ but also by a variational function space. These new objects are defined by taking the infimal convolution of the standard R\'enyi divergence with the integral probability metric (IPM) associated with the chosen function space. We derive a novel dual variational representation that can be used to construct numerically tractable divergence estimators. This representation avoids risk-sensitive terms and therefore exhibits lower variance, making it well-behaved when $\alpha>1$; this addresses a notable weakness of prior approaches. We prove several properties of these new divergences, showing that they interpolate between the classical R\'enyi divergences and IPMs. We also study the $\alpha\to\infty$ limit, which leads to a regularized worst-case-regret and a new variational representation in the classical case. Moreover, we show that the proposed regularized R\'enyi divergences inherit features from IPMs such as the ability to compare distributions that are not absolutely continuous, e.g., empirical measures and distributions with low-dimensional support. We present numerical results on both synthetic and real datasets, showing the utility of these new divergences in both estimation and GAN training applications; in particular, we demonstrate significantly reduced variance and improved training performance.