Bregman Deviations of Generic Exponential Families

TL;DR

This paper introduces a new concentration bound for exponential families using Bregman divergence and mixture methods, providing explicit confidence sets and demonstrating competitive performance in various classical distributions.

Contribution

It develops a novel, time-uniform concentration bound based on Bregman divergence and mixture techniques, extending classical information gain concepts to exponential families.

Findings

Derived explicit confidence bounds for multiple classical families.

Compared bounds to state-of-the-art methods showing competitive results.

Applied bounds to illustrative applications demonstrating practical benefits.

Abstract

We revisit the method of mixture technique, also known as the Laplace method, to study the concentration phenomenon in generic exponential families. Combining the properties of Bregman divergence associated with log-partition function of the family with the method of mixtures for super-martingales, we establish a generic bound controlling the Bregman divergence between the parameter of the family and a finite sample estimate of the parameter. Our bound is time-uniform and makes appear a quantity extending the classical information gain to exponential families, which we call the Bregman information gain. For the practitioner, we instantiate this novel bound to several classical families, e.g., Gaussian, Bernoulli, Exponential, Weibull, Pareto, Poisson and Chi-square yielding explicit forms of the confidence sets and the Bregman information gain. We further numerically compare the resulting confidence bounds to state-of-the-art alternatives for time-uniform concentration and show that this novel method yields competitive results. Finally, we highlight the benefit of our concentration bounds on some illustrative applications.