Bounds on the Excess Minimum Risk via Generalized Information Divergence Measures

TL;DR

This paper develops new bounds on the excess minimum risk in Markov chain models using generalized information divergence measures, extending previous bounds and applicable to broader distribution classes.

Contribution

It introduces a family of bounds based on Rényi, Jensen-Shannon, and Sibson divergences that generalize mutual information bounds and do not require constant sub-Gaussian parameters.

Findings

Bounds can be tighter than mutual information bounds in certain regimes.

Applicable to a broader class of distributions without constant sub-Gaussianity.

Numerical examples demonstrate the effectiveness of the generalized divergence bounds.

Abstract

Given finite-dimensional random vectors $Y$, $X$, and $Z$ that form a Markov chain in that order (i.e., $Y \to X \to Z$), we derive upper bounds on the excess minimum risk using generalized information divergence measures. Here, $Y$ is a target vector to be estimated from an observed feature vector $X$ or its stochastically degraded version $Z$. The excess minimum risk is defined as the difference between the minimum expected loss in estimating $Y$ from $X$ and from $Z$. We present a family of bounds that generalize the mutual information based bound of Gy\"orfi et al. (2023), using the R\'enyi and $\alpha$-Jensen-Shannon divergences, as well as Sibson's mutual information. Our bounds are similar to those developed by Modak et al. (2021) and Aminian et al. (2024) for the generalization error of learning algorithms. However, unlike these works, our bounds do not require the sub-Gaussian parameter to be constant and therefore apply to a broader class of joint distributions over $Y$, $X$, and $Z$. We also provide numerical examples under both constant and non-constant sub-Gaussianity assumptions, illustrating that our generalized divergence based bounds can be tighter than the one based on mutual information for certain regimes of the parameter $\alpha$.