Contraction of R\'enyi Divergences for Discrete Channels: Properties and Applications

TL;DR

This paper investigates the properties of Rényi Divergences' contraction in discrete channels, revealing how the divergence order affects contraction behavior and applying findings to Markov chain convergence and differential privacy.

Contribution

It characterizes contraction properties of Rényi Divergences across different orders and connects these properties to Markov chain convergence and privacy guarantees.

Findings

Contraction properties vary significantly for different divergence orders.

The $ abla$-Rényi Divergence exhibits unique contraction characteristics.

Results provide new bounds on Markov chain convergence rates.

Abstract

This work explores properties of Strong Data-Processing constants for R\'enyi Divergences. Parallels are made with the well-studied $\varphi$-Divergences, and it is shown that the order $\alpha$ of R\'enyi Divergences dictates whether certain properties of the contraction of $\varphi$-Divergences are mirrored or not. In particular, we demonstrate that when $\alpha>1$, the contraction properties can deviate quite strikingly from those of $\varphi$-Divergences. We also uncover specific characteristics of contraction for the $\infty$-R\'enyi Divergence and relate it to $\varepsilon$-Local Differential Privacy. The results are then applied to bound the speed of convergence of Markov chains, where we argue that the contraction of R\'enyi Divergences offers a new perspective on the contraction of $L^\alpha$-norms commonly studied in the literature.