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

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.