Contraction of Rényi Divergences for Discrete Channels: Properties and Applications

Abstract

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