Contraction of Rényi Divergences for Discrete Channels: Properties and Applications

This work explores properties of Strong Data-Processing constants for Rényi Divergences. Parallels are made with the well-studied $φ$-Divergences, and it is shown that the order $α$ of Rényi Divergences dictates whether certain properties of the contraction of $φ$-Divergences are mirrored or not. In particular, we demonstrate that when $α>1$, the contraction properties can deviate quite strikingly from those of $φ$-Divergences. We also uncover specific characteristics of contraction for the $\infty$-Rényi 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 Rényi Divergences offers a new perspective on the contraction of $L^α$-norms commonly studied in the literature.