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 ∞-Rényi Divergence and relate it to ε-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 Lα-norms commonly studied in the literature.