Contraction of Rényi Divergences for Discrete Channels

Source abstract

We investigate Strong Data-Processing Inequality (SDPI) constants for Rényi Divergences on finite spaces. We study their dependence on the Rényi order $α$, proving that they are non-decreasing and that their scaling by $(α-1)$ is convex for $α\geq1$. We also identify several support restrictions on the probability measures involved in determining these constants. In particular, in the distribution-independent setting, measures supported on a common set of at most two points suffice to evaluate the Rényi-SDPI constant. For $α\in[0,1]$, we further prove equality with the $χ^2$-SDPI constant, while at order infinity we obtain a closed-form expression. In order to link contraction over product spaces to contraction along individual coordinates, we provide tensorisation bounds for arbitrary product channels analogous to those known for $\varphi$-Divergences. At finite orders, the Rényi-SDPI constants are bounded above and below through comparisons with the $χ^2$-Divergence and Hellinger Divergences, with sharpness established in multiple cases. At order infinity, we instead relate these constants to the contraction of Total Variation Distance. Finally, our findings are applied to local differential privacy (LDP) and the analysis of Markov chains. This yields sharp contraction guarantees for pure-LDP mechanisms, and connects Rényi-LDP to Rényi-SDPI constants. For Markov chains, we derive finite-time convergence bounds and exhibit a family of chains for which Rényi-SDPIs improve on classical $χ^2$-based bounds by arbitrarily large factors.