# A Variational Characterization of R\'enyi Divergences

## TL;DR

This paper presents a new variational characterization of Re9nyi divergences between probability distributions and Markov chains, linking them to relative entropies and extending existing formulas.

## Contribution

It develops a novel variational formula for Re9nyi divergences using relative entropies, applicable to both probability distributions and Markov chains.

## Key findings

- Derived a variational formula for Re9nyi divergences between distributions.
- Extended the variational characterization to stationary finite state Markov chains.
- Connected the results with Varadhan's variational formula for spectral radius.

## Abstract

Atar, Chowdhary and Dupuis have recently exhibited a variational formula for exponential integrals of bounded measurable functions in terms of R\'enyi divergences. We develop a variational characterization of the R\'enyi divergences between two probability distributions on a measurable sace in terms of relative entropies. When combined with the elementary variational formula for exponential integrals of bounded measurable functions in terms of relative entropy, this yields the variational formula of Atar, Chowdhary and Dupuis as a corollary. We also develop an analogous variational characterization of the R\'enyi divergence rates between two stationary finite state Markov chains in terms of relative entropy rates. When combined with Varadhan's variational characterization of the spectral radius of square matrices with nonnegative entries in terms of relative entropy, this yields an analog of the variational formula of Atar, Chowdary and Dupuis in the framework of finite state Markov chains.