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arXiv · 2402.11200

Tight Bounds for Linear and Non-Linear Contraction of Divergences via Duality

Amedeo Roberto Esposito·Marco Mondelli
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Abstract

We develop a novel framework for bounding the contraction of information divergences, using duality and associated norms in Orlicz spaces. By working in the dual space, we obtain a principled approach to bounding both distribution-dependent strong data-processing inequality (SDPI) constants and \(F_\varphi\)-curves of divergences. Our bounds are either available in closed form or reducible to one-dimensional convex optimisation problems, in contrast to the infinite-dimensional optimisation problems that characterise SDPIs. These bounds depend on the densities of the reverse kernels with respect to a reference measure. To the best of our knowledge, they are the first universal closed-form bounds on distribution-dependent SDPI constants. We establish tightness for the \(\chi^2\)-divergence on several important channel classes, including full-rank binary kernels. We apply our results to several settings. In particular, we derive bounds on the mixing times of Markov chains, including chains with heavy-tailed stationary distributions; obtain improved bounds on burn-in periods for Markov chain Monte Carlo; and strengthen concentration-of-measure bounds for dependent random variables.

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Amedeo Roberto Esposito, Marco Mondelli. 2024-02-17. Tight Bounds for Linear and Non-Linear Contraction of Divergences via Duality. https://arxiv.org/abs/2402.11200

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