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.
From: Amedeo Roberto Esposito [view email]
[v1]
Sat, 17 Feb 2024 05:30:57 UTC (448 KB)
[v2]
Thu, 22 Feb 2024 07:39:46 UTC (448 KB)
[v3]
Tue, 11 Jun 2024 12:23:12 UTC (787 KB)
[v4]
Fri, 21 Nov 2025 07:44:11 UTC (383 KB)
[v5]
Wed, 2 Sep 2026 02:32:52 UTC (262 KB)
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