Mixing Time of the Proximal Sampler in Relative Fisher Information via Strong Data Processing Inequality

TL;DR

This paper proves exponential convergence of the Proximal Sampler in relative Fisher information for strongly log-concave distributions, linking it to Langevin dynamics and establishing a high-accuracy iteration complexity guarantee.

Contribution

It introduces a novel analysis using strong data processing inequalities to establish exponential convergence of the Proximal Sampler in relative Fisher information.

Findings

Proximal Sampler converges exponentially fast in relative Fisher information.

The convergence rate matches that of continuous-time Langevin dynamics.

Provides iteration complexity guarantees for high-accuracy sampling.

Abstract

We study the mixing time guarantee for sampling in relative Fisher information via the Proximal Sampler algorithm, which is an approximate proximal discretization of the Langevin dynamics. We show that when the target probability distribution is strongly log-concave, the relative Fisher information converges exponentially fast along the Proximal Sampler; this matches the exponential convergence rate of the relative Fisher information along the continuous-time Langevin dynamics for strongly log-concave target. When combined with a standard implementation of the Proximal Sampler via rejection sampling, this exponential convergence rate provides a high-accuracy iteration complexity guarantee for the Proximal Sampler in relative Fisher information when the target distribution is strongly log-concave and log-smooth. Our proof proceeds by establishing a strong data processing inequality for relative Fisher information along the Gaussian channel under strong log-concavity, and a data processing inequality along the reverse Gaussian channel for a special distribution. The forward and reverse Gaussian channels compose to form the Proximal Sampler, and these data processing inequalities imply the exponential convergence rate of the relative Fisher information along the Proximal Sampler.