Hit-and-run mixing via localization schemes

TL;DR

This paper provides an improved analysis of the hit-and-run algorithm for sampling from convex bodies, showing it mixes efficiently in terms of the isoperimetric constant, and introduces a novel localization scheme technique.

Contribution

The paper introduces a new analysis technique using localization schemes to bound the mixing time of hit-and-run in terms of the KLS constant, improving previous bounds.

Findings

Mixing time is $ ilde{O}(n^2/ ext{KLS}^2)$ for isotropic convex bodies.

Bound matches the best known bounds for the ball walk.

Uses localization schemes to analyze truncated Gaussian distributions.

Abstract

We analyze the hit-and-run algorithm for sampling uniformly from an isotropic convex body $K$ in $n$ dimensions. We show that the algorithm mixes in time $\tilde{O}(n^2/ \psi_n^2)$, where $\psi_n$ is the smallest isoperimetric constant for any isotropic logconcave distribution, also known as the Kannan-Lovasz-Simonovits (KLS) constant. Our bound improves upon previous bounds of the form $\tilde{O}(n^2 R^2/r^2)$, which depend on the ratio $R/r$ of the radii of the circumscribed and inscribed balls of $K$, gaining a factor of $n$ in the case of isotropic convex bodies. Consequently, our result gives a mixing time estimate for the hit-and-run which matches the state-of-the-art bounds for the ball walk. Our main proof technique is based on an annealing of localization schemes introduced in Chen and Eldan (2022), which allows us to reduce the problem to the analysis of the mixing time on truncated Gaussian distributions.