On the mixing time of coordinate Hit-and-Run

TL;DR

This paper proves a polynomial upper bound on the mixing time of the coordinate Hit-and-Run algorithm for sampling from convex bodies, advancing understanding of its efficiency in high-dimensional spaces.

Contribution

It establishes the first polynomial upper bound on the mixing time of coordinate Hit-and-Run on convex bodies, resolving an open question.

Findings

Polynomial upper bound on mixing time in terms of dimension, body size, and accuracy

Mixing time depends polynomially on the dimension and body parameters

Advances theoretical understanding of coordinate Hit-and-Run's efficiency

Abstract

We obtain a polynomial upper bound on the mixing time $T_{CHR}(\epsilon)$ of the coordinate Hit-and-Run random walk on an $n-$dimensional convex body, where $T_{CHR}(\epsilon)$ is the number of steps needed in order to reach within $\epsilon$ of the uniform distribution with respect to the total variation distance, starting from a warm start (i.e., a distribution which has a density with respect to the uniform distribution on the convex body that is bounded above by a constant). Our upper bound is polynomial in $n, R$ and $\frac{1}{\epsilon}$, where we assume that the convex body contains the unit $\Vert\cdot\Vert_\infty$-unit ball $B_\infty$ and is contained in its $R$-dilation $R\cdot B_\infty$. Whether coordinate Hit-and-Run has a polynomial mixing time has been an open question.