A mixing time bound for Gibbs sampling from log-smooth log-concave distributions

TL;DR

This paper establishes a polynomial mixing time bound for the Gibbs sampler when sampling from log-smooth, strongly log-concave distributions, demonstrating efficiency in high-dimensional settings.

Contribution

It provides the first explicit mixing time bound for Gibbs sampling on log-smooth, strongly log-concave distributions, highlighting dependence on dimension and condition number.

Findings

Gibbs sampler mixes in polynomial time for the specified class

Mixing time depends polynomially on dimension and condition number

Sample error can be controlled within specified total variation distance

Abstract

The Gibbs sampler, also known as the coordinate hit-and-run algorithm, is a Markov chain that is widely used to draw samples from probability distributions in arbitrary dimensions. At each iteration of the algorithm, a randomly selected coordinate is resampled from the distribution that results from conditioning on all the other coordinates. We study the behavior of the Gibbs sampler on the class of log-smooth and strongly log-concave target distributions supported on $\mathbb{R}^n$. Assuming the initial distribution is $M$-warm with respect to the target, we show that the Gibbs sampler requires at most $O^{\star}\left(\kappa^2 n^{7.5}\left(\max\left\{1,\sqrt{\frac{1}{n}\log \frac{2M}{\gamma}}\right\}\right)^2\right)$ steps to produce a sample with error no more than $\gamma$ in total variation distance from a distribution with condition number $\kappa$.