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

Neha S. Wadia

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 $\mathcal{O}^{\star}\left(\kappa^2 n^{7.5}
\left(\max \left[ 1, \sqrt{\frac{1}{n}\frac{2M}{\gamma}}\right] \right)^2
\right)$
steps to
produce a sample with error no more than $\gamma$ in total variation distance from a distribution with
condition number $\kappa$.