Hit-and-run for numerical integration

TL;DR

This paper analyzes the efficiency of hit-and-run algorithms for numerical integration of log-concave functions over convex bodies, providing refined tractability results and error bounds for MCMC methods.

Contribution

It introduces new error bounds and tractability results for hit-and-run algorithms applied to high-dimensional integration of log-concave densities.

Findings

Hit-and-run algorithms achieve refined tractability in high dimensions.

Error bounds for multi-run MCMC methods are established.

The approach applies to convex bodies and the entire Euclidean space.

Abstract

We study the numerical computation of an expectation of a bounded function with respect to a measure given by a non-normalized density on a convex body. We assume that the density is log-concave, satisfies a variability condition and is not too narrow. We consider general convex bodies or even the whole $\R^d$ and show that the integration problem satisfies a refined form of tractability. The main tools are the hit-and-run algorithm and an error bound of a multi run Markov chain Monte Carlo method.