Sampling from a log-concave distribution with compact support with proximal Langevin Monte Carlo

TL;DR

This paper provides a theoretical analysis and convergence guarantees for a proximal Langevin Monte Carlo algorithm used to sample from log-concave distributions within convex bodies, with polynomial complexity in dimension.

Contribution

It offers explicit convergence bounds and complexity analysis for the proximal Langevin Monte Carlo method applied to constrained log-concave distributions.

Findings

Established explicit convergence bounds in total variation and Wasserstein distances.

Proved polynomial complexity of the algorithm in the dimension of the space.

Compared the method with existing MCMC approaches through numerical experiments.

Abstract

This paper presents a detailed theoretical analysis of the Langevin Monte Carlo sampling algorithm recently introduced in Durmus et al. (Efficient Bayesian computation by proximal Markov chain Monte Carlo: when Langevin meets Moreau, 2016) when applied to log-concave probability distributions that are restricted to a convex body $\mathsf{K}$. This method relies on a regularisation procedure involving the Moreau-Yosida envelope of the indicator function associated with $\mathsf{K}$. Explicit convergence bounds in total variation norm and in Wasserstein distance of order $1$ are established. In particular, we show that the complexity of this algorithm given a first order oracle is polynomial in the dimension of the state space. Finally, some numerical experiments are presented to compare our method with competing MCMC approaches from the literature.