Theoretical Guarantees for Approximate Sampling from Smooth and Log-Concave Densities

Abstract

Summary Sampling from various kinds of distribution is an issue of paramount importance in statistics since it is often the key ingredient for constructing estimators, test procedures or confidence intervals. In many situations, exact sampling from a given distribution is impossible or computationally expensive and, therefore, one needs to resort to approximate sampling strategies. However, there is no well-developed theory providing meaningful non-asymptotic guarantees for the approximate sampling procedures, especially in high dimensional problems. The paper makes some progress in this direction by considering the problem of sampling from a distribution having a smooth and log-concave density defined on Rp, for some integer p > 0. We establish non-asymptotic bounds for the error of approximating the target distribution by the distribution obtained by the Langevin Monte Carlo method and its variants. We illustrate the effectiveness of the established guarantees with various experiments. Underlying our analysis are insights from the theory of continuous time diffusion processes, which may be of interest beyond the framework of log-concave densities that are considered in the present work.