Unadjusted Langevin algorithm for non-convex weakly smooth potentials

TL;DR

This paper advances the theoretical understanding of the Unadjusted Langevin Algorithm (ULA) for sampling from non-convex, weakly smooth distributions, providing convergence guarantees under broader conditions than previously known.

Contribution

It introduces a new mixture weakly smooth condition and proves convergence of ULA in various metrics for non-convex potentials, relaxing previous assumptions.

Findings

ULA converges under the new weakly smooth condition with log-Sobolev inequality.

ULA achieves convergence in $L_{2}$-Wasserstein distance for smoothed potentials.

Convergence in KL divergence is established with polynomial dependence on dimension.

Abstract

Discretization of continuous-time diffusion processes is a widely recognized method for sampling. However, the canonical Euler Maruyama discretization of the Langevin diffusion process, referred as Unadjusted Langevin Algorithm (ULA), studied mostly in the context of smooth (gradient Lipschitz) and strongly log-concave densities, is a considerable hindrance for its deployment in many sciences, including statistics and machine learning. In this paper, we establish several theoretical contributions to the literature on such sampling methods for non-convex distributions. Particularly, we introduce a new mixture weakly smooth condition, under which we prove that ULA will converge with additional log-Sobolev inequality. We also show that ULA for smoothing potential will converge in $L_{2}$-Wasserstein distance. Moreover, using convexification of nonconvex domain \citep{ma2019sampling} in combination with regularization, we establish the convergence in Kullback-Leibler (KL) divergence with the number of iterations to reach $\epsilon$-neighborhood of a target distribution in only polynomial dependence on the dimension. We relax the conditions of \citep{vempala2019rapid} and prove convergence guarantees under isoperimetry, and non-strongly convex at infinity.