Unadjusted Langevin algorithm for sampling a mixture of weakly smooth potentials

TL;DR

This paper analyzes the unadjusted Langevin algorithm for sampling from distributions with weakly smooth and dissipative potentials, establishing convergence guarantees under relaxed conditions and polynomial dependence on dimension.

Contribution

It extends convergence analysis of Langevin algorithms to weakly smooth, non-strongly convex potentials, relaxing previous smoothness and convexity assumptions.

Findings

Convergence in KL divergence with polynomial dimension dependence.

Guarantees under Poincaré inequality or non-strong convexity outside a ball.

Convergence in Wasserstein metric for smoothing potentials.

Abstract

Discretization of continuous-time diffusion processes is a widely recognized method for sampling. However, it seems to be a considerable restriction when the potentials are often required to be smooth (gradient Lipschitz). This paper studies the problem of sampling through Euler discretization, where the potential function is assumed to be a mixture of weakly smooth distributions and satisfies weakly dissipative. 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 degenerated convex at infinity conditions of \citet{erdogdu2020convergence} and prove convergence guarantees under Poincar\'{e} inequality or non-strongly convex outside the ball. In addition, we also provide convergence in $L_{\beta}$-Wasserstein metric for the smoothing potential.