Non-asymptotic convergence analysis for the Unadjusted Langevin Algorithm

TL;DR

This paper provides non-asymptotic convergence bounds for the Unadjusted Langevin Algorithm, analyzing its efficiency in high-dimensional sampling from distributions with positive density, and extends previous theoretical results.

Contribution

It offers new non-asymptotic convergence bounds for the Euler discretization of Langevin SDEs, including high-dimensional settings, improving upon prior work.

Findings

Bounds hold for both constant and decreasing step sizes.

Explicit dependence on dimension d demonstrates high-dimensional applicability.

Results extend and improve previous convergence analyses.

Abstract

In this paper, we study a method to sample from a target distribution $\pi$ over $\mathbb{R}^d$ having a positive density with respect to the Lebesgue measure, known up to a normalisation factor. This method is based on the Euler discretization of the overdamped Langevin stochastic differential equation associated with $\pi$. For both constant and decreasing step sizes in the Euler discretization, we obtain non-asymptotic bounds for the convergence to the target distribution $\pi$ in total variation distance. A particular attention is paid to the dependency on the dimension $d$, to demonstrate the applicability of this method in the high dimensional setting. These bounds improve and extend the results of (Dalalyan 2014).