Higher Order Langevin Monte Carlo Algorithm

TL;DR

This paper introduces a higher order Langevin Monte Carlo algorithm that achieves improved convergence rates in total variation and Wasserstein distances for sampling from complex distributions, even without convexity.

Contribution

The paper presents a novel unadjusted Langevin Monte Carlo method with enhanced convergence rates under weaker smoothness assumptions.

Findings

Achieves convergence rate of 1 + β/2 in Wasserstein distance.

Attains a convergence rate of 1 in total variation distance without convexity.

Provides explicit constants for strongly convex cases.

Abstract

A new (unadjusted) Langevin Monte Carlo (LMC) algorithm with improved rates in total variation and in Wasserstein distance is presented. All these are obtained in the context of sampling from a target distribution $\pi$ that has a density $\hat{\pi}$ on $\mathbb{R}^d$ known up to a normalizing constant. Moreover, $-\log \hat{\pi}$ is assumed to have a locally Lipschitz gradient and its third derivative is locally H\"{o}lder continuous with exponent $\beta \in (0,1]$. Non-asymptotic bounds are obtained for the convergence to stationarity of the new sampling method with convergence rate $1+ \beta/2$ in Wasserstein distance, while it is shown that the rate is 1 in total variation even in the absence of convexity. Finally, in the case where $-\log \hat{\pi}$ is strongly convex and its gradient is Lipschitz continuous, explicit constants are provided.