Non-asymptotic estimates for accelerated high order Langevin Monte Carlo algorithms

TL;DR

This paper introduces two new algorithms, aHOLA and aHOLLA, for high-dimensional sampling that achieve state-of-the-art convergence rates under less restrictive conditions, with theoretical guarantees and numerical validation.

Contribution

The paper develops aHOLA and aHOLLA algorithms with non-asymptotic convergence guarantees for high-dimensional sampling under weaker conditions than existing methods.

Findings

Achieve convergence rates of 1+q/2 and 1/2+q/4 in Wasserstein distances.

Demonstrate superior convergence in non-convex high-dimensional settings.

Numerical experiments confirm theoretical results.

Abstract

In this paper, we propose two new algorithms, namely, aHOLA and aHOLLA, to sample from high-dimensional target distributions with possibly super-linearly growing potentials. We establish non-asymptotic convergence bounds for aHOLA in Wasserstein-1 and Wasserstein-2 distances with rates of convergence equal to $1+q/2$ and $1/2+q/4$, respectively, under a local H\"{o}lder condition with exponent $q\in(0,1]$ and a convexity at infinity condition on the potential of the target distribution. Similar results are obtained for aHOLLA under certain global continuity conditions and a dissipativity condition. Crucially, we achieve state-of-the-art rates of convergence of the proposed algorithms in the non-convex setting which are higher than those of the existing algorithms. Examples from high-dimensional sampling and logistic regression are presented, and numerical results support our main findings.