Accelerating sampling via asymptotic relaxation enhancing flows

TL;DR

This paper introduces asymptotic relaxation enhancing flows to accelerate Langevin Monte Carlo sampling, achieving arbitrarily fast convergence while preserving the invariant measure.

Contribution

It constructs explicit finite energy flows on the full space that significantly speed up sampling convergence under natural growth conditions.

Findings

Achieves arbitrarily fast decay of relative entropy in sampling.

Constructs explicit flows on the full space without periodization.

Provides theoretical guarantees for accelerated convergence.

Abstract

In this paper, we accelerate Langevin Monte Carlo sampling from Gibbs measures $\pi\propto \exp(-U)$ by adding a large drift that preserves the invariant measure. For warm-start initial data, we characterize the sharp asymptotic decay rate of the relative entropy and introduce asymptotic relaxation enhancing flows: sequences that achieve arbitrarily fast decay. We construct such flows on the torus by scaling cellular flows and pushing them forward via diffeomorphisms, and we extend the construction to the full space using a Lyapunov function method to control behavior at infinity without periodization, obtaining explicit finite energy flows that guarantee arbitrarily fast convergence under natural growth conditions on $U$.