Double-Loop Unadjusted Langevin Algorithm

TL;DR

This paper introduces a novel annealing step-size schedule for the Unadjusted Langevin Algorithm (ULA), providing improved convergence guarantees for sampling from smooth log-concave distributions and enhanced rates for constrained cases.

Contribution

It proposes a new step-size schedule for ULA and derives a theoretical bound relating Wasserstein and total variation distances, extending convergence guarantees.

Findings

New convergence guarantees for ULA with the proposed schedule.

State-of-the-art convergence rates for constrained log-concave sampling.

Improved dimension dependence in sampling algorithms.

Abstract

A well-known first-order method for sampling from log-concave probability distributions is the Unadjusted Langevin Algorithm (ULA). This work proposes a new annealing step-size schedule for ULA, which allows to prove new convergence guarantees for sampling from a smooth log-concave distribution, which are not covered by existing state-of-the-art convergence guarantees. To establish this result, we derive a new theoretical bound that relates the Wasserstein distance to total variation distance between any two log-concave distributions that complements the reach of Talagrand T2 inequality. Moreover, applying this new step size schedule to an existing constrained sampling algorithm, we show state-of-the-art convergence rates for sampling from a constrained log-concave distribution, as well as improved dimension dependence.