Rapid Convergence of the Unadjusted Langevin Algorithm: Isoperimetry Suffices

TL;DR

This paper proves that the Unadjusted Langevin Algorithm converges efficiently under isoperimetric conditions like log-Sobolev inequalities, even without convexity, and provides bounds on bias assuming third-order smoothness.

Contribution

It establishes convergence guarantees for ULA under isoperimetric conditions without convexity and introduces bias bounds with third-order smoothness.

Findings

ULA converges in KL divergence under log-Sobolev inequality without convexity.

Convergence in Rényi divergence is achieved assuming the limit satisfies isoperimetric inequalities.

Bias of ULA's limiting distribution is bounded with third-order smoothness of the target.

Abstract

We study the Unadjusted Langevin Algorithm (ULA) for sampling from a probability distribution $\nu = e^{-f}$ on $\mathbb{R}^n$. We prove a convergence guarantee in Kullback-Leibler (KL) divergence assuming $\nu$ satisfies a log-Sobolev inequality and the Hessian of $f$ is bounded. Notably, we do not assume convexity or bounds on higher derivatives. We also prove convergence guarantees in R\'enyi divergence of order $q > 1$ assuming the limit of ULA satisfies either the log-Sobolev or Poincar\'e inequality. We also prove a bound on the bias of the limiting distribution of ULA assuming third-order smoothness of $f$, without requiring isoperimetry.