Analysis of Langevin Monte Carlo from Poincaré to Log-Sobolev

Abstract

Classically, the continuous-time Langevin diffusion converges exponentially fast to its stationary distribution $\pi$ under the sole assumption that $\pi$ satisfies a Poincaré inequality. Using this fact to provide guarantees for the discrete-time Langevin Monte Carlo (LMC) algorithm, however, is considerably more challenging due to the need for working with chi-squared or Rényi divergences, and prior works have largely focused on strongly log-concave targets. In this work, we provide the first convergence guarantees for LMC assuming that $\pi$ satisfies either a Latała--Oleszkiewicz or modified log-Sobolev inequality, which interpolates between the Poincaré and log-Sobolev settings. Unlike prior works, our results allow for weak smoothness and do not require convexity or dissipativity conditions.

Published at the journal of Foundations of Computational Mathematics (short version presented at Conference on Learning Theory)

Cited by in corpus (2)

• Convergence of Stein Variational Gradient Descent under a Weaker Smoothness Condition
• Langevin Monte Carlo for strongly log-concave distributions: Randomized midpoint revisited