Convergence of Langevin MCMC in KL-divergence

TL;DR

This paper analyzes the convergence rate of Langevin MCMC algorithms in KL-divergence, providing new bounds under smooth and strongly convex conditions, and extending to non-strongly convex cases with a gradient flow perspective.

Contribution

It establishes the convergence rate of Langevin diffusion in KL-divergence under smooth and strongly convex conditions, and offers a novel analysis using gradient flow methods.

Findings

Langevin diffusion achieves KL divergence $ o 0$ in $ ilde{O}(d/\epsilon)$ steps under strong convexity.

The analysis applies to convergence in KL-divergence, a stronger metric than total variation.

Provides a simpler proof of existing convergence results using gradient flow perspective.

Abstract

Langevin diffusion is a commonly used tool for sampling from a given distribution. In this work, we establish that when the target density $p^*$ is such that $\log p^*$ is $L$ smooth and $m$ strongly convex, discrete Langevin diffusion produces a distribution $p$ with $KL(p||p^*)\leq \epsilon$ in $\tilde{O}(\frac{d}{\epsilon})$ steps, where $d$ is the dimension of the sample space. We also study the convergence rate when the strong-convexity assumption is absent. By considering the Langevin diffusion as a gradient flow in the space of probability distributions, we obtain an elegant analysis that applies to the stronger property of convergence in KL-divergence and gives a conceptually simpler proof of the best-known convergence results in weaker metrics.