Convergence of Langevin MCMC in KL-Divergence

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 $\KL{\p}{\p^*}≤ε$ in $\tilde{O}(\frac{d}{ε})$ 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.