Sharp convergence rates for Langevin dynamics in the nonconvex setting

TL;DR

This paper establishes sharp convergence rates for Langevin dynamics algorithms in nonconvex settings, showing polynomial dependence on dimension and accuracy but exponential dependence on non-log-concavity measures.

Contribution

The paper provides the first explicit upper bounds on the iteration complexity of Langevin MCMC methods in nonconvex scenarios with mixed convexity properties.

Findings

Overdamped Langevin requires ^{cLR^2}d/\u03b5^2 iterations

Underdamped Langevin requires ^{cLR^2}^{cLR^2}a9/be iterations

Complexity is polynomial in dimension and accuracy, exponential in non-log-concavity measure

Abstract

We study the problem of sampling from a distribution $p^*(x) \propto \exp\left(-U(x)\right)$, where the function $U$ is $L$-smooth everywhere and $m$-strongly convex outside a ball of radius $R$, but potentially nonconvex inside this ball. We study both overdamped and underdamped Langevin MCMC and establish upper bounds on the number of steps required to obtain a sample from a distribution that is within $\epsilon$ of $p^*$ in $1$-Wasserstein distance. For the first-order method (overdamped Langevin MCMC), the iteration complexity is $\tilde{\mathcal{O}}\left(e^{cLR^2}d/\epsilon^2\right)$, where $d$ is the dimension of the underlying space. For the second-order method (underdamped Langevin MCMC), the iteration complexity is $\tilde{\mathcal{O}}\left(e^{cLR^2}\sqrt{d}/\epsilon\right)$ for an explicit positive constant $c$. Surprisingly, the iteration complexity for both these algorithms is only polynomial in the dimension $d$ and the target accuracy $\epsilon$. It is exponential, however, in the problem parameter $LR^2$, which is a measure of non-log-concavity of the target distribution.