Sharp convergence rates for Langevin dynamics in the nonconvex setting

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 $p<sup>*$ in $1$-Wasserstein distance. For the first-order method (overdamped Langevin MCMC), the iteration complexity is $$\tilde{\mathcal{O}}\left(e<sup>{cLR<sup>2}d/\epsilon<sup>2\right)$$, where $d$ is the dimension of the underlying space. For the second-order method (underdamped Langevin MCMC), the iteration complexity is $$\tilde{\mathcal{O}}\left(e<sup>{cLR<sup>2}\sqrt{d}/\epsilon\right)$$ 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 $LR<sup>2$, which is a measure of non-log-concavity of the target distribution.