  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.
