Sharp convergence rates for Langevin dynamics in the nonconvex setting
arXiv preprint arXiv:1805.01648, 2018•arxiv.org
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 …
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 …
We study the problem of sampling from a distribution , where the function is -smooth everywhere and -strongly convex outside a ball of radius , 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 of in -Wasserstein distance. For the first-order method (overdamped Langevin MCMC), the iteration complexity is , where is the dimension of the underlying space. For the second-order method (underdamped Langevin MCMC), the iteration complexity is for an explicit positive constant . Surprisingly, the iteration complexity for both these algorithms is only polynomial in the dimension and the target accuracy . It is exponential, however, in the problem parameter , which is a measure of non-log-concavity of the target distribution.
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