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Sharp Convergence Rates for Langevin Dynamics in the Nonconvex Setting

4 May 2018
Xiang Cheng
Niladri S. Chatterji
Yasin Abbasi-Yadkori
Peter L. Bartlett
Michael I. Jordan
ArXiv (abs)PDFHTML
Abstract

We study the problem of sampling from a distribution where the negative logarithm of the target density is LLL-smooth everywhere and mmm-strongly convex outside a ball of radius RRR, but potentially non-convex inside this ball. We study both overdamped and underdamped Langevin MCMC and prove upper bounds on the time required to obtain a sample from a distribution that is within ϵ\epsilonϵ of the target distribution in 111-Wasserstein distance. For the first-order method (overdamped Langevin MCMC), the time complexity is O~(ecLR2dϵ2)\tilde{\mathcal{O}}\left(e^{cLR^2}\frac{d}{\epsilon^2}\right)O~(ecLR2ϵ2d​), where ddd is the dimension of the underlying space. For the second-order method (underdamped Langevin MCMC), the time complexity is O~(ecLR2dϵ)\tilde{\mathcal{O}}\left(e^{cLR^2}\frac{\sqrt{d}}{\epsilon}\right)O~(ecLR2ϵd​​) for some explicit positive constant ccc. Surprisingly, the convergence rate is only polynomial in the dimension ddd and the target accuracy ϵ\epsilonϵ. It is however exponential in the problem parameter LR2LR^2LR2, which is a measure of non-logconcavity of the target distribution.

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