Langevin Monte Carlo without smoothness
Abstract
Langevin Monte Carlo (LMC) is an iterative algorithm used to generate samples from a distribution that is known only up to a normalizing constant. The nonasymptotic dependence of its mixing time on the dimension and target accuracy is understood mainly in the setting of smooth (gradient-Lipschitz) log-densities, a serious limitation for applications in machine learning. In this paper, we remove this limitation, providing polynomial-time convergence guarantees for a variant of LMC in the setting of nonsmooth log-concave distributions. At a high level, our results follow by leveraging the implicit smoothing of the log-density that comes from a small Gaussian perturbation that we add to the iterates of the algorithm and controlling the bias and variance that are induced by this perturbation.
Cite
@article{arxiv.1905.13285,
title = {Langevin Monte Carlo without smoothness},
author = {Niladri S. Chatterji and Jelena Diakonikolas and Michael I. Jordan and Peter L. Bartlett},
journal= {arXiv preprint arXiv:1905.13285},
year = {2020}
}
Comments
Updated to match the AISTATS 2020 camera ready version. Some example applications added and typos corrected