Translator Disclaimer
February 2021 On stochastic gradient Langevin dynamics with dependent data streams in the logconcave case
M. Barkhagen, N.H. Chau, É. Moulines, M. Rásonyi, S. Sabanis, Y. Zhang
Bernoulli 27(1): 1-33 (February 2021). DOI: 10.3150/19-BEJ1187

Abstract

We study the problem of sampling from a probability distribution $\pi $ on $\mathbb{R}^{d}$ which has a density w.r.t. the Lebesgue measure known up to a normalization factor $x\mapsto \mathrm{e}^{-U(x)}/\int _{\mathbb{R}^{d}}\mathrm{e}^{-U(y)}\,\mathrm{d}y$. We analyze a sampling method based on the Euler discretization of the Langevin stochastic differential equations under the assumptions that the potential $U$ is continuously differentiable, $\nabla U$ is Lipschitz, and $U$ is strongly concave. We focus on the case where the gradient of the log-density cannot be directly computed but unbiased estimates of the gradient from possibly dependent observations are available. This setting can be seen as a combination of a stochastic approximation (here stochastic gradient) type algorithms with discretized Langevin dynamics. We obtain an upper bound of the Wasserstein-2 distance between the law of the iterates of this algorithm and the target distribution $\pi $ with constants depending explicitly on the Lipschitz and strong convexity constants of the potential and the dimension of the space. Finally, under weaker assumptions on $U$ and its gradient but in the presence of independent observations, we obtain analogous results in Wasserstein-2 distance.

Citation

Download Citation

M. Barkhagen. N.H. Chau. É. Moulines. M. Rásonyi. S. Sabanis. Y. Zhang. "On stochastic gradient Langevin dynamics with dependent data streams in the logconcave case." Bernoulli 27 (1) 1 - 33, February 2021. https://doi.org/10.3150/19-BEJ1187

Information

Received: 1 March 2019; Revised: 1 September 2019; Published: February 2021
First available in Project Euclid: 20 November 2020

zbMATH: 07282840
MathSciNet: MR4177359
Digital Object Identifier: 10.3150/19-BEJ1187

Rights: Copyright © 2021 Bernoulli Society for Mathematical Statistics and Probability

JOURNAL ARTICLE
33 PAGES

This article is only available to subscribers.
It is not available for individual sale.
+ SAVE TO MY LIBRARY

SHARE
Vol.27 • No. 1 • February 2021
Back to Top