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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


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.


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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.


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

Keywords: Langevin diffusion , L-mixing , Monte Carlo methods , stochastic approximation

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


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Vol.27 • No. 1 • February 2021
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