On stochastic gradient Langevin dynamics with dependent data streams in the logconcave case

• Main Authors: Barkhagen, M, Chau, N. H, Moulines, É, Rásonyi, M, Sabanis, S, Zhang, Y
• Format: Journal Article
• Language: English
• Published: 06.12.2018
• Subjects: Mathematics - Probability; Mathematics - Statistics Theory; Statistics - Machine Learning; Statistics - Theory
• DOI: 10.48550/arxiv.1812.02709

We study the problem of sampling from a probability distribution $\pi$ on $\rset^d$ which has a density \wrt\ the Lebesgue measure known up to a normalization factor $x \mapsto \rme^{-U(x)} / \int_{\rset^d} \rme^{-U(y)} \rmd y$. 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.