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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 anormalization factor $x \mapsto \rme^{-U(x)} / \int_{\rset^d} \rme^{-U(y)} \rmdy$. We analyze a sampling method based on the Euler discretization of theLangevin stochastic differential equations under the assumptions that thepotential $U$ is continuously differentiable, $\nabla U$ is Lipschitz, and $U$is strongly concave. We focus on the case where the gradient of the log-densitycannot be directly computed but unbiased estimates of the gradient frompossibly dependent observations are available. This setting can be seen as acombination of a stochastic approximation (here stochastic gradient) typealgorithms with discretized Langevin dynamics. We obtain an upper bound of theWasserstein-2 distance between the law of the iterates of this algorithm andthe target distribution $\pi$ with constants depending explicitly on theLipschitz and strong convexity constants of the potential and the dimension ofthe space. Finally, under weaker assumptions on $U$ and its gradient but in thepresence of independent observations, we obtain analogous results inWasserstein-2 distance.

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