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

Stochastic Gradient Langevin Dynamics (SGLD) is a combination of a Robbins-Monro type algorithm with Langevin dynamics in order to perform data-driven stochastic optimization. In this paper, the SGLD method with fixed step size $\lambda$ is considered in order to sample from a logconcave target distribution $\pi$, known up to a normalisation factor. We assume that unbiased estimates of the gradient from possibly dependent observations are available. It is shown that, for all $\varepsilon>0$, the Wasserstein-$2$ distance of the $n$th iterate of the SGLD algorithm from $\pi$ is dominated by $c_1(\varepsilon)[\lambda^{1/2 - \varepsilon}+e^{-a\lambda n}]$ with appropriate constants $c_1(\varepsilon), a>0$.