Theoretical guarantees for stochastic gradient Langevin dynamics

We prove asymptotic bias bounds for stochastic gradient Langevin dynamics in Wasserstein distance of order two. We assume that the negative log-density is strongly convex with a Lipschitz gradient, and that the stochastic gradient estimator is unbiased with an error satisfying a mean-square Lipschitz condition. The bounds are of order $h$ under a fourth moment assumption on the stochastic gradient error and of order $h^{1\mathrm{/}2}$ under only a second moment assumption, where $h$ is the stepsize. A spiked-noise example shows that a second moment assumption alone is insufficient for a bound of order $h$ that is uniform over noise distributions with a fixed variance.