  We consider the geometric ergodicity of the Stochastic Gradient Langevin
Dynamics (SGLD) algorithm under nonconvexity settings. Via the technique of
reflection coupling, we prove the Wasserstein contraction of SGLD when the
target distribution is log-concave only outside some compact set. The time
discretization and the minibatch in SGLD introduce several difficulties when
applying the reflection coupling, which are addressed by a series of careful
estimates of conditional expectations. As a direct corollary, the SGLD with
constant step size has an invariant distribution and we are able to obtain its
geometric ergodicity in terms of $W_1$ distance. The generalization to
non-gradient drifts is also included.
