  In this paper, we are concerned with a non-asymptotic analysis of sampling
algorithms used in nonconvex optimization. In particular, we obtain
non-asymptotic estimates in Wasserstein-1 and Wasserstein-2 distances for a
popular class of algorithms called Stochastic Gradient Langevin Dynamics
(SGLD). In addition, the aforementioned Wasserstein-2 convergence result can be
applied to establish a non-asymptotic error bound for the expected excess risk.
Crucially, these results are obtained under a local Lipschitz condition and a
local dissipativity condition where we remove the uniform dependence in the
data stream. We illustrate the importance of this relaxation by presenting
examples from variational inference and from index tracking optimization.
