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

Nonasymptotic estimates for Stochastic Gradient Langevin Dynamics under local conditions in nonconvex optimization