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We provide a nonasymptotic analysis of the convergence of the stochasticgradient Hamiltonian Monte Carlo (SGHMC) to a target measure in Wasserstein-2distance without assuming log-concavity. Our analysis quantifies keytheoretical properties of the SGHMC as a sampler under local conditions whichsignificantly improves the findings of previous results. In particular, weprove that the Wasserstein-2 distance between the target and the law of theSGHMC is uniformly controlled by the step-size of the algorithm, thereforedemonstrate that the SGHMC can provide high-precision results uniformly in thenumber of iterations. The analysis also allows us to obtain nonasymptoticbounds for nonconvex optimization problems under local conditions and impliesthat the SGHMC, when viewed as a nonconvex optimizer, converges to a globalminimum with the best known rates. We apply our results to obtain nonasymptoticbounds for scalable Bayesian inference and nonasymptotic generalization bounds.

Nonasymptotic analysis of Stochastic Gradient Hamiltonian Monte Carlo under local conditions for nonconvex optimization