  We provide a nonasymptotic analysis of the convergence of the stochastic
gradient Hamiltonian Monte Carlo (SGHMC) to a target measure in Wasserstein-2
distance without assuming log-concavity. Our analysis quantifies key
theoretical properties of the SGHMC as a sampler under local conditions which
significantly improves the findings of previous results. In particular, we
prove that the Wasserstein-2 distance between the target and the law of the
SGHMC is uniformly controlled by the step-size of the algorithm, therefore
demonstrate that the SGHMC can provide high-precision results uniformly in the
number of iterations. The analysis also allows us to obtain nonasymptotic
bounds for nonconvex optimization problems under local conditions and implies
that the SGHMC, when viewed as a nonconvex optimizer, converges to a global
minimum with the best known rates. We apply our results to obtain nonasymptotic
bounds for scalable Bayesian inference and nonasymptotic generalization bounds.
