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.