  Stochastic gradient Hamiltonian Monte Carlo (SGHMC) is a variant of
stochastic gradient with momentum where a controlled and properly scaled
Gaussian noise is added to the stochastic gradients to steer the iterates
towards a global minimum. Many works reported its empirical success in practice
for solving stochastic non-convex optimization problems, in particular it has
been observed to outperform overdamped Langevin Monte Carlo-based methods such
as stochastic gradient Langevin dynamics (SGLD) in many applications. Although
asymptotic global convergence properties of SGHMC are well known, its
finite-time performance is not well-understood. In this work, we study two
variants of SGHMC based on two alternative discretizations of the underdamped
Langevin diffusion. We provide finite-time performance bounds for the global
convergence of both SGHMC variants for solving stochastic non-convex
optimization problems with explicit constants. Our results lead to
non-asymptotic guarantees for both population and empirical risk minimization
problems. For a fixed target accuracy level, on a class of non-convex problems,
we obtain complexity bounds for SGHMC that can be tighter than those for SGLD.
These results show that acceleration with momentum is possible in the context
of global non-convex optimization.
