Reflection coupling for unadjusted generalized Hamiltonian Monte Carlo in the nonconvex stochastic gradient case

Abstract

Contraction in Wasserstein 1-distance with explicit rates is established for generalized Hamiltonian Monte Carlo with stochastic gradients under possibly nonconvex conditions. The algorithms considered include splitting schemes of kinetic Langevin diffusion, which are commonly used in molecular dynamics simulations. To accommodate the degenerate noise structure corresponding to inertia existing in the chain, a characteristically discrete-in-time coupling and contraction proof is devised. As consequence, quantitative Gaussian concentration bounds are provided for empirical averages. Convergence in Wasserstein 2-distance and total variation are also given, together with numerical bias estimates.