Coupling and Convergence for Hamiltonian Monte Carlo
This provides theoretical guarantees for HMC convergence in multimodal distributions, addressing a bottleneck in Bayesian inference and computational statistics.
The authors tackled the problem of proving convergence rates for Hamiltonian Monte Carlo (HMC) by developing a new coupling approach to show contractivity in a Kantorovich distance, resulting in explicit bounds for steps needed to approximate stationary distributions, with HMC overcoming diffusive behavior when dynamics duration is adjusted.
Based on a new coupling approach, we prove that the transition step of the Hamiltonian Monte Carlo algorithm is contractive w.r.t. a carefully designed Kantorovich (L1 Wasserstein) distance. The lower bound for the contraction rate is explicit. Global convexity of the potential is not required, and thus multimodal target distributions are included. Explicit quantitative bounds for the number of steps required to approximate the stationary distribution up to a given error are a direct consequence of contractivity. These bounds show that HMC can overcome diffusive behaviour if the duration of the Hamiltonian dynamics is adjusted appropriately.