Probability RANDOMIZED HAMILTONIAN MONTE CARLO

Tuning the duration of the Hamiltonian flow in Hamiltonian Monte Carlo (also called Hybrid Monte Carlo) (HMC) involves a tradeoff between computational and sampling efficiency, which is typically challenging to resolve in a satisfactory way. In this article we present and analyze a randomized HMC method (RHMC), in which these durations are i.i.d. exponential random variables whose mean is a free parameter. We focus on the small time step size limit, where the algorithm is rejection-free and the computational cost is proportional to the mean duration. In this limit, we prove that RHMC is geometrically ergodic under the same conditions that imply geometric ergodicity of the solution to Langevin equations. Moreover, in the context of a multi-dimensional Gaussian distribution, we prove that the sampling efficiency of RHMC increases monotonically with the mean duration before plateauing. As a corollary, the sampling efficiency can be made arbitrarily close to optimal by choosing the mean duration sufficiently large. We numerically verify this monotonic/plateauing behavior in non-Gaussian target distributions.