Abstract
Ann. Appl. Probab. Volume 27, Number 4 (2017), 2159-2194 Tuning the durations of the Hamiltonian flow in Hamiltonian Monte Carlo (also
called Hybrid Monte Carlo) (HMC) involves a tradeoff between computational cost
and sampling quality, 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 underdamped Langevin equations. Moreover, in the context of
a multi-dimensional Gaussian distribution, we prove that the sampling
efficiency of RHMC, unlike that of constant duration HMC, behaves in a regular
way. This regularity is also verified numerically in non-Gaussian target
distributions. Finally we suggest variants of RHMC for which the time step size
is not required to be small.