2015/11/30 by Nawaf Bou-Rabee, Jesus Maria Sanz-Serna · 4 citations
Mathematics · #math.PR #msc:60J25 #msc:62D05 #msc:60H30 #msc:60H35 #msc:37A50
paper · pdf · doi:10.1214/16-aap1255
published as Ann. Appl. Probab. Volume 27, Number 4 (2017), 2159-2194 · 40 pages, to be published in Annals of Applied Probability
arxiv created 2016/10/27 · arxiv updated 2017/09/08
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.