The Annals of Applied Probability
- Ann. Appl. Probab.
- Volume 27, Number 4 (2017), 2159-2194.
Randomized Hamiltonian Monte Carlo
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 multidimensional 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.
Ann. Appl. Probab., Volume 27, Number 4 (2017), 2159-2194.
Received: November 2015
Revised: October 2016
First available in Project Euclid: 30 August 2017
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 60J25: Continuous-time Markov processes on general state spaces
Secondary: 62D05: Sampling theory, sample surveys 60J25: Continuous-time Markov processes on general state spaces 60H30: Applications of stochastic analysis (to PDE, etc.) 37A50: Relations with probability theory and stochastic processes [See also 60Fxx and 60G10]
Bou-Rabee, Nawaf; Sanz-Serna, Jesús María. Randomized Hamiltonian Monte Carlo. Ann. Appl. Probab. 27 (2017), no. 4, 2159--2194. doi:10.1214/16-AAP1255. https://projecteuclid.org/euclid.aoap/1504080029