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August 2017 Randomized Hamiltonian Monte Carlo
Nawaf Bou-Rabee, Jesús María Sanz-Serna
Ann. Appl. Probab. 27(4): 2159-2194 (August 2017). DOI: 10.1214/16-AAP1255


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.


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Nawaf Bou-Rabee. Jesús María Sanz-Serna. "Randomized Hamiltonian Monte Carlo." Ann. Appl. Probab. 27 (4) 2159 - 2194, August 2017.


Received: 1 November 2015; Revised: 1 October 2016; Published: August 2017
First available in Project Euclid: 30 August 2017

zbMATH: 1373.60129
MathSciNet: MR3693523
Digital Object Identifier: 10.1214/16-AAP1255

Primary: 60J25
Secondary: 37A50 , 60H30 , 60J25 , 62D05

Keywords: equilibrium mean squared displacement , geometric ergodicity , Hamiltonian Monte Carlo , hybrid Monte Carlo , integrated autocorrelation time , Lyapunov functions , Markov chain Monte Carlo , Randomization

Rights: Copyright © 2017 Institute of Mathematical Statistics


Vol.27 • No. 4 • August 2017
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