## The Annals of Applied Probability

- Ann. Appl. Probab.
- Volume 27, Number 4 (2017), 2159-2194.

### Randomized Hamiltonian Monte Carlo

Nawaf Bou-Rabee and Jesús María Sanz-Serna

#### Abstract

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.

#### Article information

**Source**

Ann. Appl. Probab., Volume 27, Number 4 (2017), 2159-2194.

**Dates**

Received: November 2015

Revised: October 2016

First available in Project Euclid: 30 August 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.aoap/1504080029

**Digital Object Identifier**

doi:10.1214/16-AAP1255

**Mathematical Reviews number (MathSciNet)**

MR3693523

**Zentralblatt MATH identifier**

1373.60129

**Subjects**

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]

**Keywords**

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

#### Citation

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