Geometric integrators and the Hamiltonian Monte Carlo method

TL;DR

This paper reviews the relationship between geometric numerical integrators and the Hamiltonian Monte Carlo method, emphasizing the importance of efficient, volume-preserving, and reversible integrators for high-dimensional sampling.

Contribution

It analyzes how geometric properties of integrators affect HMC efficiency and proposes potential improvements beyond the commonly used velocity Verlet method.

Findings

Velocity Verlet is effective but can be improved.

Geometric properties influence integration error and acceptance rate.

High-dimensional behavior of HMC is discussed.

Abstract

This paper surveys in detail the relations between numerical integration and the Hamiltonian (or hybrid) Monte Carlo method (HMC). Since the computational cost of HMC mainly lies in the numerical integrations, these should be performed as efficiently as possible. However, HMC requires methods that have the geometric properties of being volume-preserving and reversible, and this limits the number of integrators that may be used. On the other hand, these geometric properties have important quantitative implications on the integration error, which in turn have an impact on the acceptance rate of the proposal. While at present the velocity Verlet algorithm is the method of choice for good reasons, we argue that Verlet can be improved upon. We also discuss in detail the behavior of HMC as the dimensionality of the target distribution increases.