Identifying the Optimal Integration Time in Hamiltonian Monte Carlo

TL;DR

This paper explores how the geometric principles of Hamiltonian Monte Carlo can be used to determine the optimal integration time, improving the efficiency of the algorithm through theoretical insights and practical implementation.

Contribution

It introduces a geometric framework for identifying the optimal integration time in Hamiltonian Monte Carlo, including a new implementation called Exhaustive Hamiltonian Monte Carlo.

Findings

Optimal integration time can be derived from geometric principles

The new Exhaustive Hamiltonian Monte Carlo improves sampling efficiency

Practical examples demonstrate the benefits of the proposed approach

Abstract

By leveraging the natural geometry of a smooth probabilistic system, Hamiltonian Monte Carlo yields computationally efficient Markov Chain Monte Carlo estimation. At least provided that the algorithm is sufficiently well-tuned. In this paper I show how the geometric foundations of Hamiltonian Monte Carlo implicitly identify the optimal choice of these parameters, especially the integration time. I then consider the practical consequences of these principles in both existing algorithms and a new implementation called \emph{Exhaustive Hamiltonian Monte Carlo} before demonstrating the utility of these ideas in some illustrative examples.