Evaluating the Implicit Midpoint Integrator for Riemannian Hamiltonian Monte Carlo
Abstract
Riemannian manifold Hamiltonian Monte Carlo is traditionally carried out using the generalized leapfrog integrator. However, this integrator is not the only choice and other integrators yielding valid Markov chain transition operators may be considered. In this work, we examine the implicit midpoint integrator as an alternative to the generalized leapfrog integrator. We discuss advantages and disadvantages of the implicit midpoint integrator for Hamiltonian Monte Carlo, its theoretical properties, and an empirical assessment of the critical attributes of such an integrator for Hamiltonian Monte Carlo: energy conservation, volume preservation, and reversibility. Empirically, we find that while leapfrog iterations are faster, the implicit midpoint integrator has better energy conservation, leading to higher acceptance rates, as well as better conservation of volume and better reversibility, arguably yielding a more accurate sampling procedure.
BibTeX
@InProceedings{pmlr-v139-brofos21a,
title = {Evaluating the Implicit Midpoint Integrator for Riemannian Hamiltonian Monte Carlo},
author = {Brofos, James and Lederman, Roy R},
booktitle = {Proceedings of the 38th International Conference on Machine Learning},
pages = {1072--1081},
year = {2021},
editor = {Meila, Marina and Zhang, Tong},
volume = {139},
series = {Proceedings of Machine Learning Research},
month = {18--24 Jul},
publisher = {PMLR},
pdf = {http://proceedings.mlr.press/v139/brofos21a/brofos21a.pdf},
url = {https://proceedings.mlr.press/v139/brofos21a.html},
abstract = {Riemannian manifold Hamiltonian Monte Carlo is traditionally carried out using the generalized leapfrog integrator. However, this integrator is not the only choice and other integrators yielding valid Markov chain transition operators may be considered. In this work, we examine the implicit midpoint integrator as an alternative to the generalized leapfrog integrator. We discuss advantages and disadvantages of the implicit midpoint integrator for Hamiltonian Monte Carlo, its theoretical properties, and an empirical assessment of the critical attributes of such an integrator for Hamiltonian Monte Carlo: energy conservation, volume preservation, and reversibility. Empirically, we find that while leapfrog iterations are faster, the implicit midpoint integrator has better energy conservation, leading to higher acceptance rates, as well as better conservation of volume and better reversibility, arguably yielding a more accurate sampling procedure.}
}