Exponential Integration for Hamiltonian Monte Carlo

Wei-Lun Chao,University of Southern California,
Justin Solomon,Princeton University and Stanford University,
Dominik L. Michels,Stanford University,
Fei Sha,University of Southern California.
In Proceedings of the International Conference on Machine Learning, ICML 2015, Pages 1142-1151, Jul. 2015.


author = {Wei-Lun Chao and Justin Solomon and Dominik L.~Michels and Fei Sha},
title = {Exponential Integration for Hamiltonian Monte Carlo},
pages = {1142--1151},
booktitle = {International Conference on Machine Learning – ICML 2015},
year = {2015},
month = jul


We investigate numerical integration of ordinary differential equations (ODEs) for Hamiltonian Monte Carlo (HMC). High-quality integration is crucial for designing efficient and effective proposals for HMC. While the standard method is leapfrog (Stömer-Verlet) integration, we propose the use of an exponential integrator, which is robust to stiff ODEs with highly-oscillatory components. This oscillation is difficult to reproduce using leapfrog integration, even with carefully selected integration parameters and preconditioning. Concretely, we use a Gaussian distribution approximation to segregate stiff components of the ODE. We integrate this term analytically for stability and account for deviation from the approximation using variation of constants. We consider various ways to derive Gaussian approximations and conduct extensive empirical studies applying the proposed "exponential HMC" to several benchmarked learning problems. We compare to state-of-the-art methods for improving leapfrog HMC and demonstrate the advantages of our method in generating many effective samples with high acceptance rates in short running times.

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