A Condition Number for Hamiltonian Monte Carlo
Abstract
Hamiltonian Monte Carlo is a popular sampling technique for smooth target densities. The scale lengths of the target have long been known to influence integration error and sampling efficiency. However, quantitative measures intrinsic to the target have been lacking. In this paper, we restrict attention to the multivariate Gaussian and the leapfrog integrator, and obtain a condition number corresponding to sampling efficiency. This number, based on the spectral and Schatten norms, quantifies the number of leapfrog steps needed to efficiently sample. We demonstrate its utility by using this condition number to analyze HMC preconditioning techniques. We also find the condition number of large inverse Wishart matrices, from which we derive burn-in heuristics.
Cite
@article{arxiv.1905.09813,
title = {A Condition Number for Hamiltonian Monte Carlo},
author = {Ian Langmore and Michael Dikovsky and Scott Geraedts and Peter Norgaard and Rob Von Behren},
journal= {arXiv preprint arXiv:1905.09813},
year = {2020}
}
Comments
Significant changes: (i) Added connection to inverse Wishart ensemble, (ii) added estimation of kappa, (iii) checked and corrected proofs, (iv) re-wrote everything for clarity, (v) added authors