Preconditioning Hamiltonian Monte Carlo by minimizing Fisher Divergence

Abstract: Although Hamiltonian Monte Carlo (HMC) scales as O(d^1/4) in dimension, there is a large constant factor determined by the curvature of the target density. This constant factor can be reduced in most cases through preconditioning, the state of the art for which uses diagonal or dense penalized maximum likelihood estimation of (co)variance based on a sample of warmup draws. These estimates converge slowly in the diagonal case and scale poorly when expanded to the dense case. We propose a more effective estimator based on minimizing the sample Fisher divergence from a linearly transformed density to a standard normal distribution. We present this estimator in three forms, (a) diagonal, (b) dense, and (c) low-rank plus diagonal. Using a collection of 114 models from posteriordb, we demonstrate that the diagonal minimizer of Fisher divergence outperforms the industry-standard variance-based diagonal estimators used by Stan and PyMC by a median factor of 1.3. The low-rank plus diagonal minimizer of the Fisher divergence outperforms Stan and PyMC's diagonal estimators by a median factor of 4.