Randomized Time Riemannian Manifold Hamiltonian Monte Carlo

TL;DR

This paper introduces a randomized time approach to Hamiltonian Monte Carlo to improve sampling efficiency, especially for constrained problems, demonstrating theoretical properties and practical benefits through numerical experiments.

Contribution

It extends randomized duration techniques to constrained HMC, ensuring stationarity and ergodicity, and shows improved performance in high-dimensional applications.

Findings

Enhanced sampling efficiency in constrained HMC.

Preservation of stationary distribution and ergodicity.

Improved high-dimensional covariance estimation results.

Abstract

Hamiltonian Monte Carlo (HMC) algorithms which combine numerical approximation of Hamiltonian dynamics on finite intervals with stochastic refreshment and Metropolis correction are popular sampling schemes, but it is known that they may suffer from slow convergence in the continuous time limit. A recent paper of Bou-Rabee and Sanz-Serna (Ann. Appl. Prob., 27:2159-2194, 2017) demonstrated that this issue can be addressed by simply randomizing the duration parameter of the Hamiltonian paths. In this article, we use the same idea to enhance the sampling efficiency of a constrained version of HMC, with potential benefits in a variety of application settings. We demonstrate both the conservation of the stationary distribution and the ergodicity of the method. We also compare the performance of various schemes in numerical studies of model problems, including an application to high-dimensional covariance estimation.