Convergence of unadjusted Hamiltonian Monte Carlo for mean-field models

Abstract

We present dimension-free convergence and discretization error bounds for the unadjusted Hamiltonian Monte Carlo algorithm applied to high-dimensional probability distributions of mean-field type. These bounds require the discretization step to be sufficiently small, but do not require strong convexity of either the unary or pairwise potential terms present in the mean-field model. To handle high dimensionality, our proof uses a particlewise coupling that is contractive in a complementary particlewise metric.

arXiv comment38 pages, 4 figures

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• An entropic approach for Hamiltonian Monte Carlo: the idealized case
• Randomized Runge-Kutta-Nyström Methods for Unadjusted Hamiltonian and Kinetic Langevin Monte Carlo