On $L^q$ Convergence of the Hamiltonian Monte Carlo

TL;DR

This paper proves that Hamiltonian Monte Carlo algorithms converge in $L_q$ sense to the target distribution under mild conditions, with different modes of convergence depending on the value of q.

Contribution

It establishes $L_q$ convergence results for Hamiltonian Monte Carlo, a significant step in understanding its theoretical properties.

Findings

Strong $L_q$ convergence for $2 \,\leq\, q < \infty$.

Weak $L_q$ convergence for $1 < q < 2$.

Convergence holds under mild conditions on Hamiltonian motion.

Abstract

We establish $L_q$ convergence for Hamiltonian Monte Carlo algorithms. More specifically, under mild conditions for the associated Hamiltonian motion, we show that the outputs of the algorithms converge (strongly for $2\le q<\infty$ and weakly for $1<q<2$) to the desired target distribution.