Optimal tuning of the hybrid Monte Carlo algorithm

Abstract

We investigate the properties of the hybrid Monte Carlo algorithm (HMC) in high dimensions. HMC develops a Markov chain reversible with respect to a given target distribution $\Pi$ using separable Hamiltonian dynamics with potential $-\log\Pi$. The additional momentum variables are chosen at random from the Boltzmann distribution, and the continuous-time Hamiltonian dynamics are then discretised using the leapfrog scheme. The induced bias is removed via a Metropolis–Hastings accept/reject rule. In the simplified scenario of independent, identically distributed components, we prove that, to obtain an $\mathcal{O}(1)$ acceptance probability as the dimension $d$ of the state space tends to $\infty$, the leapfrog step size $h$ should be scaled as $h=l\times d^{-1/4}$. Therefore, in high dimensions, HMC requires $\mathcal{O}(d^{1/4})$ steps to traverse the state space. We also identify analytically the asymptotically optimal acceptance probability, which turns out to be $0.651$ (to three decimal places). This value optimally balances the cost of generating a proposal, which decreases as $l$ increases (because fewer steps are required to reach the desired final integration time), against the cost related to the average number of proposals required to obtain acceptance, which increases as $l$ increases.