Optimal tuning of the hybrid Monte Carlo algorithm

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 π using separable Hamiltonian dynamics with potential - log π. 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 O(1) acceptance probability as the dimension d of the state space tends to ∞, the leapfrog step size h should be scaled as h = l × d-1/4. Therefore, in high dimensions, HMC requires O (d1/4) 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.