Recycling intermediate steps to improve Hamiltonian Monte Carlo

Hamiltonian Monte Carlo (HMC) and related algorithms have become routinely used in Bayesian computation with their utilities highlighted by the probabilistic programming software packages Stan and PyMC. In this article, we present a simple and provably accurate method to improve the efficiency of HMC and related algorithms with essentially no extra computational cost. This is achieved by recycling the intermediate leap-frog steps used in approximating the trajectories of Hamiltonian dynamics. Standard algorithms use only the final step, and wastefully discard all the intermediate steps. Compared to the existing alternative methods for utilizing the intermediate steps, our algorithm is simpler to apply in practice and requires little programming effort beyond the usual implementations of HMC and related algorithms. Furthermore, our algorithm applies straightforwardly to No-U-Turn-Sampler, arguably the most popular variant of HMC. We show that our recycling algorithm leads to substantial gains in computational efficiency in a variety of experiments.