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Hamiltonian Monte Carlo (HMC) is a popular version of Markov chain Monte Carlo (MCMC), and variants of the algorithm drive the popular Bayesian computation software STAN. One reason for HMC's popularity is the fact that it scales to high-dimensional problems much more effectively than competitor MCMC algorithms such as Metropolis-Hastings and Langevin. In this talk I will discuss two recent results on the performance of HMC. The first estimates the efficiency of HMC on "nice" high-dimensional unimodal targets, showing that HMC has much better performance than its competitor algorithms in this regime. The second gives precise estimates of the efficiency of HMC on multimodal targets; these imply that the standard HMC algorithm often has worse performance than its competitors in this regime. Finally, I'll discuss some limits of these results and future research questions that they suggest.
This is joint work with Oren Mangoubi.
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