Implicit Hamiltonian Monte Carlo for Sampling Multiscale Distributions

Abstract

Hamiltonian Monte Carlo (HMC) has been widely adopted in the statistics community because of its ability to sample high-dimensional distributions much more efficiently than other Metropolis-based methods. Despite this, HMC often performs sub-optimally on distributions with high correlations or marginal variances on multiple scales because the resulting stiffness forces the leapfrog integrator in HMC to take an unreasonably small stepsize. We provide intuition as well as a formal analysis showing how these multiscale distributions limit the stepsize of leapfrog and we show how the implicit midpoint method can be used, together with Newton-Krylov iteration, to circumvent this limitation and achieve major efficiency gains. Furthermore, we offer practical guidelines for when to choose between implicit midpoint and leapfrog and what stepsize to use for each method, depending on the distribution being sampled. Unlike previous modifications to HMC, our method is generally applicable to highly non-Gaussian distributions exhibiting multiple scales. We illustrate how our method can provide a dramatic speedup over leapfrog in the context of the No-U-Turn sampler (NUTS) applied to several examples.

Delayed rejection Hamiltonian Monte Carlo for sampling multiscale distributions

“…Reducing the step size is not the only way of constructing delayed proposals. Another approach would be to replace the leapfrog integrator altogether for retries, for example with an implicit symplectic integrator (Pourzanjani and Petzold, 2019). Such integrators may additionally be able to deal with stiffness arising from high correlation.…”

“…The leapfrog integrator itself can be extended to higher orders (Creutz and Gocksch, 1989;Yoshida, 1990). Neal points out that a modified Euler step is valid (Neal, 2011), and recent works have proposed using other maps, such as implicit integrators (Pourzanjani and Petzold, 2019;Brofos and Lederman, 2021a) for multiscale distributions or generalizing HMC with neural networks (Levy et al, 2017). However, a lesson of the above is that approximating Hamiltonian dynamics is not necessary to have the correct invariant pdf; it is merely a convenient way to propose long-distance moves with high acceptance rates.…”

“…An alternative way to deal with high curvature is to approximate the Hamiltonian flow with an implicit symplectic integrator, which is able to naturally adjust stepping in different regions of phase space (Pourzanjani and Petzold, 2019;Brofos and Lederman, 2021b). Even simple implicit integration schemes like implicit midpoint are costly and present an algorithmic challenge for efficient and stable line search.…”