Hamiltonian Monte Carlo Swindles
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Proceedings of the Twenty Third International Conference on Artificial Intelligence and Statistics, PMLR 108:37743783, 2020.
Abstract
Hamiltonian Monte Carlo (HMC) is a powerful Markov chain Monte Carlo (MCMC) algorithm for estimating expectations with respect to continuous unnormalized probability distributions. MCMC estimators typically have higher variance than classical Monte Carlo with i.i.d. samples due to autocorrelations; most MCMC research tries to reduce these autocorrelations. In this work, we explore a complementary approach to variance reduction based on two classical Monte Carlo ’swindles’: first, running an auxiliary coupled chain targeting a tractable approximation to the target distribution, and using the auxiliary samples as control variates; and second, generating anticorrelated ("antithetic") samples by running two chains with flipped randomness. Both ideas have been explored previously in the context of Gibbs samplers and randomwalk Metropolis algorithms, but we argue that they are ripe for adaptation to HMC in light of recent coupling results from the HMC theory literature. For many posterior distributions, we find that these swindles generate effective sample sizes orders of magnitude larger than plain HMC, as well as being more efficient than analogous swindles for Metropolisadjusted Langevin algorithm and randomwalk Metropolis.
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