Unbiased Hamiltonian Monte Carlo with couplings

Abstract

We propose a methodology to parallelize Hamiltonian Monte Carlo estimators. Our approach constructs a pair of Hamiltonian Monte Carlo chains that are coupled in such a way that they meet exactly after some random number of iterations. These chains can then be combined so that resulting estimators are unbiased. This allows us to produce independent replicates in parallel and average them to obtain estimators that are consistent in the limit of the number of replicates, instead of the usual limit of the number of Markov chain iterations. We investigate the scalability of our coupling in high dimensions on a toy example. The choice of algorithmic parameters and the efficiency of our proposed methodology are then illustrated on a logistic regression with 300 covariates, and a log-Gaussian Cox point processes model with low to fine grained discretizations.

Unbiased Markov Chain Monte Carlo Methods with Couplings

“…(). We believe that the authors achieved a major breakthrough in this field by proposing a general framework for parallelizing computations, applicable to many MCMC algorithms (Heng and Jacob, ; Middleton et al ., ).…”

“…Thanks to its potential for parallelization, the framework proposed can facilitate a consideration of MCMC kernels that might be too expensive for serial implementation. For instance, one can improve MH‐within‐Gibbs samplers by performing more MH steps per component update, Hamiltonian Monte Carlo sampling by using smaller step sizes in the numerical integrator (Heng and Jacob, ) and particle MCMC sampling by using more particles in the particle filters (Andrieu et al ., ; Jacob et al ., ). We expect the optimal tuning of MCMC kernels to be different in the proposed framework from when used marginally.…”

“…Fig. 14 displays the variance reduction that is achieved in the 302‐dimensional logistic regression example of Heng and Jacob (). These results are encouraging, but more work is required to develop an understanding of gradient‐based control variates for unbiased MCMC sampling.…”

“…Going further in the use of gradient information, the Hamiltonian or hybrid Monte Carlo algorithm (Duane et al ., ; Neal, ) is a popular MCMC algorithm for large dimensional targets. In Heng and Jacob (), the framework of the present paper is applied to pairs of Hamiltonian Monte Carlo chains, with a focus of the verification of assumptions 1–3 in that context. Such couplings were analysed in detail in Mangoubi and Smith () and Bou‐Rabee et al .…”

“…() to obtain convergence rates for the underlying chains. We refer to Heng and Jacob () for more details and provide for completeness some experiments on the normal distribution target that was described above in the on‐line supplementary materials.…”