Antithetic Magnetic and Shadow Hamiltonian Monte Carlo

Abstract

Hamiltonian Monte Carlo is a Markov Chain Monte Carlo method that has been widely applied to numerous posterior inference problems within the machine learning literature. Markov Chain Monte Carlo estimators have higher variance than classical Monte Carlo estimators due to autocorrelations present between the generated samples. In this work we present three new methods for tackling the high variance problem in Hamiltonian Monte Carlo based estimators: 1) We combine antithetic and importance sampling techniques where the importance sampler is based on sampling from a modified or shadow Hamiltonian using Separable Shadow Hamiltonian Hybrid Monte Carlo, 2) We present the antithetic Magnetic Hamiltonian Monte Carlo algorithm that is based on performing antithetic sampling on the Magnetic Hamiltonian Monte Carlo algorithm and 3) We propose the antithetic Magnetic Momentum Hamiltonian Monte Carlo algorithm based on performing antithetic sampling on the Magnetic Momentum Hamiltonian Monte Carlo method. We find that the antithetic Separable Shadow Hamiltonian Hybrid Monte Carlo and antithetic Magnetic Momentum Hamiltonian Monte Carlo algorithms produce effective sample sizes that are higher than antithetic Hamiltonian Monte Carlo on all the benchmark datasets. We further find that antithetic Separable Shadow Hamiltonian Hybrid Monte Carlo and antithetic Magnetic Hamiltonian Monte Carlo produce higher effective sample sizes normalised by execution time in higher dimensions than antithetic Hamiltonian Monte Carlo. In addition, the antithetic versions of all the algorithms have higher effective sample sizes than their non-antithetic counterparts, indicating the usefulness of adding antithetic sampling to Markov Chain Monte Carlo algorithms. The methods are assessed on benchmark datasets using Bayesian logistic regression and Bayesian neural network models.INDEX TERMS Antithetic sampling, bayesian logistic regression, bayesian neural network, hamiltonian monte carlo, machine learning, magnetic hamiltonian monte carlo, shadow hamiltonian, variance reduction

Quantum-Inspired Magnetic Hamiltonian Monte Carlo

“…Magnetic Hamiltonian Monte Carlo (MHMC) is a special case of non-canonical HMC corresponding to motion of a particle in a magnetic field [ 6 , 27 ], which aims to improve the exploration of the target distribution [ 6 ]. This magnetic field adds an extra degree of freedom and can be tuned to attain significantly better sampling performance compared to HMC with very little extra computational cost [ 4 , 6 , 26 ].…”

“…MHMC utilises the same Hamiltonian as in HMC, but exploits non-canonical Hamiltonian dynamics where the canonical matrix now has a non-zero element on the diagonal [ 4 , 6 ]. The MHMC dynamics are: where G is the magnetic field.…”

“…The MHMC dynamics are: where G is the magnetic field. The update equations for the leapfrog-like integration scheme in MHMC are given as [ 6 ]: Eq (7) shows that MHMC only differs from HMC dynamics through the presence of a non-zero G [ 4 , 6 ]. MHMC and HMC have exactly the same dynamics when the magnetic component is absent.…”

“…A recent extension to HMC is MHMC which adds a magnetic field to HMC [ 6 ]. This results in faster convergence to the correct target density, and much lower auto-correlations between the generated samples when compared to HMC [ 4 , 6 , 26 ]. When the magnetic component of MHMC is absent, MHMC and HMC have the same dynamics [ 4 , 6 , 26 ].…”

“…This results in faster convergence to the correct target density, and much lower auto-correlations between the generated samples when compared to HMC [ 4 , 6 , 26 ]. When the magnetic component of MHMC is absent, MHMC and HMC have the same dynamics [ 4 , 6 , 26 ]. MHMC has similar execution times to HMC, which illustrates the close relationship between HMC and MHMC.…”