Magnetic Hamiltonian Monte Carlo With Partial Momentum Refreshment

Magnetic Hamiltonian Monte Carlo (MHMC) has been shown to provide more efficient sampling of the target posterior compared to Hamiltonian Monte Carlo (HMC). It achieves this by utilising a user specified magnetic field and the resultant non-canonical Hamiltonian dynamics. This is important for multi-modal distributions which are common in machine learning. In generating each sample in MHMC and HMC, the auxiliary momentum variable is fully regenerated from a Gaussian distribution. Partially updating the momentum has previously been employed in HMC to improve sampling behaviour. It has also been used in the context of sampling using integrator dependent shadow Hamiltonian Monte Carlo methods. In this work, we combine the sampling benefits of non-canonical Hamiltonian dynamics offered by MHMC with partial momentum refreshment to create the Magnetic Hamiltonian Monte Carlo with Partial Momentum Refreshment (PMHMC) algorithm. Numerical experiments across various target posterior distributions show that the proposed method outperforms HMC, MHMC and HMC with partial momentum refreshment across all the metrics considered.