Non-Canonical Hamiltonian Monte Carlo
This work addresses a foundational issue in Markov chain Monte Carlo methods for researchers in statistics and machine learning, offering incremental improvements through novel structures like magnetic momentum.
The paper tackles the problem of Hamiltonian Monte Carlo (HMC) being limited to canonical symplectic structures by developing a framework for HMC using non-canonical structures, and experimental results show sampling advantages such as improved performance.
Hamiltonian Monte Carlo is typically based on the assumption of an underlying canonical symplectic structure. Numerical integrators designed for the canonical structure are incompatible with motion generated by non-canonical dynamics. These non-canonical dynamics, motivated by examples in physics and symplectic geometry, correspond to techniques such as preconditioning which are routinely used to improve algorithmic performance. Indeed, recently, a special case of non-canonical structure, magnetic Hamiltonian Monte Carlo, was demonstrated to provide advantageous sampling properties. We present a framework for Hamiltonian Monte Carlo using non-canonical symplectic structures. Our experimental results demonstrate sampling advantages associated to Hamiltonian Monte Carlo with non-canonical structure. To summarize our contributions: (i) we develop non-canonical HMC from foundations in symplectic geomtry; (ii) we construct an HMC procedure using implicit integration that satisfies the detailed balance; (iii) we propose to accelerate the sampling using an {\em approximate} explicit methodology; (iv) we study two novel, randomly-generated non-canonical structures: magnetic momentum and the coupled magnet structure, with implicit and explicit integration.