Microcanonical Hamiltonian Monte Carlo

Jakob Robnik, G Bruno De Luca, Eva Silverstein, Uros Seljak

Abstract:

We develop Microcanonical Hamiltonian Monte Carlo (MCHMC), a class of models which
follow a fixed energy Hamiltonian dynamics, in contrast to Hamiltonian Monte Carlo
(HMC), which follows canonical distribution with different energy levels. MCHMC tunes
the Hamiltonian function such that the marginal of the uniform distribution on the constant-
energy-surface over the momentum variables gives the desired target distribution. We show
that MCHMC requires occasional energy conserving billiard-like momentum bounces for
ergodicity, analogous to momentum resampling in HMC. We generalize the concept of
bounces to a continuous version with partial direction preserving bounces at every step,
which gives an energy conserving underdamped Langevin-like dynamics with non-Gaussian
noise (MCLMC). MCHMC and MCLMC exhibit favorable scalings with condition number
and dimensionality. We develop an efficient hyperparameter tuning scheme that achieves
high performance and consistently outperforms NUTS HMC on several standard bench-
mark problems, in some cases by more than an order of magnitude.