Unbiased constrained sampling with Self-Concordant Barrier Hamiltonian Monte Carlo

Maxence Noble, Valentin De Bortoli, Alain Durmus

Keywords: Hamiltonian Monte Carlo, Riemannian manifold, self-concordant barrier, constrained sampling

Abstract:

In this paper, we propose Barrier Hamiltonian Monte Carlo (BHMC), a version of the
HMC algorithm which aims at sampling from a Gibbs distribution $\pi$ on a manifold
$\mathsf{M}$, endowed with a Hessian metric $\mathfrak{g}$ derived from a self-concordant
barrier. Our method relies on Hamiltonian
dynamics which comprises $\mathfrak{g}$. Therefore, it incorporates the constraints defining
$\mathsf{M}$ and is able to exploit its underlying geometry. However,
the corresponding Hamiltonian dynamics is defined via non separable Ordinary Differential Equations (ODEs) in contrast to the Euclidean case. It implies unavoidable bias in existing generalization of HMC to Riemannian manifolds. In this paper, we propose a new filter step, called ``involution checking step'', to address this problem. This step is implemented in two versions of BHMC, coined continuous BHMC (c-bHMC) and numerical BHMC (n-BHMC) respectively.
Our main results establish that these two new algorithms generate reversible Markov
chains with respect to $\pi$ and do not suffer from any bias in comparison to previous implementations. Our conclusions are supported by numerical experiments where
we consider target distributions defined on polytopes.

Submission Number: 6670