Unbiased constrained sampling with Self-Concordant Barrier Hamiltonian Monte Carlo
Main Conference Track
Authors: Maxence Noble, Valentin De Bortoli, Alain Durmus

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.