Unbiased constrained sampling with Self-Concordant Barrier Hamiltonian Monte Carlo
This addresses the issue of bias in constrained sampling for computational statistics and machine learning, though it is incremental as it builds on existing HMC methods.
The paper tackles the problem of unbiased sampling from constrained distributions on manifolds by introducing Barrier Hamiltonian Monte Carlo (BHMC) with an involution checking step, resulting in two algorithms (c-BHMC and n-BHMC) that eliminate bias and generate reversible Markov chains, as validated on polytope distributions.
In this paper, we propose Barrier Hamiltonian Monte Carlo (BHMC), a version of the HMC algorithm which aims at sampling from a Gibbs distribution $π$ on a manifold $\mathrm{M}$, 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 $\mathrm{M}$ 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 $π$ 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.