论文标题
公正的约束采样，自信屏障哈密顿蒙特卡洛
Unbiased constrained sampling with Self-Concordant Barrier Hamiltonian Monte Carlo
论文作者
论文摘要
在本文中，我们提出了障碍汉密尔顿蒙特卡洛（BHMC），该版本的HMC算法旨在从gibbs发行$π$上取样，赋予了与HESSIAN METRIC $ \ MATHFRAK {G MATHFRAK {g} $ self affers barriel-carriel-carderier-carderier-carterier-carterier-cordercordant barrier-cordercordercordant。我们的方法取决于包括$ \ mathfrak {g} $的哈密顿动力学。因此，它结合了定义$ \ mathrm {m} $的约束，并能够利用其基础几何形状。但是，相应的哈密顿动力学是通过非可分离的普通微分方程（ODE）来定义的，与欧几里得情况相反。它意味着在HMC对Riemannian流形的现有概括中不可避免的偏见。在本文中，我们提出了一个新的过滤步骤，称为“互动检查步骤”，以解决此问题。该步骤分别以两个版本的BHMC，创造的连续BHMC（C-BHMC）和数值BHMC（N-BHMC）实现。我们的主要结果表明，这两种新算法就$π$产生了可逆的马尔可夫链，并且与以前的实施相比，没有任何偏见。我们的结论得到了数值实验的支持，在这些实验中，我们考虑了在多面体上定义的目标分布。
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.
