论文标题
riemannian hamiltonian蒙特卡洛与数值集成符的条件单无关的收敛速率
Condition-number-independent convergence rate of Riemannian Hamiltonian Monte Carlo with numerical integrators
论文作者
论文摘要
我们研究了从凸体$ \ Mathcal $ \ Mathcal {m} \ subset \ subbb {r}^{n} $中，研究以$ e^{ - f（x）} $的形式从分布中抽样的融合率。我们表明，对于带有$ m $约束的多门户上的$ e^{ - α^{\ top} x} $的发行量，一个常用的集成剂家族的收敛速率独立于$ \ weft \ weft \ welet \vertα\ right \ right \ vert \ vert _ {2} $和polytoper的几何形状。特别是，隐式中点方法（IMM）和广义的leapfrog方法（LM）的混合时间为$ \ widetilde {o} \ left（mn^{3} \ right）$，以实现$ε$ $ε$的总变量距离。这些保证基于对歧管和集成符的参数的形式$ e^{ - f（x）} $密度的收敛速率的一般限制。我们的理论保证补充了[KLSV22]的经验结果，该结果表明，具有IMM的RHMC可以在实践中有效地在非常高的尺寸中采样不良条件，非平滑和约束分布。
We study the convergence rate of discretized Riemannian Hamiltonian Monte Carlo on sampling from distributions in the form of $e^{-f(x)}$ on a convex body $\mathcal{M}\subset\mathbb{R}^{n}$. We show that for distributions in the form of $e^{-α^{\top}x}$ on a polytope with $m$ constraints, the convergence rate of a family of commonly-used integrators is independent of $\left\Vert α\right\Vert _{2}$ and the geometry of the polytope. In particular, the implicit midpoint method (IMM) and the generalized Leapfrog method (LM) have a mixing time of $\widetilde{O}\left(mn^{3}\right)$ to achieve $ε$ total variation distance to the target distribution. These guarantees are based on a general bound on the convergence rate for densities of the form $e^{-f(x)}$ in terms of parameters of the manifold and the integrator. Our theoretical guarantee complements the empirical results of [KLSV22], which shows that RHMC with IMM can sample ill-conditioned, non-smooth and constrained distributions in very high dimension efficiently in practice.
