论文标题
抽样的出生死亡动力学：全球收敛，近似及其渐近学
Birth-death dynamics for sampling: Global convergence, approximations and their asymptotics
论文作者
论文摘要
受到对吉布斯（Gibbs）的挑战的挑战，我们研究了连续的出生死亡动力学。我们在以前的工作[51,57]中提高了结果，并提供了较弱的假设，在该假设下，由Kullback-Leibler Divergence控制的出生死亡的概率密度或$χ^2 $^2 $ divergence呈指数收敛至吉布斯均衡度量，其通用率与潜在的障碍无关。为了基于纯出生死亡动力学构建实用的数值采样器，我们考虑了一种相互作用的粒子系统，该系统的灵感来自梯度流结构和经典的Fokker-Planck方程，并依赖于该度量的基于内核的近似值。使用$γ$ - 梯度流的技术，我们表明，在有限的时间间隔上，核的动力学的光滑且有界的正溶液会汇聚到纯生育死亡动力学，因为内核带宽收缩至零。此外，我们提供了对与核化动力学相对应的能量最小化的偏差的定量估计。最后，我们证明了对近核动力学对吉布斯测量的渐近状态的长期渐近结果。
Motivated by the challenge of sampling Gibbs measures with nonconvex potentials, we study a continuum birth-death dynamics. We improve results in previous works [51,57] and provide weaker hypotheses under which the probability density of the birth-death governed by Kullback-Leibler divergence or by $χ^2$ divergence converge exponentially fast to the Gibbs equilibrium measure, with a universal rate that is independent of the potential barrier. To build a practical numerical sampler based on the pure birth-death dynamics, we consider an interacting particle system, which is inspired by the gradient flow structure and the classical Fokker-Planck equation and relies on kernel-based approximations of the measure. Using the technique of $Γ$-convergence of gradient flows, we show that on the torus, smooth and bounded positive solutions of the kernelized dynamics converge on finite time intervals, to the pure birth-death dynamics as the kernel bandwidth shrinks to zero. Moreover we provide quantitative estimates on the bias of minimizers of the energy corresponding to the kernelized dynamics. Finally we prove the long-time asymptotic results on the convergence of the asymptotic states of the kernelized dynamics towards the Gibbs measure.
