Particle-optimization sampling (POS) is a recently developed technique to generate high-quality samples from a target distribution by iteratively updating a set of interactive particles. A representative algorithm is the Stein variational gradient descent (SVGD). Though obtaining significant empirical success, the {\em non-asymptotic} convergence behavior of SVGD remains unknown. In this paper, we generalize POS to a stochasticity setting by injecting random noise in particle updates, called stochastic particle-optimization sampling (SPOS). Standard SVGD can be regarded as a special case of our framework. Notably, for the first time, we develop non-asymptotic convergence theory for the SPOS framework (which includes SVGD), characterizing the bias of a sample approximation w.r.t. the numbers of particles and iterations under both convex- and noncovex-energy-function settings. Remarkably, we provide theoretical understand of a pitfall of SVGD that can be avoided in the proposed SPOS framework, i.e., particles tent to collapse to a local mode in SVGD under some particular conditions. Our theory is based on the analysis of nonlinear stochastic differential equations, which serves as an extension and a complemented development to the asymptotic convergence theory for SVGD such as [1].

该论文探讨了基于粒子最优化的采样技术和 Stein 变分梯度下降算法的理论缺陷，提出了注入随机噪声的 SPOS 算法，并第一次为相关的 SPOS 框架（涉及 SVGD）开发了非渐近收敛理论，以粒子数和迭代次数的 1-Wasserstein 距离为特征，表征算法的收敛性。实验证明，与 SVGD 相同数量的更新次数（不太大）对于每个粒子，采用更多的粒子并不一定会更好地逼近目标分布，这种现象也被观察到并在合成数据实验中得到了验证。

随机粒子优化采样与非渐进收敛理论