标题： Stein 变分梯度下降作为矩匹配 显示英文标题

标题： Stein Variational Gradient Descent as Moment Matching

摘要： Stein变分梯度下降（SVGD）是一种非参数推断算法，它通过演化一组粒子来拟合感兴趣的分布。 我们分析了SVGD的非渐近性质，表明存在一个函数集合，我们称之为Stein匹配集，任何满足SVGD固定点方程的粒子集都能精确估计该集合的期望。 这个集合是将Stein算子应用于SVGD中使用的正定核的特征映射所得的结果。 我们的结果为分析不同核的SVGD性质提供了理论框架，揭示了最优核选择的见解。 特别是，我们证明使用线性核的SVDG在高斯分布上能精确估计均值和方差，而随机傅里叶特征则能实现分布逼近的概率界。 我们的结果为经典的推断问题提供了一个新的视角，即拟合Stein恒等式或求解Stein方程，这可能会激发更高效的算法。 显示英文摘要

摘要： Stein variational gradient descent (SVGD) is a non-parametric inference algorithm that evolves a set of particles to fit a given distribution of interest. We analyze the non-asymptotic properties of SVGD, showing that there exists a set of functions, which we call the Stein matching set, whose expectations are exactly estimated by any set of particles that satisfies the fixed point equation of SVGD. This set is the image of Stein operator applied on the feature maps of the positive definite kernel used in SVGD. Our results provide a theoretical framework for analyzing the properties of SVGD with different kernels, shedding insight into optimal kernel choice. In particular, we show that SVGD with linear kernels yields exact estimation of means and variances on Gaussian distributions, while random Fourier features enable probabilistic bounds for distributional approximation. Our results offer a refreshing view of the classical inference problem as fitting Stein's identity or solving the Stein equation, which may motivate more efficient algorithms.