Stein Variational Gradient Descent as Moment Matching
This work offers a theoretical framework for understanding SVGD's properties, which is incremental but useful for researchers in non-parametric inference and kernel methods.
The paper analyzes Stein variational gradient descent (SVGD) by showing that its fixed points exactly match expectations of a set of functions derived from the kernel, providing theoretical insights into kernel choice and proving exact estimation for Gaussian distributions with linear kernels.
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.