In Bayesian inference, the posterior distributions are difficult to obtain analytically for complex models such as neural networks. Variational inference usually uses a parametric distribution to approximate, from which we can easily draw samples. Recently discrete approximation by particles has attracted attention because of its expressive ability. An example is Stein variational gradient descent (SVGD), which iteratively optimizes particles. Although SVGD has been shown to be computationally efficient empirically, its theoretical properties have not been clarified yet and no finite sample bound of a convergence rate is known. Another example is Stein points (SP), which minimizes kernelized Stein discrepancy directly. The finite sample bound of SP is $\mathcal{O}(\sqrt{\log{N}/N})$ for $N$ particles, which is computationally inefficient empirically, especially in high-dimensional problems. In this paper, we propose a novel method named \emph{Frank-Wolfe Stein sampling}, which minimizes the maximum mean discrepancy in a greedy way. Our method is computationally efficient empirically and theoretically achieves a faster convergence rate, $\mathcal{O}(e^{-N})$. Numerical experiments show the superiority of our method.

本文提出了一种名为Frank-Wolfe算法最大平均差异度量（MMD-FW）的新方法，通过贪婪算法以最小化的方式最小化MMD，该方法在计算上具有实用的效率，并且我们在有限的维度中显示出其有限的样本收敛之后的线性顺序。

通过贪心粒子优化进行贝叶斯后验逼近