Stein variational gradient descent (SVGD) refers to a class of methods for Bayesian inference based on interacting particle systems. In this paper, we consider the originally proposed deterministic dynamics as well as a stochastic variant, each of which represent one of the two main paradigms in Bayesian computational statistics: variational inference and Markov chain Monte Carlo. As it turns out, these are tightly linked through a correspondence between gradient flow structures and large-deviation principles rooted in statistical physics. To expose this relationship, we develop the cotangent space construction for the Stein geometry, prove its basic properties, and determine the large-deviation functional governing the many-particle limit for the empirical measure. Moreover, we identify the Stein-Fisher information (or kernelised Stein discrepancy) as its leading order contribution in the long-time and many-particle regime in the sense of $\Gamma$-convergence, shedding some light on the finite-particle properties of SVGD. Finally, we establish a comparison principle between the Stein-Fisher information and RKHS-norms that might be of independent interest.

中文翻译：

---

斯坦因变异梯度下降：多粒子和长期渐近性

斯坦因变量梯度下降法（SVGD）是指基于相互作用粒子系统进行贝叶斯推理的一类方法。在本文中，我们考虑了最初提出的确定性动力学以及随机变量，它们分别表示贝叶斯计算统计中的两个主要范式之一：变分推理和马尔可夫链蒙特卡洛。事实证明，这些是通过梯度流结构与扎根于统计物理学的大偏差原理之间的对应关系紧密联系在一起的。为了揭示这种关系，我们开发了斯坦几何的余切空间构造，证明了其基本性质，并确定了控制多粒子极限的大偏差函数作为经验度量。而且，我们将Stein-Fisher信息（或Stein-Fisher差异）确定为它在$ \ Gamma $收敛意义上在长期多粒子系统中的主导作用，这为SVGD的有限粒子性质提供了一些启示。最后，我们在Stein-Fisher信息和RKHS规范之间建立了比较原则，这些原则可能具有独立利益。