Gradient flow in the 2-Wasserstein space is widely used to optimize functionals over probability distributions and is typically implemented using an interacting particle system with $n$ particles. Analyzing these algorithms requires showing (a) that the finite-particle system converges and/or (b) that the resultant empirical distribution of the particles closely approximates the optimal distribution (i.e., propagation of chaos). However, establishing efficient sufficient conditions can be challenging, as the finite particle system may produce heavily dependent random variables. In this work, we study the virtual particle stochastic approximation, originally introduced for Stein Variational Gradient Descent. This method can be viewed as a form of stochastic gradient descent in the Wasserstein space and can be implemented efficiently. In popular settings, we demonstrate that our algorithm's output converges to the optimal distribution under conditions similar to those for the infinite particle limit, and it produces i.i.d. samples without the need to explicitly establish propagation of chaos bounds.

翻译：2-Wasserstein空间中的梯度流被广泛用于优化概率分布上的泛函，通常通过具有$n$个粒子的相互作用粒子系统实现。分析这些算法需要证明：(a) 有限粒子系统收敛，和/或(b) 粒子生成的经验分布能紧密逼近最优分布（即混沌传播）。然而，建立高效充分条件可能具有挑战性，因为有限粒子系统可能产生高度依赖的随机变量。在本工作中，我们研究了最初为Stein变分梯度下降引入的虚拟粒子随机逼近方法。该方法可视为Wasserstein空间中的一种随机梯度下降形式，并能高效实现。在常见设定下，我们证明该算法的输出在类似于无限粒子极限的条件下收敛至最优分布，且能生成独立同分布样本，无需显式建立混沌传播界限。