Abstract
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.
Abstract (translated)
在2-Wasserstein空间中的梯度流被广泛用于优化概率分布上的泛函，并通常通过使用具有$n$个粒子的相互作用粒子系统来实现。分析这类算法需要证明（a）有限粒子系统的收敛性，以及/或(b) 粒子的最终经验分布能够接近最优分布（即传播混沌）。然而，建立有效的充分条件可能颇具挑战性，因为有限粒子系统可能会产生高度相关的随机变量。 在这项工作中，我们研究了虚拟粒子随机逼近方法，该方法最初是为Stein变分梯度下降所引入。这种方法可以被视为Wasserstein空间中的某种形式的随机梯度下降，并且可以高效地实现。在流行的应用场景中，我们证明了我们的算法输出可以在与无限粒子极限类似的条件下收敛到最优分布，并且它能够产生独立同分布（i.i.d.）样本，而无需显式建立传播混沌界。 这段文本描述了一种新的方法，即虚拟粒子随机逼近法，用于解决优化概率分布上的泛函问题。这种方法不仅理论上有效，而且在实际应用中也能高效地生成接近最优解的独立同分布样本，避免了直接处理复杂的相关性问题。
URL
https://arxiv.org/abs/2503.13115
