提出一种新算法，让粒子优化更高效且无需复杂理论证明。
Beyond Propagation of Chaos: A Stochastic Algorithm for Mean Field Optimization
- 用虚拟粒子随机逼近法实现水-瓦瑟斯坦空间的梯度下降
- 在常见场景下收敛到最优分布，条件与无限粒子极限相似
- 生成独立同分布样本，无需额外证明混沌传播
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，通常通过含n个粒子的相互作用粒子系统实现。分析这类算法需证明（a）有限粒子系统收敛，或（b）粒子经验分布能良好近似最优分布（即混沌传播）。但因有限粒子系统可能产生高度依赖的随机变量，建立有效充分条件颇具挑战。本文研究了最初为Stein变分梯度下降提出的虚拟粒子随机逼近方法。该方法可视为水-瓦瑟斯坦空间中的随机梯度下降，且可高效实现。在典型设置下，我们证明该算法输出收敛至最优分布，所需条件与无限粒子极限类似，并能生成独立同分布样本，无需显式建立混沌传播界。
原文摘要 · Abstract (English)
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.
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