将凸函数差分解引入水土空间优化，提升MMD的收敛速度与稳定性
Difference of Convex Programming in the Wasserstein Space with Applications to MMD Optimization
- 在水土空间中用凸函数差分解重构目标函数，推广经典凸凹算法
- 对MMD和能量距离实现局部收敛，迭代过程几乎达到稳定点
- 相比标准梯度下降，新方法更快更稳，适合分布匹配任务
在机器学习中，对概率测度空间中的泛函进行优化已变得普遍。直接在水土空间中进行优化是常用方法，但许多实际兴趣的泛函在水土测地线上非凸，使一阶方法分析困难。本文研究一类可在水土空间中进行凸函数差（DC）分解的目标函数，并将经典的凸凹过程（CCCP）推广至该设定。在凸分量满足光滑性和强凸性假设下，证明了所提算法迭代序列的几乎平稳性。重点研究最大均值差异（MMD）和能量距离（ED）泛函，为其开发了显式的水土空间DC分解，并在弱假设下建立了方案的局部收敛性。实验表明，合理选择的DC分解可使MMD优化的收敛速度更快、更稳定，优于水土梯度下降。
原文摘要 · Abstract (English)
Optimizing functionals over the space of probability measures is now ubiquitous in machine learning. A widely used approach is to perform the optimization directly over the Wasserstein space, but many objective functionals of practical interest are non-convex along Wasserstein geodesics, making the analysis of standard first-order methods challenging. In this work, we study a class of objectives over the Wasserstein space that admit a difference-of-convex (DC) decomposition and we lift the classical convex-concave procedure (CCCP) to this setting. Under smoothness and strong convexity assumptions on the convex components of the decomposition, we prove almost stationarity along the iterates of the resulting algorithm. Our main focus is on the Maximum Mean Discrepancy (MMD) and the Energy Distance (ED) functionals, for which we develop explicit Wasserstein DC decompositions, and establish local convergence of the scheme under mild assumptions. Empirically, we show that well-chosen DC decompositions yield faster and more stable convergence than Wasserstein gradient descent on these MMD objectives.
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