论文标题
在瓦斯堡空间中优化的一阶条件
First-order Conditions for Optimization in the Wasserstein Space
论文作者
论文摘要
我们研究了在瓦斯堡空间中受约束优化的一阶最佳条件，从而在损失了瓦斯堡距离的概率衡量标准中，试图最大程度地减少实现的函数。我们的分析结合了对瓦斯坦海域空间的几何形状和差异结构的最新见解与更古典的变化结石。我们表明，诸如“将导数设置为零”和“在最佳状态保持一致”之类的简单理由将其延伸到Wasserstein空间。我们在分配强大的优化和统计推断的设置中部署工具来研究和解决优化问题。我们方法论的一般性使我们能够自然处理功能，例如均值变化，kullback-leibler Divergence和Wasserstein距离，这些距离在传统上很难在统一的框架中研究。
We study first-order optimality conditions for constrained optimization in the Wasserstein space, whereby one seeks to minimize a real-valued function over the space of probability measures endowed with the Wasserstein distance. Our analysis combines recent insights on the geometry and the differential structure of the Wasserstein space with more classical calculus of variations. We show that simple rationales such as "setting the derivative to zero" and "gradients are aligned at optimality" carry over to the Wasserstein space. We deploy our tools to study and solve optimization problems in the setting of distributionally robust optimization and statistical inference. The generality of our methodology allows us to naturally deal with functionals, such as mean-variance, Kullback-Leibler divergence, and Wasserstein distance, which are traditionally difficult to study in a unified framework.
