Optimization in Wasserstein Spaces

Abstract

We study optimization problems whereby the optimization variable is a probability distribution. Unfortunately, the probability space is not a vector space and, thus, many classical methods available in the optimization literature (e.g., derivatives) are of little help. Thus, one typically has to resort to the abstract machinery of infinite-dimensional analysis or other ad-hoc methodologies, which are however not tailored to the probability space, generally entail projections or require convexity-type assumptions, and break when the problem changes. We believe instead that these problems call for a comprehensive methodological framework for calculus in probability spaces. In this work, we combine ideas from the optimal transport and variational analysis literature to equip the Wasserstein space (i.e., the space of probability measures endowed with the Wasserstein distance) with a variational structure, both by considerably extending existing results and introducing novel tools. Our theoretical analysis culminates in necessary optimality conditions which, perhaps surprisingly, (i) resemble rationales of Euclidean spaces, such as KKT conditions, and (ii) are intuitive, informative, and easy to study. Thereafter, we showcase how both existing and open problems of interest are considerably easier to solve. For instance, in the field of distributionally robust control and portfolio optimization. We believe this framework lays the foundation for new algorithmic and theoretical advancements in the study of optimization problems in probability spaces.