Variational Analysis in the Wasserstein Space

Abstract

We study optimization problems whereby the optimization variable is a probability measure. Since the space of probability measures is not a vector space, many classical methods for optimization (e.g., gradients) do not directly apply. Thus, one typically resorts to the abstract machinery of infinite-dimensional analysis or other ad-hoc methodologies, not tailored to the space of probability measures, which however involve projections or rely on convexity-type assumptions. We believe instead that these problems call for a comprehensive methodological framework for calculus in the space of probability measures. In this work, we combine ideas from optimal transport, variational analysis, and Wasserstein gradient flows to equip the Wasserstein space (i.e., the space of probability measures endowed with the Wasserstein distance) with a variational structure, both by combining and extending existing results and introducing novel tools. Our theoretical analysis culminates in general necessary optimality conditions. These conditions (i) resemble the optimality conditions in Euclidean spaces, such as the KKT conditions, (ii) are intuitive, informative, and easy to study, and (iii) yield closed-form solutions or can be used to design computationally attractive algorithms. We accompany our theoretical results with numerous examples and present applications to machine learning, drug discovery, and distributionally robust optimization.