Variational Analysis in the Wasserstein Space

TL;DR

This paper develops a comprehensive variational calculus framework in Wasserstein space, enabling better optimization methods for probability measures with applications in machine learning and robust optimization.

Contribution

It introduces novel tools and extends existing results to establish a variational structure in Wasserstein space, providing general optimality conditions and practical algorithms.

Findings

Derived general necessary optimality conditions resembling Euclidean KKT and Lagrange conditions.

Provided closed-form solutions and computational algorithms for optimization in probability spaces.

Demonstrated applications in machine learning, drug discovery, and distributionally robust optimization.

Abstract

We study optimization problems whereby the optimization variable is a probability measure. Since the probability space is not a vector space, many classical and powerful methods for optimization (e.g., gradients) are of little help. Thus, one typically resorts to the abstract machinery of infinite-dimensional analysis or other ad-hoc methodologies, not tailored to the probability space, 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 probability spaces. 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 very general necessary optimality conditions for optimality. Notably, our conditions (i) resemble the rationales of Euclidean spaces, such as the Karush-Kuhn-Tucker and Lagrange 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 believe this framework lays the foundation for new algorithmic and theoretical advancements in the study of optimization problems in probability spaces, which we exemplify with numerous case studies and applications to machine learning, drug discovery, and distributionally robust optimization.