Statistical Analysis of Wasserstein Distributionally Robust Estimators
We consider statistical methods that invoke a min-max distributionally robust formulation to extract good out-of-sample performance in data-driven optimization and learning problems. Acknowledging the distributional uncertainty in learning from limited samples, the min-max formulations introduce an adversarial inner player to explore unseen covariate data. The resulting distributionally robust optimization (DRO) formulations, which include Wasserstein DRO formulations (our main focus), are specified using optimal transportation phenomena. Upon describing how these infinite-dimensional min-max problems can be approached via a finite-dimensional dual reformulation, this tutorial moves into its main component, namely, explaining a generic recipe for optimally selecting the size of the adversary’s budget. This is achieved by studying the limit behavior of an optimal transport projection formulation arising from an inquiry on the smallest confidence region that includes the unknown population risk minimizer. Incidentally, this systematic prescription coincides with those in specific examples in high-dimensional statistics and results in error bounds that are free from the curse of dimensions. Equipped with this prescription, we present a central limit theorem for the DRO estimator and provide a recipe for constructing compatible confidence regions that are useful for uncertainty quantification. The rest of the tutorial is devoted to insights into the nature of the optimizers selected by the min-max formulations and additional applications of optimal transport projections.
Funding: Material in this paper is based on work supported by the Air Force Office of Scientific Research [Award FA9550-20-1-0397]. Support from the Singapore Ministry of Education (MOE) and Singapore University of Technology and Design [Research Grant SRG-ESD-2018-134] and MOE Academic Research Fund [Grant MOE2019-T2-2-163] are gratefully acknowledged, as is additional support from the National Science Foundation [Grants 1915967, 1820942, and 1838576].
Video of this TutORial from the 2021 INFORMS Annual Meeting, held virtually October 26, 2021, is available at https://youtu.be/79Py8KU4_k0.