Confidence Regions in Wasserstein Distributionally Robust Estimation
This work provides theoretical foundations for robust statistical estimation, addressing model misspecification in machine learning and optimization, though it is incremental in extending existing distributionally robust optimization theory.
The paper establishes the asymptotic normality of Wasserstein distributionally robust estimators and analyzes optimal confidence regions derived from this framework, showing that these estimators generalize regularized methods like square-root lasso and SVMs.
Wasserstein distributionally robust optimization estimators are obtained as solutions of min-max problems in which the statistician selects a parameter minimizing the worst-case loss among all probability models within a certain distance (in a Wasserstein sense) from the underlying empirical measure. While motivated by the need to identify optimal model parameters or decision choices that are robust to model misspecification, these distributionally robust estimators recover a wide range of regularized estimators, including square-root lasso and support vector machines, among others, as particular cases. This paper studies the asymptotic normality of these distributionally robust estimators as well as the properties of an optimal (in a suitable sense) confidence region induced by the Wasserstein distributionally robust optimization formulation. In addition, key properties of min-max distributionally robust optimization problems are also studied, for example, we show that distributionally robust estimators regularize the loss based on its derivative and we also derive general sufficient conditions which show the equivalence between the min-max distributionally robust optimization problem and the corresponding max-min formulation.