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Confidence regions in Wasserstein distributionally robust estimation
[Distributionally robust groupwise regularization estimator]

Author

Listed:
  • Jose Blanchet
  • Karthyek Murthy
  • Nian Si

Abstract

SummaryEstimators based on Wasserstein distributionally robust optimization 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 from the underlying empirical measure in a Wasserstein sense. 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. This paper studies the asymptotic normality of these distributionally robust estimators as well as the properties of an optimal 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.

Suggested Citation

  • Jose Blanchet & Karthyek Murthy & Nian Si, 2022. "Confidence regions in Wasserstein distributionally robust estimation [Distributionally robust groupwise regularization estimator]," Biometrika, Biometrika Trust, vol. 109(2), pages 295-315.
    • Handle: RePEc:oup:biomet:v:109:y:2022:i:2:p:295-315.
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    References listed on IDEAS

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    1. Alexander Shapiro, 1993. "Asymptotic Behavior of Optimal Solutions in Stochastic Programming," Mathematics of Operations Research, INFORMS, vol. 18(4), pages 829-845, November.
    2. Jose Blanchet & Karthyek Murthy, 2019. "Quantifying Distributional Model Risk via Optimal Transport," Mathematics of Operations Research, INFORMS, vol. 44(2), pages 565-600, May.
    3. Dariush Khezrimotlagh & Yao Chen, 2018. "The Optimization Approach," International Series in Operations Research & Management Science, in: Decision Making and Performance Evaluation Using Data Envelopment Analysis, chapter 0, pages 107-134, Springer.
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