# Confidence Regions in Wasserstein Distributionally Robust Estimation

arXiv: https://arxiv.org/abs/1906.01614 · HTML: https://mlsanity.com/paper/1906.01614/

## Key facts

- Cited by 24 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 4 Jun 2019.

## Built on by (most cited first, 24 of 24)

- [Sinkhorn Distributionally Robust Optimization](https://mlsanity.com/paper/2109.11926/) (2021)
- [Statistical Analysis of Wasserstein Distributionally Robust Estimators](https://mlsanity.com/paper/2108.02120/) (2021)
- [Outlier-Robust Wasserstein DRO](https://mlsanity.com/paper/2311.05573/) (2023)
- [Unifying Distributionally Robust Optimization via Optimal Transport Theory](https://mlsanity.com/paper/2308.05414/) (2023)
- [Generalization Bounds with Minimal Dependency on Hypothesis Class via Distributionally Robust Optimization](https://mlsanity.com/paper/2106.11180/) (2021)
- [Exact Generalization Guarantees for (Regularized) Wasserstein Distributionally Robust Models](https://mlsanity.com/paper/2305.17076/) (2023)
- [Nash Equilibria, Regularization and Computation in Optimal Transport-Based Distributionally Robust Optimization](https://mlsanity.com/paper/2303.03900/) (2023)
- [Smoothed f-Divergence Distributionally Robust Optimization](https://mlsanity.com/paper/2306.14041/) (2023)
- [On Generalization and Regularization via Wasserstein Distributionally Robust Optimization](https://mlsanity.com/paper/2212.05716/) (2022)
- [Universal generalization guarantees for Wasserstein distributionally robust models](https://mlsanity.com/paper/2402.11981/) (2024)
- [Robust Distribution Learning with Local and Global Adversarial Corruptions](https://mlsanity.com/paper/2406.06509/) (2024)
- [Evaluating Model Performance Under Worst-case Subpopulations](https://mlsanity.com/paper/2407.01316/) (2024)
- [Wasserstein Distributionally Robust Optimization with Wasserstein Barycenters](https://mlsanity.com/paper/2203.12136/) (2022)
- [Wasserstein Distributionally Robust Nonparametric Regression](https://mlsanity.com/paper/2505.07967/) (2025)
- [Mitigating distribution shift in machine learning-augmented hybrid simulation](https://mlsanity.com/paper/2401.09259/) (2024)
- [A Geometric Unification of Distributionally Robust Covariance Estimators: Shrinking the Spectrum by Inflating the Ambiguity Set](https://mlsanity.com/paper/2405.20124/) (2024)
- [Robust Generalization with Adaptive Optimal Transport Priors for Decision-Focused Learning](https://mlsanity.com/paper/2602.01427/) (2026)
- [Adjusted Wasserstein Distributionally Robust Estimator in Statistical Learning](https://mlsanity.com/paper/2303.15579/) (2023)
- [Asymptotic Behavior of Adversarial Training Estimator under ℓ_∞-Perturbation](https://mlsanity.com/paper/2401.15262/) (2024)
- [From Distributional Robustness to Robust Statistics: A Confidence Sets Perspective](https://mlsanity.com/paper/2410.14008/) (2024)
- [Improving Mixup Calibration with Wasserstein Distributionally Robust Optimization](https://mlsanity.com/paper/2506.17874/) (2025)
- [Representation-Aware Distributionally Robust Optimization: A Knowledge Transfer Framework](https://mlsanity.com/paper/2509.09371/) (2025)
- [Wasserstein Distributionally Robust Optimization Through the Lens of Structural Causal Models and Individual Fairness](https://mlsanity.com/paper/2509.26275/) (2025)
- [Statistical Guarantees for Distributionally Robust Optimization with Optimal Transport and OT-Regularized Divergences](https://mlsanity.com/paper/2603.27871/) (2026)

## Abstract (arXiv)

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.