# Statistical Analysis of Wasserstein Distributionally Robust Estimators

arXiv: https://arxiv.org/abs/2108.02120 · HTML: https://mlsanity.com/paper/2108.02120/

## Key facts

- Cited by 27 papers in the map (data through 14 Oct 2026).
- Builds on 50 of its 117 references that are in the map.
- First posted on arXiv 4 Aug 2021.

## Builds on (most cited first, 30 of 50)

- [Explaining and Harnessing Adversarial Examples](https://mlsanity.com/paper/1412.6572/) (2014)
- [Towards Deep Learning Models Resistant to Adversarial Attacks](https://mlsanity.com/paper/1706.06083/) (2017)
- [Computational Optimal Transport](https://mlsanity.com/paper/1803.00567/) (2018)
- [Optimal Transport for Domain Adaptation](https://mlsanity.com/paper/1507.00504/) (2015)
- [Generalizing to Unseen Domains via Adversarial Data Augmentation](https://mlsanity.com/paper/1805.12018/) (2018)
- [On Fairness and Calibration](https://mlsanity.com/paper/1709.02012/) (2017)
- [Data-driven Distributionally Robust Optimization Using the Wasserstein Metric: Performance Guarantees and Tractable Reformulations](https://mlsanity.com/paper/1505.05116/) (2015)
- [Generalized Sliced Wasserstein Distances](https://mlsanity.com/paper/1902.00434/) (2019)
- [Distributionally Robust Stochastic Optimization with Wasserstein Distance](https://mlsanity.com/paper/1604.02199/) (2016)
- [Statistics of Robust Optimization: A Generalized Empirical Likelihood Approach](https://mlsanity.com/paper/1610.03425/) (2016)
- [Wasserstein Distributionally Robust Optimization: Theory and Applications in Machine Learning](https://mlsanity.com/paper/1908.08729/) (2019)
- [Distributionally Robust Logistic Regression](https://mlsanity.com/paper/1509.09259/) (2015)
- [Stabilizing Training of Generative Adversarial Networks through Regularization](https://mlsanity.com/paper/1705.09367/) (2017)
- [Missing Data Imputation using Optimal Transport](https://mlsanity.com/paper/2002.03860/) (2020)
- [Subspace Robust Wasserstein Distances](https://mlsanity.com/paper/1901.08949/) (2019)
- [Regularization via Mass Transportation](https://mlsanity.com/paper/1710.10016/) (2017)
- [Tree-Sliced Variants of Wasserstein Distances](https://mlsanity.com/paper/1902.00342/) (2019)
- [Finite-Sample Guarantees for Wasserstein Distributionally Robust Optimization: Breaking the Curse of Dimensionality](https://mlsanity.com/paper/2009.04382/) (2020)
- [From Data to Decisions: Distributionally Robust Optimization is Optimal](https://mlsanity.com/paper/1704.04118/) (2017)
- [Data-driven Inverse Optimization with Imperfect Information](https://mlsanity.com/paper/1512.05489/) (2015)
- [Approximate Bayesian computation with the Wasserstein distance](https://mlsanity.com/paper/1905.03747/) (2019)
- [Recovering Best Statistical Guarantees via the Empirical Divergence-based Distributionally Robust Optimization](https://mlsanity.com/paper/1605.09349/) (2016)
- [Optimal Transport Based Distributionally Robust Optimization: Structural Properties and Iterative Schemes](https://mlsanity.com/paper/1810.02403/) (2018)
- [Wasserstein Distributionally Robust Stochastic Control: A Data-Driven Approach](https://mlsanity.com/paper/1812.09808/) (2018)
- [Robust Hypothesis Testing Using Wasserstein Uncertainty Sets](https://mlsanity.com/paper/1805.10611/) (2018)
- [Wasserstein Distributionally Robust Optimization and Variation Regularization](https://mlsanity.com/paper/1712.06050/) (2017)
- [Distributionally Robust Mean-Variance Portfolio Selection with Wasserstein Distances](https://mlsanity.com/paper/1802.04885/) (2018)
- [A Statistical Test for Probabilistic Fairness](https://mlsanity.com/paper/2012.04800/) (2020)
- [Confidence Regions in Wasserstein Distributionally Robust Estimation](https://mlsanity.com/paper/1906.01614/) (2019)
- [On Linear Optimization over Wasserstein Balls](https://mlsanity.com/paper/2004.07162/) (2020)

## Built on by (most cited first, 27 of 27)

- [Sinkhorn Distributionally Robust Optimization](https://mlsanity.com/paper/2109.11926/) (2021)
- [The Geometry of Adversarial Training in Binary Classification](https://mlsanity.com/paper/2111.13613/) (2021)
- [Sobolev Transport: A Scalable Metric for Probability Measures with Graph Metrics](https://mlsanity.com/paper/2202.10723/) (2022)
- [Unifying Distributionally Robust Optimization via Optimal Transport Theory](https://mlsanity.com/paper/2308.05414/) (2023)
- [A Short and General Duality Proof for Wasserstein Distributionally Robust Optimization](https://mlsanity.com/paper/2205.00362/) (2022)
- [Distributionally Robust Fair Principal Components via Geodesic Descents](https://mlsanity.com/paper/2202.03071/) (2022)
- [Exact Generalization Guarantees for (Regularized) Wasserstein Distributionally Robust Models](https://mlsanity.com/paper/2305.17076/) (2023)
- [On the regularized risk of distributionally robust learning over deep neural networks](https://mlsanity.com/paper/2109.06294/) (2021)
- [Optimal Transport for Measures with Noisy Tree Metric](https://mlsanity.com/paper/2310.13653/) (2023)
- [Universal generalization guarantees for Wasserstein distributionally robust models](https://mlsanity.com/paper/2402.11981/) (2024)
- [On adversarial robustness and the use of Wasserstein ascent-descent dynamics to enforce it](https://mlsanity.com/paper/2301.03662/) (2023)
- [Learning Against Distributional Uncertainty: On the Trade-off Between Robustness and Specificity](https://mlsanity.com/paper/2301.13565/) (2023)
- [Distributionally Robust Policy Evaluation under General Covariate Shift in Contextual Bandits](https://mlsanity.com/paper/2401.11353/) (2024)
- [textttskwdro: a library for Wasserstein distributionally robust machine learning](https://mlsanity.com/paper/2410.21231/) (2024)
- [Distributionally Robust Classification for Multi-source Unsupervised Domain Adaptation](https://mlsanity.com/paper/2601.21315/) (2026)
- [Wasserstein Distributionally Robust Optimization with Wasserstein Barycenters](https://mlsanity.com/paper/2203.12136/) (2022)
- [A Geometric Unification of Distributionally Robust Covariance Estimators: Shrinking the Spectrum by Inflating the Ambiguity Set](https://mlsanity.com/paper/2405.20124/) (2024)
- [Learning a Single Neuron Robustly to Distributional Shifts and Adversarial Label Noise](https://mlsanity.com/paper/2411.06697/) (2024)
- [Adjusted Wasserstein Distributionally Robust Estimator in Statistical Learning](https://mlsanity.com/paper/2303.15579/) (2023)
- [Statistical Properties of Robust Satisficing](https://mlsanity.com/paper/2405.20451/) (2024)
- [Generalizing Few Data to Unseen Domains Flexibly Based on Label Smoothing Integrated with Distributionally Robust Optimization](https://mlsanity.com/paper/2408.05082/) (2024)
- [Duality and Policy Evaluation in Distributionally Robust Bayesian Diffusion Control](https://mlsanity.com/paper/2506.19294/) (2025)
- [Representation-Aware Distributionally Robust Optimization: A Knowledge Transfer Framework](https://mlsanity.com/paper/2509.09371/) (2025)
- [Robust Learning of a Group DRO Neuron](https://mlsanity.com/paper/2601.18115/) (2026)
- [Statistical Guarantees for Distributionally Robust Optimization with Optimal Transport and OT-Regularized Divergences](https://mlsanity.com/paper/2603.27871/) (2026)
- [Duality for the Adversarial Total Variation](https://mlsanity.com/paper/2604.18540/) (2026)
- [Distributionally robust linear regression through the lens of adversarial training](https://mlsanity.com/paper/2609.39449/) (2026)

## Abstract (arXiv)

We consider statistical methods which 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, the 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.