# Robust Distribution Learning with Local and Global Adversarial Corruptions

arXiv: https://arxiv.org/abs/2406.06509 · HTML: https://mlsanity.com/paper/2406.06509/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 36 of its 62 references that are in the map.
- First posted on arXiv 10 Jun 2024.

## Builds on (most cited first, 30 of 36)

- [Fairness Without Demographics in Repeated Loss Minimization](https://mlsanity.com/paper/1806.08010/) (2018)
- [Certifying Some Distributional Robustness with Principled Adversarial Training](https://mlsanity.com/paper/1710.10571/) (2017)
- [Data-driven Distributionally Robust Optimization Using the Wasserstein Metric: Performance Guarantees and Tractable Reformulations](https://mlsanity.com/paper/1505.05116/) (2015)
- [Robust Estimators in High Dimensions without the Computational Intractability](https://mlsanity.com/paper/1604.06443/) (2016)
- [Distributionally Robust Stochastic Optimization with Wasserstein Distance](https://mlsanity.com/paper/1604.02199/) (2016)
- [Optimal Entropy-Transport problems and a new Hellinger-Kantorovich distance between positive measures](https://mlsanity.com/paper/1508.07941/) (2015)
- [Unbalanced minibatch Optimal Transport; applications to Domain Adaptation](https://mlsanity.com/paper/2103.03606/) (2021)
- [Robust Optimal Transport with Applications in Generative Modeling and Domain Adaptation](https://mlsanity.com/paper/2010.05862/) (2020)
- [Statistical and Topological Properties of Sliced Probability Divergences](https://mlsanity.com/paper/2003.05783/) (2020)
- [Being Robust (in High Dimensions) Can Be Practical](https://mlsanity.com/paper/1703.00893/) (2017)
- [Regularization via Mass Transportation](https://mlsanity.com/paper/1710.10016/) (2017)
- [DORO: Distributional and Outlier Robust Optimization](https://mlsanity.com/paper/2106.06142/) (2021)
- [Finite-Sample Guarantees for Wasserstein Distributionally Robust Optimization: Breaking the Curse of Dimensionality](https://mlsanity.com/paper/2009.04382/) (2020)
- [Projection Robust Wasserstein Distance and Riemannian Optimization](https://mlsanity.com/paper/2006.07458/) (2020)
- [Outlier-Robust Optimal Transport](https://mlsanity.com/paper/2012.07363/) (2020)
- [Robust and Heavy-Tailed Mean Estimation Made Simple, via Regret Minimization](https://mlsanity.com/paper/2007.15839/) (2020)
- [Resilience: A Criterion for Learning in the Presence of Arbitrary Outliers](https://mlsanity.com/paper/1703.04940/) (2017)
- [Convergence and Concentration of Empirical Measures under Wasserstein Distance in Unbounded Functional Spaces](https://mlsanity.com/paper/1804.10556/) (2018)
- [Statistical, Robustness, and Computational Guarantees for Sliced Wasserstein Distances](https://mlsanity.com/paper/2210.09160/) (2022)
- [Robustly Learning a Gaussian: Getting Optimal Error, Efficiently](https://mlsanity.com/paper/1704.03866/) (2017)
- [High-Dimensional Robust Mean Estimation in Nearly-Linear Time](https://mlsanity.com/paper/1811.09380/) (2018)
- [Generalized Resilience and Robust Statistics](https://mlsanity.com/paper/1909.08755/) (2019)
- [On Projection Robust Optimal Transport: Sample Complexity and Model Misspecification](https://mlsanity.com/paper/2006.12301/) (2020)
- [On Robust Optimal Transport: Computational Complexity and Barycenter Computation](https://mlsanity.com/paper/2102.06857/) (2021)
- [Confidence Regions in Wasserstein Distributionally Robust Estimation](https://mlsanity.com/paper/1906.01614/) (2019)
- [Streaming Algorithms for High-Dimensional Robust Statistics](https://mlsanity.com/paper/2204.12399/) (2022)
- [Distributionally robust stochastic programs with side information based on trimmings -- Extended version](https://mlsanity.com/paper/2009.10592/) (2020)
- [Holistic Robust Data-Driven Decisions](https://mlsanity.com/paper/2207.09560/) (2022)
- [How Hard Is Robust Mean Estimation?](https://mlsanity.com/paper/1903.07870/) (2019)
- [Outlier-Robust Wasserstein DRO](https://mlsanity.com/paper/2311.05573/) (2023)

## Built on by (most cited first, 5 of 5)

- [Distributionally Robust Optimization with Adversarial Data Contamination](https://mlsanity.com/paper/2507.10718/) (2025)
- [Bulk-Calibrated Credal Ambiguity Sets: Fast, Tractable Decision Making under Out-of-Sample Contamination](https://mlsanity.com/paper/2601.21324/) (2026)
- [Adaptive Sample-Level Framework Motivated by Distributionally Robust Optimization with Variance-Based Radius Assignment for Enhanced Neural Network Generalization Under Distribution Shift](https://mlsanity.com/paper/2511.05568/) (2025)
- [Risk-Averse Wasserstein Distributionally Robust Online Learning](https://mlsanity.com/paper/2602.20403/) (2026)
- [Wasserstein Filtering: A Sample Selection Method for Robust Distribution Learning](https://mlsanity.com/paper/2608.13418/) (2026)

## Abstract (arXiv)

We consider learning in an adversarial environment, where an varepsilon-fraction of samples from a distribution P are arbitrarily modified (global corruptions) and the remaining perturbations have average magnitude bounded by ρ (local corruptions). Given access to n such corrupted samples, we seek a computationally efficient estimator hatPn that minimizes the Wasserstein distance mathsfW1(hatPn,P). In fact, we attack the fine-grained task of minimizing mathsfW1(Pi_# hatPn, Pi_# P) for all orthogonal projections Pi ∈ ℝd × d, with performance scaling with rank(Pi) = k. This allows us to account simultaneously for mean estimation (k=1), distribution estimation (k=d), as well as the settings interpolating between these two extremes. We characterize the optimal population-limit risk for this task and then develop an efficient finite-sample algorithm with error bounded by √varepsilon k + ρ + tildeO(d√kn-1/(k lor 2)) when P has bounded covariance. This guarantee holds uniformly in k and is minimax optimal up to the sub-optimality of the plug-in estimator when ρ = varepsilon = 0. Our efficient procedure relies on a novel trace norm approximation of an ideal yet intractable 2-Wasserstein projection estimator. We apply this algorithm to robust stochastic optimization, and, in the process, uncover a new method for overcoming the curse of dimensionality in Wasserstein distributionally robust optimization.