# Minimax Statistical Estimation under Wasserstein Contamination

arXiv: https://arxiv.org/abs/2308.01853 · HTML: https://mlsanity.com/paper/2308.01853/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 42 references that are in the map.
- First posted on arXiv 3 Aug 2023.

## Builds on (most cited first, 10 of 10)

- [Optimal Transport for Domain Adaptation](https://mlsanity.com/paper/1507.00504/) (2015)
- [Data-driven Distributionally Robust Optimization Using the Wasserstein Metric: Performance Guarantees and Tractable Reformulations](https://mlsanity.com/paper/1505.05116/) (2015)
- [DeepJDOT: Deep Joint Distribution Optimal Transport for Unsupervised Domain Adaptation](https://mlsanity.com/paper/1803.10081/) (2018)
- [Statistics of Robust Optimization: A Generalized Empirical Likelihood Approach](https://mlsanity.com/paper/1610.03425/) (2016)
- [Robust Covariance and Scatter Matrix Estimation under Huber's Contamination Model](https://mlsanity.com/paper/1506.00691/) (2015)
- [Distributionally Robust Learning](https://mlsanity.com/paper/2108.08993/) (2021)
- [Generalized Resilience and Robust Statistics](https://mlsanity.com/paper/1909.08755/) (2019)
- [Minimax Confidence Intervals for the Sliced Wasserstein Distance](https://mlsanity.com/paper/1909.07862/) (2019)
- [Wasserstein Distributionally Robust Kalman Filtering](https://mlsanity.com/paper/1809.08830/) (2018)
- [Robust W-GAN-Based Estimation Under Wasserstein Contamination](https://mlsanity.com/paper/2101.07969/) (2021)

## Built on by (most cited first, 1 of 1)

- [Robust Distribution Learning with Local and Global Adversarial Corruptions](https://mlsanity.com/paper/2406.06509/) (2024)

## Abstract (arXiv)

Contaminations are a key concern in modern statistical learning, as small but systematic perturbations of all datapoints can substantially alter estimation results. Here, we study Wasserstein-r contaminations (r≥ 1) in an ℓq norm (q∈ [1,∞]), in which each observation may undergo an adversarial perturbation with bounded cost, complementing the classical Huber model, corresponding to total variation norm, where only a fraction of observations is arbitrarily corrupted. We study both independent and joint (coordinated) contaminations and develop a minimax theory under ℓqr losses. Our analysis encompasses several fundamental problems: location estimation, linear regression, and pointwise nonparametric density estimation. For joint contaminations in location estimation and for prediction in linear regression, we obtain the exact minimax risk, identify least favorable contaminations, and show that the sample mean and least squares predictor are respectively minimax optimal. For location estimation under independent contaminations, we give sharp upper and lower bounds, including exact minimaxity in the Euclidean Wasserstein contamination case, when q=r=2. For pointwise density estimation in any dimension, we derive the optimal rate, showing that it is achieved by kernel density estimation with a bandwidth that is possibly larger than the classical one. Our proofs leverage powerful tools from optimal transport developed over the last 20 years, including the dynamic Benamou-Brenier formulation. Taken together, our results suggest that in contrast to the Huber contamination model, for norm-based Wasserstein contaminations, classical estimators may be nearly optimally robust.