Minimax Statistical Estimation under Wasserstein Contamination
This work addresses robustness in statistical learning for scenarios with systematic perturbations, providing theoretical guarantees that classical methods can be near-optimal, which is incremental but important for applications in data security and reliability.
The paper tackles the problem of statistical estimation under adversarial Wasserstein contaminations, where each observation may be perturbed with bounded cost, and develops a minimax theory for location estimation, linear regression, and pointwise density estimation, showing that classical estimators like the sample mean and least squares are minimax optimal in certain cases.
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 ($r\ge 1$) in an $\ell_q$ norm ($q\in [1,\infty]$), 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 $\ell_q^r$ 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.