Generalized resilience and robust statistics

Robust statistics traditionally focuses on outliers, or perturbations in total variation distance. However, a dataset could be maliciously corrupted in many other ways, such as systematic measurement errors and missing covariates. We consider corruption in either TV or Wasserstein distance, and show that robust estimation is possible whenever the true population distribution satisfies a property called generalized resilience, which holds under moment or hypercontractive conditions. For TV corruption model, our finite-sample analysis improves over previous results for mean estimation with bounded kth moment, linear regression, and joint mean and covariance estimation. For W1 corruption, we provide the first finite-sample guarantees for second moment estimation and linear regression. Technically, our robust estimators are a generalization of minimum distance (MD) functionals, which project the corrupted distribution onto a given set of well-behaved distributions. The error of these MD functionals is bounded by a certain modulus of continuity, and we provide a systematic method for upper bounding this modulus for the class of generalized resilient distributions, which usually gives sharp population-level results and good finite-sample guarantees.