# Generalized Resilience and Robust Statistics

arXiv: https://arxiv.org/abs/1909.08755 · HTML: https://mlsanity.com/paper/1909.08755/

## Key facts

- Cited by 37 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 19 Sep 2019.

## Built on by (most cited first, 30 of 37)

- [Robust and Differentially Private Mean Estimation](https://mlsanity.com/paper/2102.09159/) (2021)
- [Differential privacy and robust statistics in high dimensions](https://mlsanity.com/paper/2111.06578/) (2021)
- [DORO: Distributional and Outlier Robust Optimization](https://mlsanity.com/paper/2106.06142/) (2021)
- [Improved Rates for Differentially Private Stochastic Convex Optimization with Heavy-Tailed Data](https://mlsanity.com/paper/2106.01336/) (2021)
- [Statistical, Robustness, and Computational Guarantees for Sliced Wasserstein Distances](https://mlsanity.com/paper/2210.09160/) (2022)
- [Private Robust Estimation by Stabilizing Convex Relaxations](https://mlsanity.com/paper/2112.03548/) (2021)
- [Byzantine-Robust Federated Learning with Optimal Statistical Rates and Privacy Guarantees](https://mlsanity.com/paper/2205.11765/) (2022)
- [Outlier-Robust Optimal Transport: Duality, Structure, and Statistical Analysis](https://mlsanity.com/paper/2111.01361/) (2021)
- [Corruption Robust Offline Reinforcement Learning with Human Feedback](https://mlsanity.com/paper/2402.06734/) (2024)
- [Holistic Robust Data-Driven Decisions](https://mlsanity.com/paper/2207.09560/) (2022)
- [Outlier-Robust Wasserstein DRO](https://mlsanity.com/paper/2311.05573/) (2023)
- [On the power of adaptivity in statistical adversaries](https://mlsanity.com/paper/2111.10352/) (2021)
- [Robust W-GAN-Based Estimation Under Wasserstein Contamination](https://mlsanity.com/paper/2101.07969/) (2021)
- [Near Optimal Private and Robust Linear Regression](https://mlsanity.com/paper/2301.13273/) (2023)
- [Robust Estimation for Random Graphs](https://mlsanity.com/paper/2111.05320/) (2021)
- [High Dimensional Distributed Gradient Descent with Arbitrary Number of Byzantine Attackers](https://mlsanity.com/paper/2307.13352/) (2023)
- [Trimmed Maximum Likelihood Estimation for Robust Learning in Generalized Linear Models](https://mlsanity.com/paper/2206.04777/) (2022)
- [Stochastic Re-weighted Gradient Descent via Distributionally Robust Optimization](https://mlsanity.com/paper/2306.09222/) (2023)
- [General Synthetic-Powered Inference](https://mlsanity.com/paper/2509.20345/) (2025)
- [Solving a Research Problem in Mathematical Statistics with AI Assistance](https://mlsanity.com/paper/2511.18828/) (2025)
- [Outcome Assumptions and Duality Theory for Balancing Weights](https://mlsanity.com/paper/2203.09557/) (2022)
- [Robust Distribution Learning with Local and Global Adversarial Corruptions](https://mlsanity.com/paper/2406.06509/) (2024)
- [Robust estimation algorithms don't need to know the corruption level](https://mlsanity.com/paper/2202.05453/) (2022)
- [Robust Estimation for Nonparametric Families via Generative Adversarial Networks](https://mlsanity.com/paper/2202.01269/) (2022)
- [Wasserstein Distributionally Robust Nonparametric Regression](https://mlsanity.com/paper/2505.07967/) (2025)
- [Influence Based Defense Against Data Poisoning Attacks in Online Learning](https://mlsanity.com/paper/2104.13230/) (2021)
- [The Broader Landscape of Robustness in Algorithmic Statistics](https://mlsanity.com/paper/2412.02670/) (2024)
- [Minimax Statistical Estimation under Wasserstein Contamination](https://mlsanity.com/paper/2308.01853/) (2023)
- [Noisy Computing of the mathsfOR and mathsfMAX Functions](https://mlsanity.com/paper/2309.03986/) (2023)
- [Automatic Outlier Rectification via Optimal Transport](https://mlsanity.com/paper/2403.14067/) (2024)

## Abstract (arXiv)

Robust statistics traditionally focuses on outliers, or perturbations in total variation distance. However, a dataset could be corrupted in many other ways, such as systematic measurement errors and missing covariates. We generalize the robust statistics approach to consider perturbations under any Wasserstein distance, and show that robust estimation is possible whenever a distribution's population statistics are robust under a certain family of friendly perturbations. This generalizes a property called resilience previously employed in the special case of mean estimation with outliers. We justify the generalized resilience property by showing that it holds under moment or hypercontractive conditions. Even in the total variation case, these subsume conditions in the literature for mean estimation, regression, and covariance estimation; the resulting analysis simplifies and sometimes improves these known results in both population limit and finite-sample rate. Our robust estimators are based on minimum distance (MD) functionals (Donoho and Liu, 1988), which project onto a set of distributions under a discrepancy related to the perturbation. We present two approaches for designing MD estimators with good finite-sample rates: weakening the discrepancy and expanding the set of distributions. We also present connections to Gao et al. (2019)'s recent analysis of generative adversarial networks for robust estimation.