Robust Estimation Under Heterogeneous Corruption Rates

TL;DR

This paper develops tight minimax rates for robust estimation in settings with heterogeneous corruption, revealing how optimal estimators handle varying corruption levels across samples.

Contribution

It introduces the first tight minimax rates for robust estimation under non-uniform corruption, extending analysis to multivariate and univariate distributions, and identifies thresholds for sample discard.

Findings

Established minimax rates for mean estimation under heterogeneous corruption.

Derived bounds for multivariate Gaussian mean estimation and linear regression.

Showed that optimal estimators discard samples beyond a corruption threshold.

Abstract

We study the problem of robust estimation under heterogeneous corruption rates, where each sample may be independently corrupted with a known but non-identical probability. This setting arises naturally in distributed and federated learning, crowdsourcing, and sensor networks, yet existing robust estimators typically assume uniform or worst-case corruption, ignoring structural heterogeneity. For mean estimation for multivariate bounded distributions and univariate gaussian distributions, we give tight minimax rates for all heterogeneous corruption patterns. For multivariate gaussian mean estimation and linear regression, we establish the minimax rate for squared error up to a factor of $\sqrt{d}$, where $d$ is the dimension. Roughly, our findings suggest that samples beyond a certain corruption threshold may be discarded by the optimal estimators -- this threshold is determined by the empirical distribution of the corruption rates given.