Robust Statistical Estimators with Bounded Empirical Sensitivity
Provides a fundamental characterization of the trade-off between accuracy and robustness for statistical estimators, relevant to the robust statistics community.
The paper introduces a new robustness measure called empirical sensitivity and proves tight lower bounds for Gaussian mean estimation, showing that any estimator achieving optimal error must have empirical sensitivity at least Ω(η + √(ηd/n)), which is tight up to log factors.
We introduce a new measure of robustness for statistical estimators, which we call \emph{empirical sensitivity}. An estimator $\hat θ$ has bounded empirical sensitivity if, with high probability over a dataset $X = (X_1, \dots, X_n) \sim \mathcal{D}^{\otimes n}$, for any dataset $Y$ obtained by modifying at most $ηn$ points in $X$, we have that $\hat θ(Y)$ is close to $\hat θ(X)$. We study bounds on this quantity for the prototypical problem of Gaussian mean estimation. We prove new lower bounds, showing that for any estimator $\hat μ$ which achieves an optimal $\ell_2$-error bound of $O\left(\sqrt{d/n}\right)$, the empirical sensitivity is at least $Ω\left(η+ \sqrt{ηd/n}\right)$. The two terms arise due to obstructions on the mean and variance (via an Efron-Stein argument) of such an estimator. We show that this bound is tight up to logarithmic factors, by employing recent results for robust empirical mean estimation.