# Robust Statistical Estimators with Bounded Empirical Sensitivity

arXiv: https://arxiv.org/abs/2605.21860 · HTML: https://mlsanity.com/paper/2605.21860/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 14 of its 27 references that are in the map.
- First posted on arXiv 21 May 2026.

## Builds on (most cited first, 14 of 14)

- [Robust Estimators in High Dimensions without the Computational Intractability](https://mlsanity.com/paper/1604.06443/) (2016)
- [Agnostic Estimation of Mean and Covariance](https://mlsanity.com/paper/1604.06968/) (2016)
- [Robustness Implies Privacy in Statistical Estimation](https://mlsanity.com/paper/2212.05015/) (2022)
- [Resilience: A Criterion for Learning in the Presence of Arbitrary Outliers](https://mlsanity.com/paper/1703.04940/) (2017)
- [From Robustness to Privacy and Back](https://mlsanity.com/paper/2302.01855/) (2023)
- [Near Instance-Optimality in Differential Privacy](https://mlsanity.com/paper/2005.10630/) (2020)
- [Recursive Mechanism: Towards Node Differential Privacy and Unrestricted Joins [Full Version, Draft 0.1]](https://mlsanity.com/paper/1304.4795/) (2013)
- [On the power of adaptivity in statistical adversaries](https://mlsanity.com/paper/2111.10352/) (2021)
- [Adaptive and oblivious statistical adversaries are equivalent](https://mlsanity.com/paper/2410.13548/) (2024)
- [Is nasty noise actually harder than malicious noise?](https://mlsanity.com/paper/2511.09763/) (2025)
- [Stability and Accuracy Trade-offs in Statistical Estimation](https://mlsanity.com/paper/2601.11701/) (2026)
- [Wasserstein-Cram'er-Rao Theory of Unbiased Estimation](https://mlsanity.com/paper/2511.07414/) (2025)
- [On the Learnability of Distribution Classes with Adaptive Adversaries](https://mlsanity.com/paper/2509.05137/) (2025)
- [Computation-Utility-Privacy Tradeoffs in Bayesian Estimation](https://mlsanity.com/paper/2603.18254/) (2026)

## Abstract (arXiv)

We introduce a new measure of robustness for statistical estimators, which we call empirical sensitivity. An estimator hat θ has bounded empirical sensitivity if, with high probability over a dataset X = (X1, dots, Xn) ~ Dotimes n, for any dataset Y obtained by modifying at most η n 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 ℓ2-error bound of Oleft(√d/nright), the empirical sensitivity is at least Omegaleft(η + √η d/nright). 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.