# Uniform bounds for robust mean estimators

## TL;DR

This paper introduces a unified approach to robust mean estimation that provides strong, non-asymptotic guarantees even with contaminated data, extending existing methods and achieving optimal bounds in adversarial settings.

## Contribution

It offers a new unified framework for robust mean estimators, proving uniform deviation bounds and extending results to adversarial contamination with optimal dependence.

## Key findings

- Unified bounds for robust mean estimators
- Extension to adversarially contaminated data
- Optimal dependence on contamination proportion

## Abstract

This paper is devoted to the estimators of the mean that provide strong non-asymptotic guarantees under minimal assumptions on the underlying distribution. The main ideas behind proposed techniques are based on bridging the notions of symmetry and robustness. We show that existing methods, such as median-of-means and Catoni's estimators, can often be viewed as special cases of our construction. The main contribution of the paper is the proof of uniform bounds for the deviations of the stochastic process defined by proposed estimators. Moreover, we extend our results to the case of adversarial contamination where a constant fraction of the observations is arbitrarily corrupted. Finally, we apply our methods to the problem of robust multivariate mean estimation and show that obtained inequalities achieve optimal dependence on the proportion of corrupted samples.