Uniform bounds for robust mean estimators

We study estimators of the means of a family of random variables $\{f\left(X\right),f\in F\}$ that admit uniform, over the class $F$ of real-valued functions, non-asymptotic error bounds under minimal moment assumptions on the underlying distribution. We show that known robust methods, such as the median-of-means and Catoni’s estimators, can often be viewed as special cases of our construction. The paper’s primary contribution lies in establishing uniform bounds for the deviations of stochastic processes defined by the proposed estimators. Furthermore, we analyze the stability of these estimators within the context of the ‘adversarial contamination’ framework. Finally, we demonstrate the applicability of our methods to the problem of robust multivariate mean estimation, showing that the resulting inequalities achieve optimal dependence on the parameters of the problem.