Near-optimal mean estimators with respect to general norms

TL;DR

This paper introduces a near-optimal mean estimator for high-dimensional vectors under general norms, achieving optimal accuracy-confidence tradeoff with minimal assumptions, based on a median-of-means approach.

Contribution

It presents a novel estimator tailored for arbitrary norms that attains near-optimal performance with only covariance assumptions, extending median-of-means techniques.

Findings

Achieves near-optimal accuracy-confidence tradeoff.

Works under minimal assumptions, only requiring a covariance matrix.

Introduces a uniform median-of-means estimator for general norms.

Abstract

We study the problem of estimating the mean of a random vector in $\mathbb{R}^d$ based on an i.i.d.\ sample, when the accuracy of the estimator is measured by a general norm on $\mathbb{R}^d$. We construct an estimator (that depends on the norm) that achieves an essentially optimal accuracy/confidence tradeoff under the only assumption that the random vector has a well-defined covariance matrix. The estimator is based on the construction of a uniform median-of-means estimator in a class of real valued functions that may be of independent interest.