Error bounds of Median-of-means estimators with VC-dimension

TL;DR

This paper derives upper error bounds for median-of-means estimators in high-dimensional settings with heavy-tailed data, using VC dimension instead of traditional complexity measures.

Contribution

It introduces a new robust estimator based on MOM for covariance and mean estimation without strong distributional assumptions, leveraging VC dimension.

Findings

Derived error bounds for MOM estimators under weak moment conditions.

Implemented MOM in covariance estimation without sub-Gaussian assumptions.

Proposed a MOM version of halfspace depth with error bounds.

Abstract

We obtain the upper error bounds of robust estimators for mean vector, using the median-of-means (MOM) method. The method is designed to handle data with heavy tails and contamination, with only a finite second moment, which is weaker than many others, relying on the VC dimension rather than the Rademacher complexity to measure statistical complexity. This allows us to implement MOM in covariance estimation, without imposing conditions such as $L$-sub-Gaussian or $L_{4}-L_{2}$ norm equivalence. In particular, we derive a new robust estimator, the MOM version of the halfspace depth, along with error bounds for mean estimation in any norm.