Error Bounds of Median-of-Means Estimators with VC-Dimension

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 \(L\) $L$ -sub-Gaussian or \(L_{4}-L_{2}\) $L_4-L_2$ norm equivalence. In particular, we derive a new robust estimator, the MOM version of the half-space depth, along with error bounds for mean estimation in any norm.