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

Yuxuan Wang, Yiming Chen, Hanchao Wang, Lixin Zhang
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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.
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具有 VC 维度的均值中值估计器的误差边界
我们利用均值中值(MOM)方法获得了均值向量稳健估计器的误差上限。该方法设计用于处理尾部和污染严重的数据,只有有限的第二矩,比许多其他方法更弱,依靠 VC 维度而不是拉德马赫复杂性来衡量统计复杂性。这样,我们就可以实现 MOMin 协方差估计,而无需强加诸如 $L$-sub-Gaussian 或 $L_{4}-L_{2}$ norm equivalence 等条件。特别是,我们推导出了一种新的稳健估计器--半空间深度的 MOM 版本,以及在任何规范下进行均值估计的误差边界。
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