A Robust Univariate Mean Estimator is All You Need

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Adarsh Prasad, Sivaraman Balakrishnan, Pradeep Ravikumar ;
Proceedings of the Twenty Third International Conference on Artificial Intelligence and Statistics, PMLR 108:4034-4044, 2020.

Abstract

We study the problem of designing estimators when the data has heavy-tails and is corrupted by outliers. In such an adversarial setup, we aim to design statistically optimal estimators for flexible non-parametric distribution classes such as distributions with bounded-2k moments and symmetric distributions. Our primary workhorse is a conceptually simple reduction from multivariate estimation to univariate estimation. Using this reduction, we design estimators which are optimal in both heavy-tailed and contaminated settings. Our estimators achieve an optimal dimension independent bias in the contaminated setting, while also simultaneously achieving high-probability error guarantees with optimal sample complexity. These results provide some of the first such estimators for a broad range of problems including Mean Estimation, Sparse Mean Estimation, Covariance Estimation, Sparse Covariance Estimation and Sparse PCA.

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