A Robust Univariate Mean Estimator is All You Need

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.

Cite this Paper


BibTeX
@InProceedings{pmlr-v108-prasad20a, title = {A Robust Univariate Mean Estimator is All You Need}, author = {Prasad, Adarsh and Balakrishnan, Sivaraman and Ravikumar, Pradeep}, booktitle = {Proceedings of the Twenty Third International Conference on Artificial Intelligence and Statistics}, pages = {4034--4044}, year = {2020}, editor = {Chiappa, Silvia and Calandra, Roberto}, volume = {108}, series = {Proceedings of Machine Learning Research}, month = {26--28 Aug}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v108/prasad20a/prasad20a.pdf}, url = {http://proceedings.mlr.press/v108/prasad20a.html}, 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.} }
Endnote
%0 Conference Paper %T A Robust Univariate Mean Estimator is All You Need %A Adarsh Prasad %A Sivaraman Balakrishnan %A Pradeep Ravikumar %B Proceedings of the Twenty Third International Conference on Artificial Intelligence and Statistics %C Proceedings of Machine Learning Research %D 2020 %E Silvia Chiappa %E Roberto Calandra %F pmlr-v108-prasad20a %I PMLR %P 4034--4044 %U http://proceedings.mlr.press/v108/prasad20a.html %V 108 %X 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.
APA
Prasad, A., Balakrishnan, S. & Ravikumar, P.. (2020). A Robust Univariate Mean Estimator is All You Need. Proceedings of the Twenty Third International Conference on Artificial Intelligence and Statistics, in Proceedings of Machine Learning Research 108:4034-4044 Available from http://proceedings.mlr.press/v108/prasad20a.html.

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