The Sample Complexity of Robust Covariance Testing

Ilias Diakonikolas, Daniel M. Kane
Proceedings of Thirty Fourth Conference on Learning Theory, PMLR 134:1511-1521, 2021.

Abstract

We study the problem of testing the covariance matrix of a high-dimensional Gaussian in a robust setting, where the input distribution has been corrupted in Huber’s contamination model. Specifically, we are given i.i.d. samples from a distribution of the form $Z = (1-\epsilon) X + \epsilon B$, where $X$ is a zero-mean and unknown covariance Gaussian $\mathcal{N}(0, \Sigma)$, $B$ is a fixed but unknown noise distribution, and $\epsilon>0$ is an arbitrarily small constant representing the proportion of contamination. We want to distinguish between the cases that $\Sigma$ is the identity matrix versus $\gamma$-far from the identity in Frobenius norm. In the absence of contamination, prior work gave a simple tester for this hypothesis testing task that uses $O(d)$ samples. Moreover, this sample upper bound was shown to be best possible, within constant factors. Our main result is that the sample complexity of covariance testing dramatically increases in the contaminated setting. In particular, we prove a sample complexity lower bound of $\Omega(d^2)$ for $\epsilon$ an arbitrarily small constant and $\gamma = 1/2$. This lower bound is best possible, as $O(d^2)$ samples suffice to even robustly {\em learn} the covariance. The conceptual implication of our result is that, for the natural setting we consider, robust hypothesis testing is at least as hard as robust estimation.

Cite this Paper

BibTeX
@InProceedings{pmlr-v134-diakonikolas21a, title = {The Sample Complexity of Robust Covariance Testing}, author = {Diakonikolas, Ilias and Kane, Daniel M.}, booktitle = {Proceedings of Thirty Fourth Conference on Learning Theory}, pages = {1511--1521}, year = {2021}, editor = {Belkin, Mikhail and Kpotufe, Samory}, volume = {134}, series = {Proceedings of Machine Learning Research}, month = {15--19 Aug}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v134/diakonikolas21a/diakonikolas21a.pdf}, url = {https://proceedings.mlr.press/v134/diakonikolas21a.html}, abstract = {We study the problem of testing the covariance matrix of a high-dimensional Gaussian in a robust setting, where the input distribution has been corrupted in Huber’s contamination model. Specifically, we are given i.i.d. samples from a distribution of the form $Z = (1-\epsilon) X + \epsilon B$, where $X$ is a zero-mean and unknown covariance Gaussian $\mathcal{N}(0, \Sigma)$, $B$ is a fixed but unknown noise distribution, and $\epsilon>0$ is an arbitrarily small constant representing the proportion of contamination. We want to distinguish between the cases that $\Sigma$ is the identity matrix versus $\gamma$-far from the identity in Frobenius norm. In the absence of contamination, prior work gave a simple tester for this hypothesis testing task that uses $O(d)$ samples. Moreover, this sample upper bound was shown to be best possible, within constant factors. Our main result is that the sample complexity of covariance testing dramatically increases in the contaminated setting. In particular, we prove a sample complexity lower bound of $\Omega(d^2)$ for $\epsilon$ an arbitrarily small constant and $\gamma = 1/2$. This lower bound is best possible, as $O(d^2)$ samples suffice to even robustly {\em learn} the covariance. The conceptual implication of our result is that, for the natural setting we consider, robust hypothesis testing is at least as hard as robust estimation.} }
Endnote
%0 Conference Paper %T The Sample Complexity of Robust Covariance Testing %A Ilias Diakonikolas %A Daniel M. Kane %B Proceedings of Thirty Fourth Conference on Learning Theory %C Proceedings of Machine Learning Research %D 2021 %E Mikhail Belkin %E Samory Kpotufe %F pmlr-v134-diakonikolas21a %I PMLR %P 1511--1521 %U https://proceedings.mlr.press/v134/diakonikolas21a.html %V 134 %X We study the problem of testing the covariance matrix of a high-dimensional Gaussian in a robust setting, where the input distribution has been corrupted in Huber’s contamination model. Specifically, we are given i.i.d. samples from a distribution of the form $Z = (1-\epsilon) X + \epsilon B$, where $X$ is a zero-mean and unknown covariance Gaussian $\mathcal{N}(0, \Sigma)$, $B$ is a fixed but unknown noise distribution, and $\epsilon>0$ is an arbitrarily small constant representing the proportion of contamination. We want to distinguish between the cases that $\Sigma$ is the identity matrix versus $\gamma$-far from the identity in Frobenius norm. In the absence of contamination, prior work gave a simple tester for this hypothesis testing task that uses $O(d)$ samples. Moreover, this sample upper bound was shown to be best possible, within constant factors. Our main result is that the sample complexity of covariance testing dramatically increases in the contaminated setting. In particular, we prove a sample complexity lower bound of $\Omega(d^2)$ for $\epsilon$ an arbitrarily small constant and $\gamma = 1/2$. This lower bound is best possible, as $O(d^2)$ samples suffice to even robustly {\em learn} the covariance. The conceptual implication of our result is that, for the natural setting we consider, robust hypothesis testing is at least as hard as robust estimation.
APA
Diakonikolas, I. & Kane, D.M.. (2021). The Sample Complexity of Robust Covariance Testing. Proceedings of Thirty Fourth Conference on Learning Theory, in Proceedings of Machine Learning Research 134:1511-1521 Available from https://proceedings.mlr.press/v134/diakonikolas21a.html.

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