Robustly Learning a Gaussian: Getting Optimal Error, Efficiently
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1.9781611975031.171.pdf
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Author(s)
Stewart, Alistair
Diakonikolas, Ilias
Kamath, Gautam Chetan
Kane, Daniel M
Li, Jerry Zheng
Moitra, Ankur
Date Issued
January 2018
Journal
Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms
Publisher
Society for Industrial and Applied Mathematics
Citation
Diakonikolas, Ilias, Gautam Kamath, Daniel M. Kane, Jerry Li, Ankur Moitra, and Alistair Stewart. “Robustly Learning a Gaussian: Getting Optimal Error, Efficiently.” Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms (January 2018): 2683–2702.
Version
Final published version
Abstract
We study the fundamental problem of learning the parameters of a high-dimensional Gaussian in the presence of noise | where an "-fraction of our samples were chosen by an adversary. We give robust estimators that achieve estimation error O(ϵ) in the total variation distance, which is optimal up to a universal constant that is independent of the dimension. In the case where just the mean is unknown, our robustness guarantee is optimal up to a factor of p 2 and the running time is polynomial in d and 1/ϵ. When both the mean and covariance are unknown, the running time is polynomial in d and quasipolynomial in 1/ϵ. Moreover all of our algorithms require only a polynomial number of samples. Our work shows that the same sorts of error guarantees that were established over fifty years ago in the one-dimensional setting can also be achieved by efficient algorithms in high-dimensional settings.
MIT Department
Massachusetts Institute of Technology. Computer Science and Artificial Intelligence Laboratory
Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science
Massachusetts Institute of Technology. Department of Mathematics
Massachusetts Institute of Technology. Department of Physics
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Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use.
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DOI of Published Version
https://doi.org/10.1137/1.9781611975031.171
