License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ITCS.2022.98
URN: urn:nbn:de:0030-drops-156942
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2022/15694/
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Lee, Jasper C.H. ; Valiant, Paul

Optimal Sub-Gaussian Mean Estimation in Very High Dimensions

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LIPIcs-ITCS-2022-98.pdf (0.7 MB)

Abstract

We address the problem of mean estimation in very high dimensions, in the high probability regime parameterized by failure probability δ. For a distribution with covariance Σ, let its "effective dimension" be d_eff = {Tr(Σ)}/{λ_{max}(Σ)}. For the regime where d_eff = ω(log^2 (1/δ)), we show the first algorithm whose sample complexity is optimal to within 1+o(1) factor. The algorithm has a surprisingly simple structure: 1) re-center the samples using a known sub-Gaussian estimator, 2) carefully choose an easy-to-compute positive integer t and then remove the t samples farthest from the origin and 3) return the sample mean of the remaining samples. The core of the analysis relies on a novel vector Bernstein-type tail bound, showing that under general conditions, the sample mean of a bounded high-dimensional distribution is highly concentrated around a spherical shell.

BibTeX - Entry

@InProceedings{lee_et_al:LIPIcs.ITCS.2022.98,
  author =	{Lee, Jasper C.H. and Valiant, Paul},
  title =	{{Optimal Sub-Gaussian Mean Estimation in Very High Dimensions}},
  booktitle =	{13th Innovations in Theoretical Computer Science Conference (ITCS 2022)},
  pages =	{98:1--98:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-217-4},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{215},
  editor =	{Braverman, Mark},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2022/15694},
  URN =		{urn:nbn:de:0030-drops-156942},
  doi =		{10.4230/LIPIcs.ITCS.2022.98},
  annote =	{Keywords: High-dimensional mean estimation}
}

Keywords: High-dimensional mean estimation
Collection: 13th Innovations in Theoretical Computer Science Conference (ITCS 2022)
Issue Date: 2022
Date of publication: 25.01.2022

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