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Optimal Sub-Gaussian Mean Estimation in Very High Dimensions

Authors Jasper C.H. Lee, Paul Valiant

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Subject Classification

ACM Subject Classification
  • Mathematics of computing → Nonparametric statistics
  • Mathematics of computing → Multivariate statistics
  • Theory of computation → Sample complexity and generalization bounds
  • Theory of computation → Streaming, sublinear and near linear time algorithms
Keywords
  • High-dimensional mean estimation

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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.

Cite As Get BibTex

Jasper C.H. Lee and Paul Valiant. Optimal Sub-Gaussian Mean Estimation in Very High Dimensions. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 98:1-98:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022) https://doi.org/10.4230/LIPIcs.ITCS.2022.98

Author Details

Jasper C.H. Lee
  • University of Wisconsin-Madison, WI, USA
Paul Valiant
  • Purdue University, West Lafayette, IN, USA

Funding

  • Lee, Jasper C.H.: Supported in part by the generous funding of a Croucher Fellowship for Postdoctoral Research and by NSF award DMS-2023239. Part of this work was done during Jasper’s visit at the Simons Institute for the Theory of Computing.
  • Valiant, Paul: Supported in part by NSF award CCF-2127806 and IIS-1562657. Part of this work was done at the Institute for Advanced Study, partially supported by NSF award DMS-1926686, and indirectly supported by NSF award CCF-1900460.

Acknowledgements

We thank Avi Wigderson for insightful discussions on geometric intuitions for high-dimensional inequalities.

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