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General Gaussian Noise Mechanisms and Their Optimality for Unbiased Mean Estimation

Authors Aleksandar Nikolov, Haohua Tang

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Aleksandar Nikolov
  • University of Toronto, Canada
Haohua Tang
  • University of Toronto, Canada

Funding

This research was supported by an NSERC Discovery Grant.

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Aleksandar Nikolov and Haohua Tang. General Gaussian Noise Mechanisms and Their Optimality for Unbiased Mean Estimation. In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 85:1-85:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.ITCS.2024.85

Abstract

We investigate unbiased high-dimensional mean estimators in differential privacy. We consider differentially private mechanisms whose expected output equals the mean of the input dataset, for every dataset drawn from a fixed bounded domain K in ℝ^d. A classical approach to private mean estimation is to compute the true mean and add unbiased, but possibly correlated, Gaussian noise to it. In the first part of this paper, we study the optimal error achievable by a Gaussian noise mechanism for a given domain K, when the error is measured in the 𝓁_p norm for some p ≥ 2. We give algorithms that compute the optimal covariance for the Gaussian noise for a given K under suitable assumptions, and prove a number of nice geometric properties of the optimal error. These results generalize the theory of factorization mechanisms from domains K that are symmetric and finite (or, equivalently, symmetric polytopes) to arbitrary bounded domains. In the second part of the paper we show that Gaussian noise mechanisms achieve nearly optimal error among all private unbiased mean estimation mechanisms in a very strong sense. In particular, for every input dataset, an unbiased mean estimator satisfying concentrated differential privacy introduces approximately at least as much error as the best Gaussian noise mechanism. We extend this result to local differential privacy, and to approximate differential privacy, but for the latter the error lower bound holds either for a dataset or for a neighboring dataset, and this relaxation is necessary.

Subject Classification

ACM Subject Classification
  • Theory of computation → Theory and algorithms for application domains
Keywords
  • differential privacy
  • mean estimation
  • unbiased estimator
  • instance optimality

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