A Central Limit Theorem for Differentially Private Query Answering

Jinshuo Dong, Weijie J. Su, Linjun Zhang

Research output: Chapter in Book/Report/Conference proceedingConference contribution

4 Scopus citations

Abstract

Perhaps the single most important use case for differential privacy is to privately answer numerical queries, which is usually achieved by adding noise to the answer vector. The central question is, therefore, to understand which noise distribution optimizes the privacy-accuracy trade-off, especially when the dimension of the answer vector is high. Accordingly, an extensive literature has been dedicated to the question and the upper and lower bounds have been successfully matched up to constant factors [BUV18, SU17]. In this paper, we take a novel approach to address this important optimality question. We first demonstrate an intriguing central limit theorem phenomenon in the high-dimensional regime. More precisely, we prove that a mechanism is approximately Gaussian Differentially Private [DRS21] if the added noise satisfies certain conditions. In particular, densities proportional to e-kxkα p , where kxkp is the standard ℓp-norm, satisfies the conditions. Taking this perspective, we make use of the Cramer–Rao inequality and show a “uncertainty principle”-style result: the product of privacy parameter and the ℓ2-loss of the mechanism is lower bounded by the dimension. Furthermore, the Gaussian mechanism achieves the constant-sharp optimal privacy-accuracy trade-off among all such noises. Our findings are corroborated by numerical experiments.

Original languageEnglish (US)
Title of host publicationAdvances in Neural Information Processing Systems 34 - 35th Conference on Neural Information Processing Systems, NeurIPS 2021
EditorsMarc'Aurelio Ranzato, Alina Beygelzimer, Yann Dauphin, Percy S. Liang, Jenn Wortman Vaughan
PublisherNeural information processing systems foundation
Pages14759-14770
Number of pages12
ISBN (Electronic)9781713845393
StatePublished - 2021
Externally publishedYes
Event35th Conference on Neural Information Processing Systems, NeurIPS 2021 - Virtual, Online
Duration: Dec 6 2021Dec 14 2021

Publication series

NameAdvances in Neural Information Processing Systems
Volume18
ISSN (Print)1049-5258

Conference

Conference35th Conference on Neural Information Processing Systems, NeurIPS 2021
CityVirtual, Online
Period12/6/2112/14/21

All Science Journal Classification (ASJC) codes

  • Computer Networks and Communications
  • Information Systems
  • Signal Processing

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