Goodness-of-Fit Testing for Hölder Continuous Densities Under Local Differential Privacy

Amandine Dubois; Thomas B. Berrett; Cristina Butucea

doi:10.1007/978-3-031-30114-8_2

Goodness-of-Fit Testing for Hölder Continuous Densities Under Local Differential Privacy

Amandine Dubois
, Thomas B. Berrett
, Cristina Butucea

ENSAI Ecole Nationale de la Statistique et de lAnalyse de lInformation
University of Warwick

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

Abstract

We address the problem of goodness-of-fit testing for Hölder continuous densities under local differential privacy constraints. We study minimax separation rates when only non-interactive privacy mechanisms are allowed to be used and when both non-interactive and sequentially interactive can be used for privatisation. We propose privacy mechanisms and associated testing procedures whose analysis enables us to obtain upper bounds on the minimax rates. These results are complemented with lower bounds. By comparing these bounds, we show that the proposed privacy mechanisms and tests are optimal up to at most a logarithmic factor for several choices of f₀ including densities from uniform, normal, Beta, Cauchy, Pareto, exponential distributions. In particular, we observe that the results are deteriorated in the private setting compared to the non-private one. Moreover, we show that sequentially interactive mechanisms improve upon the results obtained when considering only non-interactive privacy mechanisms.

Original language	English
Title of host publication	Foundations of Modern Statistics - Festschrift in Honor of Vladimir Spokoiny
Editors	Denis Belomestny, Cristina Butucea, Enno Mammen, Eric Moulines, Markus Reiß, Vladimir V. Ulyanov
Publisher	Springer
Pages	53-119
Number of pages	67
ISBN (Print)	9783031301131
DOIs	https://doi.org/10.1007/978-3-031-30114-8_2
Publication status	Published - 1 Jan 2023
Event	International conference on Foundations of Modern Statistics, FMS 2019 - Berlin, Germany Duration: 6 Nov 2019 → 8 Nov 2019

Publication series

Name	Springer Proceedings in Mathematics and Statistics
Volume	425
ISSN (Print)	2194-1009
ISSN (Electronic)	2194-1017

Conference

Conference	International conference on Foundations of Modern Statistics, FMS 2019
Country/Territory	Germany
City	Berlin
Period	6/11/19 → 8/11/19

Keywords

Goodness-of-fit test
Local differential privacy
Minimax separation rates
Privacy mechanims
Total variation separation distance

Access to Document

10.1007/978-3-031-30114-8_2

Cite this

Dubois, A., Berrett, T. B., & Butucea, C. (2023). Goodness-of-Fit Testing for Hölder Continuous Densities Under Local Differential Privacy. In D. Belomestny, C. Butucea, E. Mammen, E. Moulines, M. Reiß, & V. V. Ulyanov (Eds.), Foundations of Modern Statistics - Festschrift in Honor of Vladimir Spokoiny (pp. 53-119). (Springer Proceedings in Mathematics and Statistics; Vol. 425). Springer. https://doi.org/10.1007/978-3-031-30114-8_2

Dubois, Amandine ; Berrett, Thomas B. ; Butucea, Cristina. / Goodness-of-Fit Testing for Hölder Continuous Densities Under Local Differential Privacy. Foundations of Modern Statistics - Festschrift in Honor of Vladimir Spokoiny. editor / Denis Belomestny ; Cristina Butucea ; Enno Mammen ; Eric Moulines ; Markus Reiß ; Vladimir V. Ulyanov. Springer, 2023. pp. 53-119 (Springer Proceedings in Mathematics and Statistics).

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title = "Goodness-of-Fit Testing for H{\"o}lder Continuous Densities Under Local Differential Privacy",

abstract = "We address the problem of goodness-of-fit testing for H{\"o}lder continuous densities under local differential privacy constraints. We study minimax separation rates when only non-interactive privacy mechanisms are allowed to be used and when both non-interactive and sequentially interactive can be used for privatisation. We propose privacy mechanisms and associated testing procedures whose analysis enables us to obtain upper bounds on the minimax rates. These results are complemented with lower bounds. By comparing these bounds, we show that the proposed privacy mechanisms and tests are optimal up to at most a logarithmic factor for several choices of f0 including densities from uniform, normal, Beta, Cauchy, Pareto, exponential distributions. In particular, we observe that the results are deteriorated in the private setting compared to the non-private one. Moreover, we show that sequentially interactive mechanisms improve upon the results obtained when considering only non-interactive privacy mechanisms.",

keywords = "Goodness-of-fit test, Local differential privacy, Minimax separation rates, Privacy mechanims, Total variation separation distance",

author = "Amandine Dubois and Berrett, \{Thomas B.\} and Cristina Butucea",

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Dubois, A, Berrett, TB & Butucea, C 2023, Goodness-of-Fit Testing for Hölder Continuous Densities Under Local Differential Privacy. in D Belomestny, C Butucea, E Mammen, E Moulines, M Reiß & VV Ulyanov (eds), Foundations of Modern Statistics - Festschrift in Honor of Vladimir Spokoiny. Springer Proceedings in Mathematics and Statistics, vol. 425, Springer, pp. 53-119, International conference on Foundations of Modern Statistics, FMS 2019, Berlin, Germany, 6/11/19. https://doi.org/10.1007/978-3-031-30114-8_2

Goodness-of-Fit Testing for Hölder Continuous Densities Under Local Differential Privacy. / Dubois, Amandine; Berrett, Thomas B.; Butucea, Cristina.
Foundations of Modern Statistics - Festschrift in Honor of Vladimir Spokoiny. ed. / Denis Belomestny; Cristina Butucea; Enno Mammen; Eric Moulines; Markus Reiß; Vladimir V. Ulyanov. Springer, 2023. p. 53-119 (Springer Proceedings in Mathematics and Statistics; Vol. 425).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

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AU - Butucea, Cristina

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N2 - We address the problem of goodness-of-fit testing for Hölder continuous densities under local differential privacy constraints. We study minimax separation rates when only non-interactive privacy mechanisms are allowed to be used and when both non-interactive and sequentially interactive can be used for privatisation. We propose privacy mechanisms and associated testing procedures whose analysis enables us to obtain upper bounds on the minimax rates. These results are complemented with lower bounds. By comparing these bounds, we show that the proposed privacy mechanisms and tests are optimal up to at most a logarithmic factor for several choices of f0 including densities from uniform, normal, Beta, Cauchy, Pareto, exponential distributions. In particular, we observe that the results are deteriorated in the private setting compared to the non-private one. Moreover, we show that sequentially interactive mechanisms improve upon the results obtained when considering only non-interactive privacy mechanisms.

AB - We address the problem of goodness-of-fit testing for Hölder continuous densities under local differential privacy constraints. We study minimax separation rates when only non-interactive privacy mechanisms are allowed to be used and when both non-interactive and sequentially interactive can be used for privatisation. We propose privacy mechanisms and associated testing procedures whose analysis enables us to obtain upper bounds on the minimax rates. These results are complemented with lower bounds. By comparing these bounds, we show that the proposed privacy mechanisms and tests are optimal up to at most a logarithmic factor for several choices of f0 including densities from uniform, normal, Beta, Cauchy, Pareto, exponential distributions. In particular, we observe that the results are deteriorated in the private setting compared to the non-private one. Moreover, we show that sequentially interactive mechanisms improve upon the results obtained when considering only non-interactive privacy mechanisms.

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KW - Minimax separation rates

KW - Privacy mechanims

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Dubois A, Berrett TB, Butucea C. Goodness-of-Fit Testing for Hölder Continuous Densities Under Local Differential Privacy. In Belomestny D, Butucea C, Mammen E, Moulines E, Reiß M, Ulyanov VV, editors, Foundations of Modern Statistics - Festschrift in Honor of Vladimir Spokoiny. Springer. 2023. p. 53-119. (Springer Proceedings in Mathematics and Statistics). doi: 10.1007/978-3-031-30114-8_2

Goodness-of-Fit Testing for Hölder Continuous Densities Under Local Differential Privacy

Abstract

Publication series

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Keywords

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