A Differentially Private Linear-Time fPTAS for the Minimum Enclosing Ball Problem

Bar Mahpud; Or Sheffet

A Differentially Private Linear-Time fPTAS for the Minimum Enclosing Ball Problem

Bar-Ilan University - The Alexander Kofkin Faculty of Engineering

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

Abstract

The Minimum Enclosing Ball (MEB) problem is one of the most fundamental problems in clustering, with applications in operations research, statistics and computational geometry. In this works, we give the first linear time differentially private (DP) fPTAS for the Minimum Enclosing Ball problem, improving both on the runtime and the utility bound of the best known DP-PTAS for the problem, of Ghazi et al [21]. Given n points in R^d that are covered by the ball B(θ_opt, r_opt), our simple iterative DP-algorithm returns a ball B(θ, r) where r ≤ (1 + γ)r_opt and which leaves at most Õ((equation presented)) points uncovered in Õ(^n/γ2)-time. We also give a local-model version of our algorithm, that leaves at most Õ((equation presented)) points uncovered, improving on the n^0.67-bound of Nissim and Stemmer [31] (at the expense of other parameters). Lastly, we test our algorithm empirically and discuss open problems.

Original language	English
Title of host publication	Advances in Neural Information Processing Systems 35 - 36th Conference on Neural Information Processing Systems, NeurIPS 2022
Editors	S. Koyejo, S. Mohamed, A. Agarwal, D. Belgrave, K. Cho, A. Oh
Publisher	Neural information processing systems foundation
ISBN (Electronic)	9781713871088
State	Published - 2022
Event	36th Conference on Neural Information Processing Systems, NeurIPS 2022 - New Orleans, United States Duration: 28 Nov 2022 → 9 Dec 2022

Publication series

Name	Advances in Neural Information Processing Systems
Volume	35
ISSN (Print)	1049-5258

Conference

Conference	36th Conference on Neural Information Processing Systems, NeurIPS 2022
Country/Territory	United States
City	New Orleans
Period	28/11/22 → 9/12/22

Bibliographical note

Funding Information:
O.S. is supported by the BIU Center for Research in Applied Cryptography and Cyber Security in conjunction with the Israel National Cyber Bureau in the Prime Ministers Office, and by ISF grant no. 2559/20. Both authors thank the anonymous reviewers for terrific suggestions and advice on improving this paper.

Publisher Copyright:
© 2022 Neural information processing systems foundation. All rights reserved.

Cite this

Mahpud, B., & Sheffet, O. (2022). A Differentially Private Linear-Time fPTAS for the Minimum Enclosing Ball Problem. In S. Koyejo, S. Mohamed, A. Agarwal, D. Belgrave, K. Cho, & A. Oh (Eds.), Advances in Neural Information Processing Systems 35 - 36th Conference on Neural Information Processing Systems, NeurIPS 2022 (Advances in Neural Information Processing Systems; Vol. 35). Neural information processing systems foundation.

Mahpud, Bar ; Sheffet, Or. / A Differentially Private Linear-Time fPTAS for the Minimum Enclosing Ball Problem. Advances in Neural Information Processing Systems 35 - 36th Conference on Neural Information Processing Systems, NeurIPS 2022. editor / S. Koyejo ; S. Mohamed ; A. Agarwal ; D. Belgrave ; K. Cho ; A. Oh. Neural information processing systems foundation, 2022. (Advances in Neural Information Processing Systems).

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abstract = "The Minimum Enclosing Ball (MEB) problem is one of the most fundamental problems in clustering, with applications in operations research, statistics and computational geometry. In this works, we give the first linear time differentially private (DP) fPTAS for the Minimum Enclosing Ball problem, improving both on the runtime and the utility bound of the best known DP-PTAS for the problem, of Ghazi et al [21]. Given n points in Rd that are covered by the ball B(θopt, ropt), our simple iterative DP-algorithm returns a ball B(θ, r) where r ≤ (1 + γ)ropt and which leaves at most {\~O}((equation presented)) points uncovered in {\~O}(n/γ2)-time. We also give a local-model version of our algorithm, that leaves at most {\~O}((equation presented)) points uncovered, improving on the n0.67-bound of Nissim and Stemmer [31] (at the expense of other parameters). Lastly, we test our algorithm empirically and discuss open problems.",

author = "Bar Mahpud and Or Sheffet",

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Mahpud, B & Sheffet, O 2022, A Differentially Private Linear-Time fPTAS for the Minimum Enclosing Ball Problem. in S Koyejo, S Mohamed, A Agarwal, D Belgrave, K Cho & A Oh (eds), Advances in Neural Information Processing Systems 35 - 36th Conference on Neural Information Processing Systems, NeurIPS 2022. Advances in Neural Information Processing Systems, vol. 35, Neural information processing systems foundation, 36th Conference on Neural Information Processing Systems, NeurIPS 2022, New Orleans, United States, 28/11/22.

A Differentially Private Linear-Time fPTAS for the Minimum Enclosing Ball Problem. / Mahpud, Bar; Sheffet, Or.
Advances in Neural Information Processing Systems 35 - 36th Conference on Neural Information Processing Systems, NeurIPS 2022. ed. / S. Koyejo; S. Mohamed; A. Agarwal; D. Belgrave; K. Cho; A. Oh. Neural information processing systems foundation, 2022. (Advances in Neural Information Processing Systems; Vol. 35).

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

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Mahpud B, Sheffet O. A Differentially Private Linear-Time fPTAS for the Minimum Enclosing Ball Problem. In Koyejo S, Mohamed S, Agarwal A, Belgrave D, Cho K, Oh A, editors, Advances in Neural Information Processing Systems 35 - 36th Conference on Neural Information Processing Systems, NeurIPS 2022. Neural information processing systems foundation. 2022. (Advances in Neural Information Processing Systems).

A Differentially Private Linear-Time fPTAS for the Minimum Enclosing Ball Problem

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Bibliographical note

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