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Unconditional differentially private mechanisms for linear queries

Published: 19 May 2012 Publication History

Abstract

We investigate the problem of designing differentially private mechanisms for a set of d linear queries over a database, while adding as little error as possible. Hardt and Talwar [HT10] related this problem to geometric properties of a convex body defined by the set of queries and gave a O(log3 d)-approximation to the minimum l22 error, assuming a conjecture from convex geometry called the Slicing or Hyperplane conjecture. In this work we give a mechanism that works unconditionally, and also gives an improved O(log2 d) approximation to the expected l22 error. We remove the dependence on the Slicing conjecture by using a result of Klartag [Kla06] that shows that any convex body is close to one for which the conjecture holds; our main contribution is in making this result constructive by using recent techniques of Dadush, Peikert and Vempala [DPV10]. The improvement in approximation ratio relies on a stronger lower bound we derive on the optimum. This new lower bound goes beyond the packing argument that has traditionally been used in Differential Privacy and allows us to add the packing lower bounds obtained from orthogonal subspaces. We are able to achieve this via a symmetrization argument which argues that there always exists a near optimal differentially private mechanism which adds noise that is independent of the input database! We believe this result should be of independent interest, and also discuss some interesting consequences.

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References

[1]
Keith M. Ball. Logarithmically concave functions and sections of convex sets in Rn. Studia Mathematica, 1988:69--84.
[2]
7}BarakCDKMT07Boaz Barak, Kamalika Chaudhuri, Cynthia Dwork, Satyen Kale, Frank McSherry, and Kunal Talwar. Privacy, accuracy, and consistency too: a holistic solution to contingency table release. In Proc. 26th PODS, pages 273--282. ACM, 2007.
[3]
Avrim Blum, Katrina Ligett, and Aaron Roth. A learning theory approach to non-interactive database privacy. In STOC '08: Proceedings of the 40th annual ACM symposium on Theory of computing, pages 609--618, New York, NY, USA, 2008. ACM.
[4]
Hai Brenner and Kobbi Nissim. Impossibility of differentially private universally optimal mechanisms. In Proceedings of the 2010 IEEE 51st Annual Symposium on Foundations of Computer Science, FOCS '10, pages 71--80, Washington, DC, USA, 2010. IEEE Computer Society.
[5]
Anindya De. Lower bounds in differential privacy. CoRR, abs/1107.2183, 2011.
[6]
Martin E. Dyer, Alan M. Frieze, and Ravi Kannan. A random polynomial time algorithm for approximating the volume of convex bodies. J. ACM, 38(1):1--17, 1991.
[7]
Alexander Dinghas. Über eine Klasse superadditiver Mengenfunktionale von Brunn-Minkowski-Lusternikschem Typus. Math. Z., 68:111--125, 1957.
[8]
Cynthia Dwork, Frank McSherry, Kobbi Nissim, and Adam Smith. Calibrating noise to sensitivity in private data analysis. In Proc. 3rd TCC, pages 265--284. Springer, 2006.
[9]
Cynthia Dwork, Frank McSherry, and Kunal Talwar. The price of privacy and the limits of LP decoding. In Proc. 39th STOC, pages 85--94. ACM, 2007.
[10]
Irit Dinur and Kobbi Nissim. Revealing information while preserving privacy. In Proc. 22nd PODS, pages 202--210. ACM, 2003.
[11]
9}DworkNRRV09Cynthia Dwork, Moni Naor, Omer Reingold, Guy N. Rothblum, and Salil Vadhan. On the complexity of differentially private data release: efficient algorithms and hardness results. In Proceedings of the 41st annual ACM symposium on Theory of computing, STOC '09, pages 381--390, New York, NY, USA, 2009. ACM.
[12]
Daniel Dadush, Chris Peikert, and Santosh Vempala. Enumerative algorithms for the shortest and closest lattice vector problems in any norm via m-ellipsoid coverings. CoRR, abs/1011.5666, 2010.
[13]
Cynthia Dwork, Guy N. Rothblum, and Salil Vadhan. Boosting and differential privacy. In Proceedings of the 2010 IEEE 51st Annual Symposium on Foundations of Computer Science, FOCS '10, pages 51--60, Washington, DC, USA, 2010. IEEE Computer Society.
[14]
Cynthia Dwork and Sergey Yekhanin. New efficient attacks on statistical disclosure control mechanisms. In Proc. 28th CRYPTO, pages 469--480. Springer, 2008.
[15]
Apostolos Giannopoulos. Notes on isotropic convex bodies., 2003.
[16]
Arpita Ghosh, Tim Roughgarden, and Mukund Sundararajan. Universally utility-maximizing privacy mechanisms. In STOC, pages 351--360, 2009.
[17]
Mangesh Gupte and Mukund Sundararajan. Universally optimal privacy mechanisms for minimax agents. In Proceedings of the twenty-ninth ACM SIGMOD-SIGACT-SIGART symposium on Principles of database systems, PODS '10, pages 135--146, New York, NY, USA, 2010. ACM.
[18]
Moritz Hardt and Guy N. Rothblum. A multiplicative weights mechanism for privacy-preserving data analysis. In Proceedings of the 2010 IEEE 51st Annual Symposium on Foundations of Computer Science, FOCS '10, pages 61--70, Washington, DC, USA, 2010. IEEE Computer Society.
[19]
Moritz Hardt and Kunal Talwar. On the geometry of differential privacy. In Proceedings of the 42nd ACM symposium on Theory of computing, STOC '10, pages 705--714, New York, NY, USA, 2010. ACM.
[20]
B. Klartag. An isomorphic version of the slicing problem. Journal of Functional Analysis, 218(2):372 -- 394, 2005.
[21]
B. Klartag. On convex perturbations with a bounded isotropic constant. Geometric And Functional Analysis, 16:1274--1290, 2006. 10.1007/s00039-006-0588--1.
[22]
Boaz Klartag and Vitali Milman. Geometry of log-concave functions and measures. Geom. Dedicata, 170:169--182, 2005.
[23]
Shiva Prasad Kasiviswanathan, Mark Rudelson, Adam Smith, and Jonathan Ullman. The price of privately releasing contingency tables and the spectra of random matrices with correlated rows. In Proceedings of the 42nd ACM symposium on Theory of computing, STOC '10, pages 775--784, New York, NY, USA, 2010. ACM.
[24]
0}LiHRMM10Chao Li, Michael Hay, Vibhor Rastogi, Gerome Miklau, and Andrew McGregor. Optimizing linear counting queries under differential privacy. In Proceedings of the twenty-ninth ACM SIGMOD-SIGACT-SIGART symposium on Principles of database systems, PODS '10, pages 123--134, New York, NY, USA, 2010. ACM.
[25]
László Lovász and Miklós Simonovits. Random walks in a convex body and an improved volume algorithm. Random Struct. Algorithms, 4(4):359--412, 1993.
[26]
László Lovász and Santosh Vempala. Simulated annealing in convex bodies and an *(4) volume algorithm. J. Comput. Syst. Sci., 72(2):392--417, 2006.
[27]
Frank McSherry and Ilya Mironov. Differentially private recommender systems: building privacy into the net. In Proc. 15th KDD, pages 627--636. ACM, 2009.
[28]
V. Milman and A. Pajor. Isotropic position and inertia ellipsoids and zonoids of the unit ball of a normed n-dimensional space. Geometric Aspects of Functional Analysis, 1376:64--104, 1989.
[29]
V.D. Milman and A. Pajor. Isotropic position and inertia ellipsoids and zonoids of the unit ball of a normed n-dimensional space. Geometric Aspects of Functional Analysis, 1376:64--104, 1989.
[30]
Frank McSherry and Kunal Talwar. Mechanism design via differential privacy. In Proc. 48th FOCS, pages 94--103. IEEE, 2007.
[31]
Kobbi Nissim, Sofya Raskhodnikova, and Adam Smith. Smooth sensitivity and sampling in private data analysis. In STOC '07: Proceedings of the thirty-ninth annual ACM symposium on Theory of computing, pages 75--84, New York, NY, USA, 2007. ACM.
[32]
Vibhor Rastogi, Sungho Hong, and Dan Suciu. The boundary between privacy and utility in data publishing. In Christoph Koch, Johannes Gehrke, Minos N. Garofalakis, Divesh Srivastava, Karl Aberer, Anand Deshpande, Daniela Florescu, Chee Yong Chan, Venkatesh Ganti, Carl-Christian Kanne, Wolfgang Klas, and Erich J. Neuhold, editors, VLDB, pages 531--542. ACM, 2007.
[33]
Aaron Roth and Tim Roughgarden. Interactive privacy via the median mechanism. In Proceedings of the 42nd ACM symposium on Theory of computing, STOC '10, pages 765--774, New York, NY, USA, 2010. ACM.
[34]
Santosh Vempala. Geometric random walks: a survey. MSRI Volume on Combinatorial and Computational Geometry, 52:577--616, 2005.

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    cover image ACM Conferences
    STOC '12: Proceedings of the forty-fourth annual ACM symposium on Theory of computing
    May 2012
    1310 pages
    ISBN:9781450312455
    DOI:10.1145/2213977
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    Published: 19 May 2012

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    Author Tags

    1. differential privacy
    2. slicing conjecture

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    May 19 - 22, 2012
    New York, New York, USA

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    • (2023)On computing pairwise statistics with local differential privacyProceedings of the 37th International Conference on Neural Information Processing Systems10.5555/3666122.3667303(27129-27146)Online publication date: 10-Dec-2023
    • (2023)Differentially Private Data Release over Multiple TablesProceedings of the 42nd ACM SIGMOD-SIGACT-SIGAI Symposium on Principles of Database Systems10.1145/3584372.3588665(207-219)Online publication date: 18-Jun-2023
    • (2023)Differentially Private Range Query on Shortest PathsAlgorithms and Data Structures10.1007/978-3-031-38906-1_23(340-370)Online publication date: 28-Jul-2023
    • (2021)A central limit theorem for differentially private query answeringProceedings of the 35th International Conference on Neural Information Processing Systems10.5555/3540261.3541392(14759-14770)Online publication date: 6-Dec-2021
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    • (2021)Mobility-Aware Differentially Private Trajectory for Privacy-Preserving Continual CrowdsourcingIEEE Access10.1109/ACCESS.2021.30582119(26362-26376)Online publication date: 2021
    • (2020)A workload-adaptive mechanism for linear queries under local differential privacyProceedings of the VLDB Endowment10.14778/3407790.340779813:12(1905-1918)Online publication date: 14-Sep-2020
    • (2020)The power of factorization mechanisms in local and central differential privacyProceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing10.1145/3357713.3384297(425-438)Online publication date: 22-Jun-2020
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