A Framework of Metrics for Differential Privacy from Local Sensitivity

Peeter Laud 1 , Alisa Pankova 2  and Martin Pettai 3
  • 1 Cybernetica AS,
  • 2 Cybernetica AS,
  • 3 Cybernetica AS,

Abstract

The meaning of differential privacy (DP) is tightly bound with the notion of distance on databases, typically defined as the number of changed rows. Considering the semantics of data, this metric may be not the most suitable one, particularly when a distance comes out as larger than the data owner desired (which would undermine privacy). In this paper, we give a mechanism to specify continuous metrics that depend on the locations and amounts of changes in a much more nuanced manner. Our metrics turn the set of databases into a Banach space. In order to construct DP information release mechanisms based on our metrics, we introduce derivative sensitivity, an analogue to local sensitivity for continuous functions. We use this notion in an analysis that determines the amount of noise to be added to the result of a database query in order to obtain a certain level of differential privacy, and demonstrate that derivative sensitivity allows us to employ powerful mechanisms from calculus to perform the analysis for a variety of queries. We have implemented the analyzer and evaluated its efficiency and precision.

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  • [1] TPC BENCHMARKTM H, revision 2.17.3. Transaction Processing Performance Council, 2017. http://www.tpc.org/TPC_Documents_Current_Versions/pdf/tpc-h_v2.17.3.pdf.

  • [2] Myrto Arapinis, Diego Figueira, and Marco Gaboardi. Sensitivity of counting queries. In Ioannis Chatzigiannakis, Michael Mitzenmacher, Yuval Rabani, and Davide Sangiorgi, editors, 43rd International Colloquium on Automata, Languages, and Programming, ICALP 2016, July 11-15, 2016, Rome, Italy, volume 55 of LIPIcs, pages 120:1–120:13. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2016.

  • [3] Jordan Awan and Aleksandra Slavkovic. Structure and sensitivity in differential privacy: Comparing k-norm mechanisms. arXiv preprint arXiv:1801.09236, 2018.

  • [4] L.W. Baggett. Functional Analysis: A Primer. Chapman & Hall Pure and Applied Mathematics. Taylor & Francis, 1991.

  • [5] Konstantinos Chatzikokolakis, Miguel E. Andrés, Nicolás Emilio Bordenabe, and Catuscia Palamidessi. Broadening the scope of differential privacy using metrics. In Emiliano De Cristofaro and Matthew Wright, editors, Privacy Enhancing Technologies - 13th International Symposium, PETS 2013, Bloomington, IN, USA, July 10-12, 2013. Proceedings, volume 7981 of Lecture Notes in Computer Science, pages 82–102. Springer, 2013.

  • [6] Konstantinos Chatzikokolakis, Catuscia Palamidessi, and Marco Stronati. Geo-indistinguishability: A principled approach to location privacy. In Raja Natarajan, Gautam Barua, and Manas Ranjan Patra, editors, Distributed Computing and Internet Technology - 11th International Conference, ICDCIT 2015, Bhubaneswar, India, February 5-8, 2015. Proceedings, volume 8956 of Lecture Notes in Computer Science, pages 49–72. Springer, 2015.

  • [7] Yan Chen, Ashwin Machanavajjhala, Michael Hay, and Gerome Miklau. Pegasus: Data-adaptive differentially private stream processing. In Bhavani M. Thuraisingham, David Evans, Tal Malkin, and Dongyan Xu, editors, Proceedings of the 2017 ACM SIGSAC Conference on Computer and Communications Security, CCS 2017, Dallas, TX, USA, October 30 - November 03, 2017, pages 1375–1388. ACM, 2017.

  • [8] Cynthia Dwork. Differential privacy. In Michele Bugliesi, Bart Preneel, Vladimiro Sassone, and Ingo Wegener, editors, Automata, Languages and Programming, 33rd International Colloquium, ICALP 2006, Venice, Italy, July 10-14, 2006, Proceedings, Part II, volume 4052 of Lecture Notes in Computer Science, pages 1–12. Springer, 2006.

  • [9] Cynthia Dwork, Frank McSherry, Kobbi Nissim, and Adam D. Smith. Calibrating noise to sensitivity in private data analysis. In Shai Halevi and Tal Rabin, editors, Theory of Cryptography, Third Theory of Cryptography Conference, TCC 2006, New York, NY, USA, March 4-7, 2006, Proceedings, volume 3876 of Lecture Notes in Computer Science, pages 265–284. Springer, 2006.

  • [10] Cynthia Dwork, Moni Naor, Toniann Pitassi, and Guy N. Rothblum. Differential privacy under continual observation. In Schulman [27], pages 715–724.

  • [11] Hamid Ebadi and David Sands. Featherweight PINQ. Journal of Privacy and Security, 7(2):159–184, 2016.

  • [12] Hamid Ebadi, David Sands, and Gerardo Schneider. Differential privacy: Now it’s getting personal. In Sriram K. Rajamani and David Walker, editors, Proceedings of the 42nd Annual ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages, POPL 2015, Mumbai, India, January 15-17, 2015, pages 69–81. ACM, 2015.

  • [13] Ehab ElSalamouny, Konstantinos Chatzikokolakis, and Catuscia Palamidessi. Generalized differential privacy: Regions of priors that admit robust optimal mechanisms. In Franck van Breugel, Elham Kashefi, Catuscia Palamidessi, and Jan Rutten, editors, Horizons of the Mind. A Tribute to Prakash Panangaden - Essays Dedicated to Prakash Panangaden on the Occasion of His 60th Birthday, volume 8464 of Lecture Notes in Computer Science, pages 292–318. Springer, 2014.

  • [14] Marco Gaboardi, Andreas Haeberlen, Justin Hsu, Arjun Narayan, and Benjamin C. Pierce. Linear dependent types for differential privacy. In Roberto Giacobazzi and Radhia Cousot, editors, The 40th Annual ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages, POPL ’13, Rome, Italy - January 23 - 25, 2013, pages 357–370. ACM, 2013.

  • [15] Moritz Hardt and Kunal Talwar. On the geometry of differential privacy. In Schulman [27], pages 705–714.

  • [16] Michael Hay, Ashwin Machanavajjhala, Gerome Miklau, Yan Chen, and Dan Zhang. Principled evaluation of differentially private algorithms using dpbench. In Fatma Özcan, Georgia Koutrika, and Sam Madden, editors, Proceedings of the 2016 International Conference on Management of Data, SIGMOD Conference 2016, San Francisco, CA, USA, June 26 - July 01, 2016, pages 139–154. ACM, 2016.

  • [17] Xi He, Ashwin Machanavajjhala, and Bolin Ding. Blowfish privacy: tuning privacy-utility trade-offs using policies. In Curtis E. Dyreson, Feifei Li, and M. Tamer Özsu, editors, International Conference on Management of Data, SIGMOD 2014, Snowbird, UT, USA, June 22-27, 2014, pages 1447–1458. ACM, 2014.

  • [18] Noah M. Johnson, Joseph P. Near, and Dawn Song. Towards practical differential privacy for SQL queries. Proceedings of the VLDB Endowment, 11(5):526–539, 2018.

  • [19] Shiva Prasad Kasiviswanathan, Kobbi Nissim, Sofya Raskhodnikova, and Adam Smith. Analyzing graphs with node differential privacy. In Amit Sahai, editor, Theory of Cryptography, pages 457–476, Berlin, Heidelberg, 2013. Springer Berlin Heidelberg.

  • [20] Daniel Kifer and Ashwin Machanavajjhala. Pufferfish: A framework for mathematical privacy definitions. ACM Trans. Database Syst., 39(1):3:1–3:36, 2014.

  • [21] Ker-I. Ko and Harvey Friedman. Computational complexity of real functions. Theoretical Computer Science, 20:323–352, 1982.

  • [22] Jaewoo Lee and Chris Clifton. How much is enough? choosing ∈ for differential privacy. In Xuejia Lai, Jianying Zhou, and Hui Li, editors, Information Security, 14th International Conference, ISC 2011, Xi’an, China, October 26-29, 2011. Proceedings, volume 7001 of Lecture Notes in Computer Science, pages 325–340. Springer, 2011.

  • [23] Frank McSherry. Privacy integrated queries: an extensible platform for privacy-preserving data analysis. In Ugur Çetintemel, Stanley B. Zdonik, Donald Kossmann, and Nesime Tatbul, editors, Proceedings of the ACM SIGMOD International Conference on Management of Data, SIGMOD 2009, Providence, Rhode Island, USA, June 29 - July 2, 2009, pages 19–30. ACM, 2009.

  • [24] Kobbi Nissim, Sofya Raskhodnikova, and Adam D. Smith. Smooth sensitivity and sampling in private data analysis. In David S. Johnson and Uriel Feige, editors, Proceedings of the 39th Annual ACM Symposium on Theory of Computing, San Diego, California, USA, June 11-13, 2007, pages 75–84. ACM, 2007.

  • [25] Catuscia Palamidessi and Marco Stronati. Differential privacy for relational algebra: Improving the sensitivity bounds via constraint systems. In Herbert Wiklicky and Mieke Massink, editors, Proceedings 10th Workshop on Quantitative Aspects of Programming Languages and Systems, QAPL 2012, Tallinn, Estonia, 31 March and 1 April 2012., volume 85 of EPTCS, pages 92–105, 2012.

  • [26] Jason Reed and Benjamin C. Pierce. Distance makes the types grow stronger: a calculus for differential privacy. In Paul Hudak and Stephanie Weirich, editors, Proceeding of the 15th ACM SIGPLAN international conference on Functional programming, ICFP 2010, Baltimore, Maryland, USA, September 27-29, 2010, pages 157–168. ACM, 2010.

  • [27] Leonard J. Schulman, editor. Proceedings of the 42nd ACM Symposium on Theory of Computing, STOC 2010, Cambridge, Massachusetts, USA, 5-8 June 2010. ACM, 2010.

  • [28] Xi Wu, Fengan Li, Arun Kumar, Kamalika Chaudhuri, Somesh Jha, and Jeffrey F. Naughton. Bolt-on differential privacy for scalable stochastic gradient descent-based analytics. In Semih Salihoglu, Wenchao Zhou, Rada Chirkova, Jun Yang, and Dan Suciu, editors, Proceedings of the 2017 ACM International Conference on Management of Data, SIGMOD Conference 2017, Chicago, IL, USA, May 14-19, 2017, pages 1307–1322. ACM, 2017.

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