Iterative Constructions and Private Data Release

  • Anupam Gupta
  • Aaron Roth
  • Jonathan Ullman
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7194)

Abstract

In this paper we study the problem of approximately releasing the cut function of a graph while preserving differential privacy, and give new algorithms (and new analyses of existing algorithms) in both the interactive and non-interactive settings.

Our algorithms in the interactive setting are achieved by revisiting the problem of releasing differentially private, approximate answers to a large number of queries on a database. We show that several algorithms for this problem fall into the same basic framework, and are based on the existence of objects which we call iterative database construction algorithms. We give a new generic framework in which new (efficient) IDC algorithms give rise to new (efficient) interactive private query release mechanisms. Our modular analysis simplifies and tightens the analysis of previous algorithms, leading to improved bounds. We then give a new IDC algorithm (and therefore a new private, interactive query release mechanism) based on the Frieze/Kannan low-rank matrix decomposition. This new release mechanism gives an improvement on prior work in a range of parameters where the size of the database is comparable to the size of the data universe (such as releasing all cut queries on dense graphs).

We also give a non-interactive algorithm for efficiently releasing private synthetic data for graph cuts with error O(|V|1.5). Our algorithm is based on randomized response and a non-private implementation of the SDP-based, constant-factor approximation algorithm for cut-norm due to Alon and Naor. Finally, we give a reduction based on the IDC framework showing that an efficient, private algorithm for computing sufficiently accurate rank-1 matrix approximations would lead to an improved efficient algorithm for releasing private synthetic data for graph cuts. We leave finding such an algorithm as our main open problem.

References

  1. 1.
    Alon, N., Naor, A.: Approximating the cut-norm via Grothendieck’s inequality. SIAM J. Comput. 35(4), 787–803 (2006) (electronic) MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Barak, B., Chaudhuri, K., Dwork, C., Kale, S., McSherry, F., Talwar, K.: Privacy, accuracy, and consistency too: a holistic solution to contingency table release. In: PODS, pp. 273–282 (2007)Google Scholar
  3. 3.
    Bhaskara, A., Krishnaswamy, R., Talwar, K.: Unconditional differentially private mechanisms for linear queries (2011) (manuscript) Google Scholar
  4. 4.
    Blum, A., Dwork, C., McSherry, F., Nissim, K.: Practical privacy: the SuLQ framework. In: PODS, pp. 128–138 (2005)Google Scholar
  5. 5.
    Blum, A., Ligett, K., Roth, A.: A learning theory approach to non-interactive database privacy. In: STOC, pp. 609–618 (2008)Google Scholar
  6. 6.
    Chawla, S., Dwork, C., McSherry, F., Smith, A., Wee, H.: Toward Privacy in Public Databases. In: Kilian, J. (ed.) TCC 2005. LNCS, vol. 3378, pp. 363–385. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  7. 7.
    Dwork, C., McSherry, F., Nissim, K., Smith, A.: Calibrating Noise to Sensitivity in Private Data Analysis. In: Halevi, S., Rabin, T. (eds.) TCC 2006. LNCS, vol. 3876, pp. 265–284. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  8. 8.
    Dwork, C., Naor, M., Reingold, O., Rothblum, G., Vadhan, S.: On the complexity of differentially private data release: efficient algorithms and hardness results. In: STOC, pp. 381–390 (2009)Google Scholar
  9. 9.
    Dwork, C., Rothblum, G., Vadhan, S.: Boosting and differential privacy. In: FOCS, pp. 51–60 (2010)Google Scholar
  10. 10.
    Frieze, A., Kannan, R.: Quick approximation to matrices and applications. Combinatorica 19(2), 175–220 (1999)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Frieze, A., Kannan, R.: A simple algorithm for constructing Szemerédi’s regularity partition. Electron. J. Combin. 6, Research Paper 17, 7 (1999)Google Scholar
  12. 12.
    Gupta, A., Hardt, M., Roth, A., Ullman, J.: Privately Releasing Conjunctions and the Statistical Query Barrier. In: STOC. ACM, New York (2011)Google Scholar
  13. 13.
    Gupta, A., Roth, A., Ullman, J.: Iterative constructions and private data release, Arxiv preprint arXiv:1107.3731 (2011)Google Scholar
  14. 14.
    Hardt, M., Ligett, K., McSherry, F.: A simple and practical algorithm for differentially private data release, Arxiv preprint arXiv:1012.4763 (2011)Google Scholar
  15. 15.
    Hardt, M., Rothblum, G.: A multiplicative weights mechanism for privacy-preserving data analysis. In: FOCS, pp. 61–70 (2010)Google Scholar
  16. 16.
    Hardt, M., Talwar, K.: On the Geometry of Differential Privacy. In: STOC (2010)Google Scholar
  17. 17.
    Kasiviswanathan, S., Lee, H., Nissim, K., Raskhodnikova, S., Smith, A.: What Can We Learn Privately? In: FOCS, pp. 531–540 (2008)Google Scholar
  18. 18.
    McSherry, F., Talwar, K.: Mechanism design via differential privacy. In: FOCS (2007)Google Scholar
  19. 19.
    Roth, A., Roughgarden, T.: Interactive Privacy via the Median Mechanism. In: STOC (2010)Google Scholar
  20. 20.
    Trevisan, L., Tulsiani, M., Vadhan, S.P.: Regularity, boosting, and efficiently simulating every high-entropy distribution. In: CCC (2009)Google Scholar
  21. 21.
    Ullman, J., Vadhan, S.: PCPs and the Hardness of Generating Private Synthetic Data. In: Ishai, Y. (ed.) TCC 2011. LNCS, vol. 6597, pp. 400–416. Springer, Heidelberg (2011)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Anupam Gupta
    • 1
  • Aaron Roth
    • 2
  • Jonathan Ullman
    • 3
  1. 1.Computer Science DepartmentCarnegie Mellon UniversityPittsburghUSA
  2. 2.Department of Computer and Information ScienceUniversity of PennsylvaniaPhiladelphiaUSA
  3. 3.School of Engineering and Applied SciencesHarvard UniversityCambridgeUK

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