Advertisement

Abstract

We study local filters for two properties of functions f:{0,1} d  → ℝ: the Lipschitz property and monotonicity. A local filter with additive error a is a randomized algorithm that is given black-box access to a function f and a query point x in the domain of f. Its output is a value F(x), such that (i) the reconstructed function F(x) satisfies the property (in our case, is Lipschitz or monotone) and (ii) if the input function f satisfies the property, then for every point x in the domain (with high constant probability) the reconstructed value F(x) differs from f(x) by at most a. Local filters were introduced by Saks and Seshadhri (SICOMP 2010) and the relaxed definition we study is due to Bhattacharyya et al.(RANDOM 2010), except that we further relax it by allowing additive error. Local filters for Lipschitz and monotone functions have applications to areas such as data privacy.

We show that every local filter for Lipschitz or monotone functions runs in time exponential in the dimension d, even when the filter is allowed significant additive error. Prior lower bounds (for local filters with no additive error, i.e., with a = 0) applied only to more restrictive class of filters, e.g., nonadaptive filters. To prove our lower bounds, we construct families of hard functions and show that lookups of a local filter on these functions are captured by a combinatorial object that we call a c-connector. Then we present a lower bound on the maximum outdegree of a c-connector, and show that it implies the desired bounds on the running time of local filters. Our lower bounds, in particular, imply the same bound on the running time for a class of privacy mechanisms.

Keywords

Monotone Function Lipschitz Function Additive Error Query Point Random Seed 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Ailon, N., Chazelle, B., Comandur, S., Liu, D.: Property-preserving data reconstruction. Algorithmica 51(2), 160–182 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Alon, N., Rubinfeld, R., Vardi, S., Xie, N.: Space-efficient local computation algorithms. In: Rabani, Y. (ed.) SODA, pp. 1132–1139. SIAM (2012)Google Scholar
  3. 3.
    Awasthi, P., Jha, M., Molinaro, M., Raskhodnikova, S.: Limitations of local filters of lipschitz and monotone functions. Electronic Colloquium on Computational Complexity (ECCC) TR12-075 (2012)Google Scholar
  4. 4.
    Bhaskar, R., Bhowmick, A., Goyal, V., Laxman, S., Thakurta, A.: Noiseless Database Privacy. In: Lee, D.H., Wans, X. (eds.) ASIACRYPT 2011. LNCS, vol. 7073, pp. 215–232. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  5. 5.
    Bhattacharyya, A., Grigorescu, E., Jha, M., Jung, K., Raskhodnikova, S., Woodruff, D.P.: Lower bounds for local monotonicity reconstruction from transitive-closure spanners. SIAM J. Discrete Math. 26(2), 618–646 (2012)CrossRefGoogle Scholar
  6. 6.
    Bhattacharyya, A., Grigorescu, E., Jung, K., Raskhodnikova, S., Woodruff, D.P.: Transitive-closure spanners. In: SODA, pp. 932–941 (2009)Google Scholar
  7. 7.
    Blum, M., Luby, M., Rubinfeld, R.: Self-testing/correcting with applications to numerical problems. J. Comput. Syst. Sci. 47(3), 549–595 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Dwork, C., Kenthapadi, K., McSherry, F., Mironov, I., Naor, M.: Our Data, Ourselves: Privacy Via Distributed Noise Generation. In: Vaudenay, S. (ed.) EUROCRYPT 2006. LNCS, vol. 4004, pp. 486–503. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  9. 9.
    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
  10. 10.
    Goldreich, O., Goldwasser, S., Ron, D.: Property testing and its connection to learning and approximation. J. ACM 45(4), 653–750 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Jha, M., Raskhodnikova, S.: Testing and reconstruction of Lipschitz functions with applications to data privacy. In: IEEE FOCS, pp. 433–442 (2011) full version available at, http://eccc.hpi-web.de/report/2011/057/
  12. 12.
    Katz, J., Trevisan, L.: On the efficiency of local decoding procedures for error-correcting codes. In: STOC, pp. 80–86 (2000)Google Scholar
  13. 13.
    Raskhodnikova, S.: Transitive-Closure Spanners: A Survey. In: Goldreich, O. (ed.) Property Testing. LNCS, vol. 6390, pp. 167–196. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  14. 14.
    Rubinfeld, R., Sudan, M.: Robust characterization of polynomials with applications to program testing. SIAM J. Comput. 25(2), 252–271 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Rubinfeld, R., Tamir, G., Vardi, S., Xie, N.: Fast local computation algorithms. In: ICS, pp. 223–238 (2011)Google Scholar
  16. 16.
    Saks, M.E., Seshadhri, C.: Local monotonicity reconstruction. SIAM J. Comput. 39(7), 2897–2926 (2010)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Pranjal Awasthi
    • 1
  • Madhav Jha
    • 2
  • Marco Molinaro
    • 1
  • Sofya Raskhodnikova
    • 2
  1. 1.Carnegie Mellon UniversityUSA
  2. 2.Pennsylvania State UniversityUSA

Personalised recommendations