Limitations of local filters of Lipschitz and monotone functions

Pranjal Awasthi, Madhav Jha, Marco Molinaro, Sofya Raskhodnikova

Research output: Chapter in Book/Report/Conference proceedingConference contribution

2 Scopus citations

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.

Original languageEnglish (US)
Title of host publicationApproximation, Randomization, and Combinatorial Optimization
Subtitle of host publicationAlgorithms and Techniques - 15th International Workshop, APPROX 2012, and 16th International Workshop, RANDOM 2012, Proceedings
Pages374-386
Number of pages13
DOIs
StatePublished - Aug 28 2012
Event15th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2012 and the 16th International Workshop on Randomization and Computation, RANDOM 2012 - Cambridge, MA, United States
Duration: Aug 15 2012Aug 17 2012

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume7408 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

Other

Other15th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2012 and the 16th International Workshop on Randomization and Computation, RANDOM 2012
CountryUnited States
CityCambridge, MA
Period8/15/128/17/12

All Science Journal Classification (ASJC) codes

  • Theoretical Computer Science
  • Computer Science(all)

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    Awasthi, P., Jha, M., Molinaro, M., & Raskhodnikova, S. (2012). Limitations of local filters of Lipschitz and monotone functions. In Approximation, Randomization, and Combinatorial Optimization: Algorithms and Techniques - 15th International Workshop, APPROX 2012, and 16th International Workshop, RANDOM 2012, Proceedings (pp. 374-386). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 7408 LNCS). https://doi.org/10.1007/978-3-642-32512-0_32