TY - GEN
T1 - Limitations of local filters of Lipschitz and monotone functions
AU - Awasthi, Pranjal
AU - Jha, Madhav
AU - Molinaro, Marco
AU - Raskhodnikova, Sofya
PY - 2012
Y1 - 2012
N2 - 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.
AB - 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.
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U2 - 10.1007/978-3-642-32512-0_32
DO - 10.1007/978-3-642-32512-0_32
M3 - Conference contribution
AN - SCOPUS:84865305380
SN - 9783642325113
T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
SP - 374
EP - 386
BT - Approximation, Randomization, and Combinatorial Optimization
T2 - 15th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2012 and the 16th International Workshop on Randomization and Computation, RANDOM 2012
Y2 - 15 August 2012 through 17 August 2012
ER -