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Algorithmic and Analysis Techniques in Property Testing
"... Property testing algorithms are “ultra”efficient algorithms that decide whether a given object (e.g., a graph) has a certain property (e.g., bipartiteness), or is significantly different from any object that has the property. To this end property testing algorithms are given the ability to perform ..."
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Cited by 48 (7 self)
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Property testing algorithms are “ultra”efficient algorithms that decide whether a given object (e.g., a graph) has a certain property (e.g., bipartiteness), or is significantly different from any object that has the property. To this end property testing algorithms are given the ability to perform (local) queries to the input, though the decision they need to make usually concern properties with a global nature. In the last two decades, property testing algorithms have been designed for many types of objects and properties, amongst them, graph properties, algebraic properties, geometric properties, and more. In this article we survey results in property testing, where our emphasis is on common analysis and algorithmic techniques. Among the techniques surveyed are the following: • The selfcorrecting approach, which was mainly applied in the study of property testing of algebraic properties; • The enforce and test approach, which was applied quite extensively in the analysis of algorithms for testing graph properties (in the densegraphs model), as well as in other contexts;
Improved approximation of linear threshold functions
 In Proc. 24nd Annual IEEE Conference on Computational Complexity (CCC
, 2009
"... We prove two main results on how arbitrary linear threshold functions f(x) = sign(w · x − θ) over the ndimensional Boolean hypercube can be approximated by simple threshold functions. Our first result shows that every nvariable threshold function f is ɛclose to a threshold function depending only ..."
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Cited by 19 (12 self)
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We prove two main results on how arbitrary linear threshold functions f(x) = sign(w · x − θ) over the ndimensional Boolean hypercube can be approximated by simple threshold functions. Our first result shows that every nvariable threshold function f is ɛclose to a threshold function depending only on Inf(f) 2 · poly(1/ɛ) many variables, where Inf(f) denotes the total influence or average sensitivity of f. This is an exponential sharpening of Friedgut’s wellknown theorem [Fri98], which states that every Boolean function f is ɛclose to a function depending only on 2 O(Inf(f)/ɛ) many variables, for the case of threshold functions. We complement this upper bound by showing that Ω(Inf(f) 2 + 1/ɛ 2) many variables are required for ɛapproximating threshold functions. Our second result is a proof that every nvariable threshold function is ɛclose to a threshold function with integer weights at most poly(n) · 2 Õ(1/ɛ2/3). This is an improvement, in the dependence on the error parameter ɛ, on an earlier result of [Ser07] which gave a poly(n) · 2 Õ(1/ɛ2) bound. Our improvement is obtained via a new proof technique that uses strong anticoncentration bounds from probability theory. The new technique also gives a simple and modular proof of the original [Ser07] result, and extends to give lowweight approximators for threshold functions under a range of probability distributions other than the uniform distribution.
Active Property Testing
, 2011
"... One of the motivations for property testing of boolean functions is the idea that testing can serve as a preprocessing step before learning. However, in most machine learning applications, the ability to query functions at arbitrary points in the input space is considered highly unrealistic. Instead ..."
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Cited by 6 (0 self)
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One of the motivations for property testing of boolean functions is the idea that testing can serve as a preprocessing step before learning. However, in most machine learning applications, the ability to query functions at arbitrary points in the input space is considered highly unrealistic. Instead, the dominant query paradigm in applied machine learning has been that of active learning, where the algorithm may ask for examples to be labeled, but only from among those that exist in nature. That is, the algorithm may make a polynomial number of draws from the underlying distribution D and then query for labels, but only of points in its sample. In this work, we bring this wellstudied model in learning to the domain of testing. We show that for a number of important properties for learning, testing can still yield substantial benefits in this setting. This includes testing whether data satisfies the “cluster assumption”, testing linear separators, testing the largemargin assumption in lowdimensional spaces, and testing unions of intervals. In most of these cases, we show active testing requires substantially fewer label requests than passive testing (where the algorithm must pay for labels on every example drawn from D), or active or passive learning. For example, testing the cluster assumption can be done with O(1) label requests using active testing, but requires Ω ( √ N) labeled examples for passive testing and Ω(N) for learning, where N is the number of clusters; a similar pattern holds for unions of
Testing the Lipschitz property over product distributions with applications to data privacy
 In Proceedings, Theory of Cryptography Conference (TCC
"... Analysis of statistical data privacy has emerged as an important area of research. In this work we design algorithms to test privacy guarantees of a given AlgorithmA executing on a data setD which contains potentially sensitive information about individuals. We design an efficient algorithm Atest w ..."
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Cited by 4 (4 self)
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Analysis of statistical data privacy has emerged as an important area of research. In this work we design algorithms to test privacy guarantees of a given AlgorithmA executing on a data setD which contains potentially sensitive information about individuals. We design an efficient algorithm Atest which can verify whether A satisfies generalized differential privacy guarantee. Generalized differential privacy [BBG+11] is a relaxation of the notion of differential privacy initially proposed by [DMNS06]. By now differential privacy is the most widely accepted notion of statistical data privacy. To design Algorithm Atest, we show a new connection between the differential privacy guarantee and Lipschitzness property of a given function. More specifically, we show that an efficient algorithm for testing of Lipschitz property can be transformed into Atest which can test for generalized differential privacy. Lipschitz property testing and its variants, first studied by [JR11], has been explored by many works [JR11, AJMR12b, AJMR12a, CS12] because of its intrinsic connection to data privacy as highlighted by [JR11]. To develop a Lipschitz property tester with an explicit application in privacy has been an intriguing problem since the work of [JR11]. In our work, we present such a direct application of lipschitz tester to testing privacy. We provide concrete instantiations of Lipschitz testers (over both the hypercube and the hypergrid domains) which are used
On samplebased testers
 Electronic Colloquium on Computational Complexity (ECCC
"... The standard definition of property testing endows the tester with the ability to make arbitrary queries to “elements ” of the tested object. In contrast, samplebased testers only obtain independently distributed elements (a.k.a. labeled samples) of the tested object. While samplebased testers were ..."
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Cited by 3 (0 self)
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The standard definition of property testing endows the tester with the ability to make arbitrary queries to “elements ” of the tested object. In contrast, samplebased testers only obtain independently distributed elements (a.k.a. labeled samples) of the tested object. While samplebased testers were defined by Goldreich, Goldwasser, and Ron (JACM 1998), most research in property testing is focused on querybased testers. In this work, we advance the study of samplebased property testers by providing several general positive results as well as by revealing relations between variants of this testing model. In particular: • We show that certain types of querybased testers yield samplebased testers of sublinear sample complexity. For example, this holds for a natural class of proximity oblivious testers. • We study the relation between distributionfree samplebased testers and onesided error samplebased testers w.r.t the uniform distribution. While most of this work ignores the time complexity of testing, one part of it does focus on this aspect. The main result in this part is a sublineartime samplebased tester for kColorability, for any k ≥ 2.
Distributionfree testing algorithms for monomials with a sublinear number of queries
 In Proceedings of the 13th international conference on Approximation, and 14 the International conference on Randomization, and combinatorial optimization: algorithms and techniques, APPROX/RANDOM’10
, 2010
"... We consider the problem of distributionfree testing of the class of monotone monomials and the class of monomials over n variables. While there are very efcient algorithms for testing a variety of functions classes when the underlying distribution is uniform, designing distributionfree algorithms ..."
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We consider the problem of distributionfree testing of the class of monotone monomials and the class of monomials over n variables. While there are very efcient algorithms for testing a variety of functions classes when the underlying distribution is uniform, designing distributionfree algorithms (which must work under any arbitrary and unknown distribution), tends to be a more challenging task. When the underlying distribution is uniform, Parnas et al. (SIAM Journal on Discrete Math, 2002) give an algorithm for testing (monotone) monomials whose query complexity does not depends on n, and whose dependence on the distance parameter is (inverse) linear. In contrast, Glasner and Servedio (in Proceedings of RANDOM, 2007) prove that every distributionfree testing algorithm for monotone monomials as well as for general monomials must have query complexity ~ (n1=5) (for a constant distance parameter ). In this paper we present distributionfree testing algorithms for these classes where the query complexity of the algorithms is ~O(n1=2=). We note that as opposed to previous results for distributionfree testing, our algorithms do not build on the algorithms that work under the uniform distribution. Rather, we dene and exploit certain structural properties of monomials (and functions that differ from them in a nonnegligible manner), which were not used in previous work on property testing. Research supported by the Israel Science Foundation (grant No. 246/08) i
DistributionFree Testing for Monomials with a Sublinear Number of Queries
, 2011
"... We consider the problem of distributionfree testing of the class of monotone monomials and the class of monomials over n variables. While there are very efficient testers for a variety of classes of functions when the underlying distribution is uniform, designing distributionfree testers (which ..."
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Cited by 2 (0 self)
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We consider the problem of distributionfree testing of the class of monotone monomials and the class of monomials over n variables. While there are very efficient testers for a variety of classes of functions when the underlying distribution is uniform, designing distributionfree testers (which must work under an arbitrary and unknown distribution) tends to be more challenging. When the underlying distribution is uniform, Parnas et al. (SIAM J. Discr. Math., 2002) give a tester for (monotone) monomials whose query complexity does not depend on n, and whose dependence on the distance parameter is (inverse) linear. In contrast, Glasner and Servedio (Theory of Computing, 2009) prove that every distributionfree tester for monotone monomials as well as for general monomials must have query complexity ˜ Ω(n 1/5) (for a constant distance parameter ε). In this paper we present distributionfree testers for these classes with query complexity Õ(n 1/2 /ε). We note that in contrast to previous results for distributionfree testing, our testers do not build on the testers that work under the uniform distribution. Rather, we define and exploit certain structural properties of monomials (and functions that differ from them on a nonnegligible part of the input space), which were not used in previous work on property testing.
Property Testing on Product Distributions: Optimal Testers for Bounded Derivative Properties
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