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37
Efficient testing of sparse GF(2) polynomials
 In Automata, Languages and Programming: ThirtyFifth International Colloquium (ICALP
, 2008
"... Abstract. We give the first algorithm that is both queryefficient and timeefficient for testing whether an unknown function f: {0, 1} n →{−1, 1} is an ssparse GF(2) polynomial versus ǫfar from every such polynomial. Our algorithm makes poly(s,1/ǫ) blackbox queries to f and runs in time n · poly ..."
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Abstract. We give the first algorithm that is both queryefficient and timeefficient for testing whether an unknown function f: {0, 1} n →{−1, 1} is an ssparse GF(2) polynomial versus ǫfar from every such polynomial. Our algorithm makes poly(s,1/ǫ) blackbox queries to f and runs in time n · poly(s,1/ǫ). The only previous algorithm for this testing problem [DLM + 07] used poly(s,1/ǫ) queries, but had running time exponential in s and superpolynomial in 1/ǫ. Our approach significantly extends the “testing by implicit learning ” methodology of [DLM + 07]. The learning component of that earlier work was a bruteforce exhaustive search over a concept class to find a hypothesis consistent with a sample of random examples. In this work, the learning component is a sophisticated exact learning algorithm for sparse GF(2) polynomials due to Schapire and Sellie [SS96]. A crucial element of this work, which enables us to simulate the membership queries required by [SS96], is an analysis establishing new properties of how sparse GF(2) polynomials simplify under certain restrictions of “lowinfluence ” sets of variables. 1
Efficient sample extractors for juntas with applications
 Automata, Languages and Programming
, 2011
"... We develop a queryefficient sample extractor for juntas, that is, a probabilistic algorithm that can simulate random samples from the core of a kjunta f: {0, 1} n → {0, 1} given oracle access to a function f ′ : {0, 1} n → {0, 1} that is only close to f. After a preprocessing step, which takes Õ(k ..."
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We develop a queryefficient sample extractor for juntas, that is, a probabilistic algorithm that can simulate random samples from the core of a kjunta f: {0, 1} n → {0, 1} given oracle access to a function f ′ : {0, 1} n → {0, 1} that is only close to f. After a preprocessing step, which takes Õ(k) queries, generating each sample to the core of f takes only one query to f ′. We then plug in our sample extractor in the “testing by implicit learning ” framework of Diakonikolas et al. [DLM + 07], improving the query complexity of testers for various Boolean function classes. In particular, for some of the classes considered in [DLM + 07], such as sterm DNF formulas, sizes decision trees, sizes Boolean formulas, ssparse polynomials over F2, and sizes branching programs, the query complexity is reduced from Õ(s4 /ɛ 2) to Õ(s/ɛ2). This shows that, using the new sample extractor, testing by implicit learning can lead to testers having better query complexity than those tailored to a specific problem, such as the tester of Parnas et al. [PRS02] for the class of monotone sterm DNF formulas. In terms of techniques, we extend the tools used in [CGM11] for testing function isomorphism to juntas. Specifically, while the original analysis in [CGM11] allowed queryefficient noisy sampling
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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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
Learning and Lower Bounds for AC0 with Threshold Gates. Available at http://eccc.hpiweb.de/report/2010/074
, 2010
"... Abstract. In 2002 Jackson et al. [JKS02] asked whether AC0 circuits augmented with a threshold gate at the output can be efficiently learned from uniform random examples. We answer this question affirmatively by showing that such circuits have fairly strong Fourier concentration; hence the lowdegre ..."
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Abstract. In 2002 Jackson et al. [JKS02] asked whether AC0 circuits augmented with a threshold gate at the output can be efficiently learned from uniform random examples. We answer this question affirmatively by showing that such circuits have fairly strong Fourier concentration; hence the lowdegree algorithm of Linial, Mansour and Nisan [LMN93] learns such circuits in subexponential time. Under a conjecture of Gotsman and Linial [GL94] which upper bounds the total influence of lowdegree polynomial threshold functions, the running time is quasipolynomial. Our results extend to AC0 circuits augmented with a small superconstant number of threshold gates at arbitrary locations in the circuit. We also establish some new structural properties of AC0 circuits augmented with threshold gates, which allow us to prove a range of separation results and lower bounds.
Testing by implicit learning: a brief survey
 Property Testing
, 2010
"... We give a highlevel survey of the “testing by implicit learning ” paradigm, and explain some of the property testing results for various Boolean function classes that have been obtained using this approach. 1 ..."
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We give a highlevel survey of the “testing by implicit learning ” paradigm, and explain some of the property testing results for various Boolean function classes that have been obtained using this approach. 1
Nearly tight bounds for testing function isomorphism
, 2011
"... We study the problem of testing isomorphism (equivalence up to relabeling of the input variables) between Boolean functions. We prove that: • For most functions f: {0, 1} n → {0, 1}, the query complexity of testing isomorphism to f is Ω(n). Moreover, the query complexity of testing isomorphism to mo ..."
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We study the problem of testing isomorphism (equivalence up to relabeling of the input variables) between Boolean functions. We prove that: • For most functions f: {0, 1} n → {0, 1}, the query complexity of testing isomorphism to f is Ω(n). Moreover, the query complexity of testing isomorphism to most kjuntas f: {0, 1} n → {0, 1} is Ω(k). • Isomorphism to any kjunta f: {0, 1} n → {0, 1} can be tested with O(k log k) queries. • For some kjuntas f: {0, 1} n → {0, 1}, testing isomorphism to f with onesided error requires Ω(k log(n/k)) queries. In particular, testing if f: {0, 1} n → {0, 1} is a kparity with onesided error requires Ω(k log(n/k)) queries. • The query complexity of testing isomorphism between two unknown functions f, g: {0, 1} n → {0, 1} is ˜ Θ(2 n/2). These bounds are tight up to logarithmic factors, and they significantly strengthen the bounds proved by Fischer et al. (FOCS 2002) and Blais and O’Donnell (CCC 2010).
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
Testing Computability by Width Two OBDDs
"... Property testing is concerned with deciding whether an object (e.g. a graph or a function) has a certain property or is “far ” (for some definition of far) from every object with that property. In this paper we give lower and upper bounds for testing functions for the property of being computable by ..."
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Property testing is concerned with deciding whether an object (e.g. a graph or a function) has a certain property or is “far ” (for some definition of far) from every object with that property. In this paper we give lower and upper bounds for testing functions for the property of being computable by a readonce width2 Ordered Binary Decision Diagram (OBDD), also known as a branching program, where the order of the variables is known. Width2 OBDDs generalize two classes of functions that have been studied in the context of property testing linear functions (over GF (2)) and monomials. In both these cases membership can be tested in time that is linear in 1/ɛ. Interestingly, unlike either of these classes, in which the query complexity of the testing algorithm does not depend on the number, n, of variables in the tested function, we show that (onesided error) testing for computability by a width2 OBDD requires Ω(log(n)) queries, and give an algorithm (with onesided error) that tests for this property and performs Property testing is concerned with deciding whether an object (e.g. a graph or a function) has a certain property or is “far ” (for some definition of far) from every object with that property [RS96,
Robust characterizations of kwise independence over product spaces and related testing results. Random Structures and Algorithms
, 2012
"... A discrete distribution D over Σ1 × · · · × Σn is called (nonuniform) kwise independent if for any subset of k indices {i1,..., ik} and for any z1 ∈ Σi1,..., zk ∈ Σik, PrX∼D[Xi1 · · ·Xik = z1 · · · zk] = PrX∼D[Xi1 = z1] · · ·PrX∼D[Xik = zk]. We study the problem of testing (nonuniform) k ..."
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A discrete distribution D over Σ1 × · · · × Σn is called (nonuniform) kwise independent if for any subset of k indices {i1,..., ik} and for any z1 ∈ Σi1,..., zk ∈ Σik, PrX∼D[Xi1 · · ·Xik = z1 · · · zk] = PrX∼D[Xi1 = z1] · · ·PrX∼D[Xik = zk]. We study the problem of testing (nonuniform) kwise independent distributions over product spaces. For the uniform case we show an upper bound on the distance between a distribution D from kwise independent distributions in terms of the sum of Fourier coefficients of D at vectors of weight at most k. Such a bound was previously known only when the underlying domain is {0, 1}n. For the nonuniform case, we give a new characterization of distributions being kwise independent and further show that such a characterization is robust based on our results for the uniform case. These results greatly generalize those of Alon et al. [STOC’07, pp. 496–505] on uniform kwise independence over the Boolean cubes to nonuniform kwise independence over product spaces. Our results yield natural testing algorithms for kwise independence with time and sample complexity sublinear in terms of the support size of the distribution when k is a constant. The main technical tools employed include discrete Fourier transform and the theory of linear systems of congruences.