Distribution-Free Testing Algorithms for Monomials with a Sublinear Number of Queries

Dolev, Elya; Ron, Dana

doi:10.1007/978-3-642-15369-3_40

Cited by 9 publications

(32 citation statements)

References 18 publications

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“…We first obtain an efficient algorithm that is one-sided and makes onlyÕ((n 1/3 /ǫ 5 )) queries. When 1/ǫ is small compared to n, it improves the previous bestÕ(n 1/2 /ǫ)-query algorithm of Dolev and Ron [DR11]. Theorem 1.1.…”

Section: Introductionmentioning

confidence: 84%

“…2 If it fails on x, then f is not a monotone conjunction; otherwise, let h(x) denote the index found, called the representative index of x [DR11]. Roughly speaking, the algorithm of Dolev and Ron draws n 1/2 samples from D and uses the binary search procedure to compute the representative index h(x) of each sample x from f −1 (0).…”

Section: The Approach Of Our Algorithmmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…Below we start with some intuition behind our new algorithm and lower bound construction for monotone conjunctions, and compare our approaches and techniques with those of [GS09] and [DR11].…”

Section: Introductionmentioning

confidence: 99%

“…The distance between f and C in the standard testing model is measured with respect to the uniform distribution. Equivalently, it is the smallest fraction of entries of f one needs to flip to make it a member of C. A natural generalization of the standard model, called distribution-free property testing, was first introduced by Goldreich, Goldwasser and Ron [GGR98] and has been studied in [AC06,HK07,HK08a, HK08b,GS09,DR11].For the distribution-free model, there is an unknown distribution D over {0, 1} n in addition to the unknown f . The goal of an algorithm is to determine whether f is in C versus far from C with respect to D, given both blackbox access to f and sampling access to D. The model of distribution-free property testing is well motivated by scenarios where the distance being of interest is measured * Supported in part by NSF grants CCF-1149257 and CCF-1423100.…”

mentioning

confidence: 99%

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Tight Bounds for the Distribution-Free Testing of Monotone Conjunctions

Chen

Xie

2015

Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms

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We improve both upper and lower bounds for the distributionfree testing of monotone conjunctions. Given oracle access to an unknown Boolean function f : {0, 1} n → {0, 1} and sampling oracle access to an unknown distribution D over {0, 1} n , we present anÕ(n 1/3 /ǫ 5 )-query algorithm that tests whether f is a monotone conjunction versus ǫ-far from any monotone conjunction with respect to D. This improves the previous best upper bound ofÕ(n 1/2 /ǫ) by Dolev and Ron [DR11], when 1/ǫ is small compared to n. For some constant ǫ0 > 0, we also prove a lower bound ofΩ(n 1/3 ) for the query complexity, improving the previous best lower bound ofΩ(n 1/5 ) by Glasner and Servedio [GS09]. Our upper and lower bounds are tight, up to a polylogarithmic factor, when the distance parameter ǫ is a constant. Furthermore, the same upper and lower bounds can be extended to the distribution-free testing of general conjunctions, and the lower bound can be extended to that of decision lists and linear threshold functions. IntroductionThe field of property testing analyzes the resources an algorithm requires to determine whether an unknown object satisfies a certain property versus far from satisfying the property. It was introduced in [RS96], after prior work in [BFL91,BLR93], and has been studied extensively during the past two decades (e.g., see surveys in [Gol98,Fis01,Ron01,AS05,Rub06]). For the purpose of this paper we consider a Boolean function f : {0, 1} n → {0, 1}, and a class C of Boolean functions, viewed as a property. The distance between f and C in the standard testing model is measured with respect to the uniform distribution. Equivalently, it is the smallest fraction of entries of f one needs to flip to make it a member of C. A natural generalization of the standard model, called distribution-free property testing, was first introduced by Goldreich, Goldwasser and Ron [GGR98] and has been studied in [AC06,HK07,HK08a, HK08b,GS09,DR11].For the distribution-free model, there is an unknown distribution D over {0, 1} n in addition to the unknown f . The goal of an algorithm is to determine whether f is in C versus far from C with respect to D, given both blackbox access to f and sampling access to D. The model of distribution-free property testing is well motivated by scenarios where the distance being of interest is measured * Supported in part by NSF grants CCF-1149257 and CCF-1423100.† xichen@cs.columbia.edu, Columbia University ‡ jinyu@cs.columbia.edu, Columbia University with respect to an unknown distribution D. It is also inspired by similar models in computational learning theory (e.g., the distribution-free PAC learning model [Val84] with membership queries). It was observed [GGR98] that any proper distribution-free PAC learning algorithm can be used for distribution-free property testing.In this paper we study the distribution-free testing of monotone conjunctions (or monotone monomials): f is a monotone conjunction if f (z) = i∈S z i , for some S ⊆ [n]. We first obtain an efficient algorithm that is one-s...

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Section: Introductionmentioning

confidence: 84%

Section: The Approach Of Our Algorithmmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

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Tight Bounds for the Distribution-Free Testing of Monotone Conjunctions

Chen

Xie

2015

Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms

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Distribution-Free Testing Algorithms for Monomials with a Sublinear Number of Queries

Dolev

Ron

2010

Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques

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We consider the problem of distribution-free testing of the class of monotone monomials and the class of monomials over n variables. While there are very efficient algorithms for testing a variety of functions classes when the underlying distribution is uniform, designing distribution-free 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 distribution-free testing algorithm for monotone monomials as well as for general monomials must have query complexityΩ(n

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Active Property Testing

Balcan

Blais

Blum

et al. 2012

2012 IEEE 53rd Annual Symposium on Foundations of Computer Science

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One of the motivations for property testing of boolean functions is the idea that testing can provide a fast preprocessing step before learning. However, in most machine learning applications, it is not possible to request for labels of fictitious examples constructed by the algorithm. Instead, the dominant query paradigm in applied machine learning, called active learning, is one where the algorithm may query for labels, but only on points in a given polynomial-sized (unlabeled) sample, drawn from some underlying distribution D. In this work, we bring this well-studied model in learning to the domain of testing.We develop both general results for this active testing model as well as efficient testing algorithms for a number of important properties for learning, demonstrating that testing can still yield substantial benefits in this restricted setting. For example, we show that testing unions of d intervals can be done with O(1) label requests in our setting, whereas it is known to require Ω(d) labeled examples for learning (and Ω( √ d) for passive testing [41] where the algorithm must pay for every example drawn from D). In fact, our results for testing unions of intervals also yield improvements on prior work in both the classic query model (where any point in the domain can be queried) and the passive testing model as well. For the problem of testing linear separators in R n over the Gaussian distribution, we show that both active and passive testing can be done with O( √ n) queries, substantially less than the Ω(n) needed for learning, with near-matching lower bounds. We also present a general combination result in this model for building testable properties out of others, which we then use to provide testers for a number of assumptions used in semi-supervised learning. In addition to the above results, we also develop a general notion of the testing dimension of a given property with respect to a given distribution, that we show characterizes (up to constant factors) the intrinsic number of label requests needed to test that property. We develop such notions for both the active and passive testing models. We then use these dimensions to prove a number of lower bounds, including for linear separators and the class of dictator functions.Our results show that testing can be a powerful tool in realistic models for learning, and further that active testing exhibits an interesting and rich structure. Our work in addition brings together tools from a range of areas including U-statistics, noise-sensitivity, self-correction, and spectral analysis of random matrices, and develops new tools that may be of independent interest.

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Distribution-Free Testing Algorithms for Monomials with a Sublinear Number of Queries

Cited by 9 publications

References 18 publications

Tight Bounds for the Distribution-Free Testing of Monotone Conjunctions

Tight Bounds for the Distribution-Free Testing of Monotone Conjunctions

Distribution-Free Testing Algorithms for Monomials with a Sublinear Number of Queries

Active Property Testing

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