Testing Problems with Sublearning Sample Complexity

Kearns, Michael; Ron, Dana

doi:10.1006/jcss.1999.1656

Cited by 41 publications

(30 citation statements)

References 11 publications

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“…Namely, the algorithm rejects functions that are -far from every 4k-junta rather than -far from every k-junta. Similar relaxations have been considered both in the standard testing model (e.g., [32,26,28]) and in the tolerant testing model [33]. Although one may hope for a stronger statement (distinguishing functions close to k-junta from those far from k-junta), we note that for most practical purposes the relaxation above is more than sufficient.…”

Section: Our Resultsmentioning

confidence: 84%

Tolerant Junta Testing and the Connection to Submodular Optimization and Function Isomorphism

Blais

Canonne

Eden

et al. 2018

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

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The function f : {−1, 1} n → {−1, 1} is a k-junta if it depends on at most k of its variables. We consider the problem of tolerant testing of k-juntas, where the testing algorithm must accept any function that is -close to some k-junta and reject any function that is -far from every k -junta for some = O( ) and k = O(k).Our first result is an algorithm that solves this problem with query complexity polynomial in k and 1/ . This result is obtained via a new polynomialtime approximation algorithm for submodular function minimization (SFM) under large cardinality constraints, which holds even when only given an approximate oracle access to the function.Our second result considers the case where k = k. We show how to obtain a smooth tradeoff between the amount of tolerance and the query complexity in this setting. Specifically, we design an algorithm that given ρ ∈ (0, 1/2) accepts any function that is ρ 8 -close to some k-junta and rejects any function that is -far from every k-junta. The query complexity of the algorithm is OFinally, we show how to apply the second result to the problem of tolerant isomorphism testing between two unknown Boolean functions f and g. We give an algorithm for this problem whose query complexity only depends on the (unknown) smallest k such that either f or g is close to being a k-junta.

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Section: Our Resultsmentioning

confidence: 84%

Tolerant Junta Testing and the Connection to Submodular Optimization and Function Isomorphism

Blais

Canonne

Eden

et al. 2018

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

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“…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.…”

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confidence: 99%

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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“…Much research has been done on understanding the relation between property testing algorithms and learning algorithms, see, e. g., [16,18] and the lengthy survey [24]. As Goldreich has noted [15], an often-invoked motivation for property testing is that (inexpensive) testing algorithms can be used as a "preliminary diagnostic" to determine whether it is appropriate to run a (more expensive) learning algorithm.…”

Section: Discussionmentioning

confidence: 99%