Active Property Testing

Abstract

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.

On Sample-Based Testers

“…This amounts to presenting high lower bounds on the sample complexity of sample-based testers for that property. A few positive results on sample-based testers have been obtained in previous work [8,18,2], and they are further discussed in Subsection 1.3. We also discuss the relation to testing properties of distributions in Subsection 1.2.1.…”

“…See further discussion following Theorem 1.1. 2 In fact, the main definition in [9] refers to (distributionfree) sample-based testers (cf. [9, Definition 2.1]).…”

Testing Surface Area

“…This significantly improves on the result in [KR98]. By taking η = ϵ and applying Fact 1.1, we can also achieve no approximation factor (i.e., κ = 1) with an O(1/ϵ 3.5 )-query tester, slightly improving the result from [BBBY12]. For n = 2 we can achieve approximation factor 1.081 using O(1/ϵ) queries; for n = 3 we can achieve approximation factor 1.126 using O(1/ϵ) queries.…”

“…Thus we have a tester whose completeness is ensured by the Buffon's Needle Theorem. We remark that the ideas so far are precisely those used in the n = 1 tester of Balcan et al [BBBY12]. The hope is that this underestimate is close to the truth if δ is "small enough".…”

“…In the 1-dimensional case studied in [BBBY12] this task is very easy: the partial definition of G fixes some intervals to be contained in G and some intervals to be excluded from G. At this point, any sensible method of filling in the gaps yields appropriately small "surface area" (number of endpoints). In 2 or more dimensions, though, the problem is more complicated; indeed it seems very hard to give a construction for this gap-filling problem.…”