In this paper, we consider the question of determining whether a function f has property P or is e-far from any function with property P. A property testing algorithm is given a sample of the value of f on instances drawn according to some distribution. In some cases, it is also allowed to query f on instances of its choice. We study this question for different properties and establish some connections to problems in learning theory and approximation.In particular, we focus our attention on testing graph properties. Given access to a graph G in the form of being able to query whether an edge exists or not between a pair of vertices, we devise algorithms to test whether the underlying graph has properties such as being bipartite, k-Colorable, or having a p-Clique (clique of density p with respect to the vertex set). Our graph property testing algorithms are probabilistic and make assertions that are correct with high probability, while making a number of queries that is independent of the size of the graph. Moreover, the property testing algorithms can be used to efficiently (i.e., in time linear in the number of vertices) construct partitions of the graph that correspond to the property being tested, if it holds for the input graph.

Property Testing and its connection to Learning and Approximation

Property testing is concerned with the design of super-fast algorithms for the structural analysis of large quantities of data. The aim is to unveil global features of the data, such as determining whether the data has a particular property or estimating global parameters. Remarkably, it is possible for decisions to be made by accessing only a small portion of the data. Property testing focuses on properties and parameters that go beyond simple statistics. This book provides an extensive and authoritative introduction to property testing. It provides a wide range of algorithmic techniques for the design and analysis of tests for algebraic properties, properties of Boolean functions, graph properties, and properties of distributions.

Introduction to Property Testing

We present improved algorithms for testing monotonicity of functions. Namely, given the ability to query an unknown function f: Σ n ↦ Ξ, where Σ and Ξ are finite ordered sets, the test always accepts a monotone f, and rejects f with high probability if it is e-far from being monotone (i.e., every monotone function differs from f on more than an e fraction of the domain). For any e > 0, the query complexity of the test is O((n/e) · log ∣Σ ∣ · log ∣Ξ∣). The previous best known bound was \(\tilde{O}((n^2/\epsilon) \cdot \vert\Sigma\vert^2 \cdot \vert\Xi\vert)\).

Improved Testing Algorithms for Monotonicity.

One motivation 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 arbitrary examples constructed by an 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 to the domain of testing. We develop both general results for this *active testing* model as well as efficient testing algorithms for several 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 Omega(d) labeled examples for learning (and Omega(sqrt{d}) for passive testing [KR00] 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(sqrt{n}) queries, substantially less than the Omega(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.

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Active Property Testing

The primary problem in property testing is to decide whether a given function satisfies a certain property or is far from any function satisfying it. This crucially requires a notion of distance between functions. The most prevalent notion is the Hamming distance over the uniform distribution on the domain. This restriction to uniformity is rather limiting, and it is important to investigate distances induced by more general distributions.In this article, we provide simple and optimal testers for bounded derivative properties over arbitrary product distributions. Bounded derivative properties include fundamental properties, such as monotonicity and Lipschitz continuity. Our results subsume almost all known results (upper and lower bounds) on monotonicity and Lipschitz testing over arbitrary ranges.We prove an intimate connection between bounded derivative property testing and binary search trees (BSTs). We exhibit a tester whose query complexity is the sum of expected depths of optimal BSTs for each marginal. Furthermore, we show that this sum-of-depths is also a lower bound. A technical contribution of our work is an optimal dimension reduction theorem for all bounded derivative properties that relates the distance of a function from the property to the distance of restrictions of the function to random lines. Such a theorem has been elusive even for monotonicity, and our theorem is an exponential improvement to the previous best-known result.

Property testing on product distributions: optimal testers for bounded derivative properties

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 testers for a variety of classes of functions when the underlying distribution is uniform, designing distribution-free 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 distribution-free tester for monotone monomials as well as for general monomials must have query complexity e W(n 1=5 ) (for a constant distance parameter e). In this paper we present distribution-free testers for these classes with query complexity e O(n 1=2 =e). We note that in contrast to previous results for distribution-free 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 non-negligible part of the input space), which were not used in previous work on property testing.

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

We consider the problem of distribution-free testing of the class of monotone monomials and the class of monomials over n variables.0 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 complexity Ω(n1/5) (for a constant distance parameter e).

In this paper we present distribution-free testing algorithms for these classes where the query complexity of the algorithms is O(n1/2/e). We note that as opposed to previous results for distribution-free testing, our algorithms do not build on the algorithms that work under the uniform distribution. Rather, we define 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.

Elya Dolev

Papers

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

Distribution-free testing algorithms for monomials with a sublinear number of queries