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Property Testing: A Learning Theory Perspective
"... Property testing deals with tasks where the goal is to distinguish between the case that an object (e.g., function or graph) has a prespecified property (e.g., the function is linear or the graph is bipartite) and the case that it differs significantly from any such object. The task should be perfor ..."
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Cited by 49 (9 self)
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Property testing deals with tasks where the goal is to distinguish between the case that an object (e.g., function or graph) has a prespecified property (e.g., the function is linear or the graph is bipartite) and the case that it differs significantly from any such object. The task should be performed by observing only a very small part of the object, in particular by querying the object, and the algorithm is allowed a small failure probability. One view of property testing is as a relaxation of learning the object (obtaining an approximate representation of the object). Thus property testing algorithms can serve as a preliminary step to learning. That is, they can be applied in order to select, very efficiently, what hypothesis class to use for learning. This survey takes the learningtheory point of view and focuses on results for testing properties of functions that are of interest to the learning theory community. In particular, we cover results for testing algebraic properties of functions such as linearity, testing properties defined by concise representations, such as having a small DNF representation, and more. 1
Algorithmic and Analysis Techniques in Property Testing
"... Property testing algorithms are “ultra”efficient algorithms that decide whether a given object (e.g., a graph) has a certain property (e.g., bipartiteness), or is significantly different from any object that has the property. To this end property testing algorithms are given the ability to perform ..."
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Cited by 48 (7 self)
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Property testing algorithms are “ultra”efficient algorithms that decide whether a given object (e.g., a graph) has a certain property (e.g., bipartiteness), or is significantly different from any object that has the property. To this end property testing algorithms are given the ability to perform (local) queries to the input, though the decision they need to make usually concern properties with a global nature. In the last two decades, property testing algorithms have been designed for many types of objects and properties, amongst them, graph properties, algebraic properties, geometric properties, and more. In this article we survey results in property testing, where our emphasis is on common analysis and algorithmic techniques. Among the techniques surveyed are the following: • The selfcorrecting approach, which was mainly applied in the study of property testing of algebraic properties; • The enforce and test approach, which was applied quite extensively in the analysis of algorithms for testing graph properties (in the densegraphs model), as well as in other contexts;
Property Testing Lower Bounds Via Communication Complexity
, 2011
"... We develop a new technique for proving lower bounds in property testing, by showing a strong connection between testing and communication complexity. We give a simple scheme for reducing communication problems to testing problems, thus allowing us to use known lower bounds in communication complexit ..."
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Cited by 34 (8 self)
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We develop a new technique for proving lower bounds in property testing, by showing a strong connection between testing and communication complexity. We give a simple scheme for reducing communication problems to testing problems, thus allowing us to use known lower bounds in communication complexity to prove lower bounds in testing. This scheme is general and implies a number of new testing bounds, as well as simpler proofs of several known bounds. For the problem of testing whether a boolean function is klinear (a parity function on k variables), we achieve a lower bound of Ω(k) queries, even for adaptive algorithms with twosided error, thus confirming a conjecture of Goldreich [25]. The same argument behind this lower bound also implies a new proof of known lower bounds for testing related classes such as kjuntas. For some classes, such as the class of monotone functions and the class of ssparse GF(2) polynomials, we significantly strengthen the best known bounds.
Testing halfspaces
 IN PROC. 20TH ANNUAL SYMPOSIUM ON DISCRETE ALGORITHMS (SODA
, 2009
"... This paper addresses the problem of testing whether a Booleanvalued function f is a halfspace, i.e. a function of the form f(x) = sgn(w ·x−θ). We consider halfspaces over the continuous domain R n (endowed with the standard multivariate Gaussian distribution) as well as halfspaces over the Boolean ..."
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Cited by 34 (15 self)
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This paper addresses the problem of testing whether a Booleanvalued function f is a halfspace, i.e. a function of the form f(x) = sgn(w ·x−θ). We consider halfspaces over the continuous domain R n (endowed with the standard multivariate Gaussian distribution) as well as halfspaces over the Boolean cube {−1, 1} n (endowed with the uniform distribution). In both cases we give an algorithm that distinguishes halfspaces from functions that are ǫfar from any halfspace using only poly ( 1) queries, independent of ǫ the dimension n. Two simple structural results about halfspaces are at the heart of our approach for the Gaussian distribution: the first gives an exact relationship between the expected value of a halfspace f and the sum of the squares of f’s degree1 Hermite coefficients, and the second shows that any function that approximately satisfies this relationship is close to a halfspace. We prove analogous results for the Boolean cube {−1, 1} n (with Fourier coefficients in place of Hermite coefficients) for balanced halfspaces in which all degree1 Fourier coefficients are small. Dealing with general halfspaces over {−1, 1} n poses significant additional complications and requires other ingredients. These include “crossconsistency ” versions of the results mentioned above for pairs of halfspaces with the same weights but different thresholds; new structural results relating the largest degree1 Fourier coefficient and the largest weight in unbalanced halfspaces; and algorithmic techniques from recent work on testing juntas [FKR+ 02].
Testing Fourier dimensionality and sparsity
"... Abstract. We present a range of new results for testing properties of Boolean functions that are defined in terms of the Fourier spectrum. Broadly speaking, our results show that the property of a Boolean function having a concise Fourier representation is locally testable. We first give an efficien ..."
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Abstract. We present a range of new results for testing properties of Boolean functions that are defined in terms of the Fourier spectrum. Broadly speaking, our results show that the property of a Boolean function having a concise Fourier representation is locally testable. We first give an efficient algorithm for testing whether the Fourier spectrum of a Boolean function is supported in a lowdimensional subspace of F n 2 (equivalently, for testing whether f is a junta over a small number of parities). We next give an efficient algorithm for testing whether a Boolean function has a sparse Fourier spectrum (small number of nonzero coefficients). In both cases we also prove lower bounds showing that any testing algorithm — even an adaptive one — must have query complexity within a polynomial factor of our algorithms, which are nonadaptive. Finally, we give an “implicit learning ” algorithm that lets us test any subproperty of Fourier concision. Our technical contributions include new structural results about sparse Boolean functions and new analysis of the pairwise independent hashing of Fourier coefficients from [13]. 1
A unified framework for testing linearinvariant properties
 In Proceedings of the 51st Annual IEEE Symposium on Foundations of Computer Science
, 2010
"... In the history of property testing, a particularly important role has been played by linearinvariant properties, i.e., properties of Boolean functions on the hypercube which are closed under linear transformations of the domain. Examples of such properties include linearity, ReedMuller codes, and F ..."
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Cited by 18 (6 self)
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In the history of property testing, a particularly important role has been played by linearinvariant properties, i.e., properties of Boolean functions on the hypercube which are closed under linear transformations of the domain. Examples of such properties include linearity, ReedMuller codes, and Fourier sparsity. In this work, we describe a framework that can lead to a unified analysis of the testability of all linearinvariant properties, drawing on techniques from additive combinatorics and from graph theory. Our main contributions here are the following: 1. We introduce a simple combinatorial condition, which we call subspaceheredity, and conjecture that any property of Boolean functions satisfying it can be efficiently tested. Verifying this conjecture will unify many individual results in this area. 2. We show that if our conjecture holds, then one can obtain a simple combinatorial characterization of properties of Boolean functions that can be efficiently tested with onesided error, thus addressing a challenge posed by Sudan recently. 3. We introduce a new technique for proving the testability of Boolean functions. Using it, we verify a special case of the conjecture. Our approach here is motivated by techniques that proved to be very successful previously in studying the testability of graph properties.
Invariance in property testing
 Electronic Colloquium on Computational Complexity (ECCC
"... Property testing considers the task of testing rapidly (in particular, with very few samples into the data), if some massive data satisfies some given property, or is far from satisfying the property. For “global properties”, i.e., properties that really depend somewhat on every piece of the data, o ..."
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Cited by 14 (2 self)
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Property testing considers the task of testing rapidly (in particular, with very few samples into the data), if some massive data satisfies some given property, or is far from satisfying the property. For “global properties”, i.e., properties that really depend somewhat on every piece of the data, one could ask how it can be tested by so few samples? We suggest that for “natural ” properties, this should happen because the property is invariant under “nice ” set of “relabellings ” of the data. We refer to this set of relabellings as the “invariance class ” of the property and advocate explicit identification of the invariance class of locally testable properties. Our hope is the explicit knowledge of the invariance class may lead to more general, broader, results. After pointing out the invariance classes associated with some the basic classes of testable properties, we focus on “algebraic properties ” which seem to be characterized by the fact that the properties are themselves vector spaces, while their domains are also vector spaces and the properties are invariant under affine transformations of the domain. We survey recent results (obtained with Tali Kaufman, Elena Grigorescu and Eli BenSasson) that give broad conditions that are sufficient for local testability among this class of properties, and some structural theorems that attempt to describe which properties exhibit the sufficient conditions. 1
Improved bounds for testing juntas
 In Proc. 12th Workshop RANDOM
, 2008
"... Abstract. We consider the problem of testing functions for the property of being a kjunta (i.e., of depending on at most k variables). Fischer, Kindler, Ron, Safra, and Samorodnitsky (J. Comput. Sys. Sci., 2004) showed that Õ(k2)/ɛ queries are sufficient to test kjuntas, and conjectured that this ..."
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Cited by 13 (5 self)
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Abstract. We consider the problem of testing functions for the property of being a kjunta (i.e., of depending on at most k variables). Fischer, Kindler, Ron, Safra, and Samorodnitsky (J. Comput. Sys. Sci., 2004) showed that Õ(k2)/ɛ queries are sufficient to test kjuntas, and conjectured that this bound is optimal for nonadaptive testing algorithms. Our main result is a nonadaptive algorithm for testing kjuntas with Õ(k 3/2)/ɛ queries. This algorithm disproves the conjecture of Fischer et al. We also show that the query complexity of nonadaptive algorithms for testing juntas has a lower bound of min ` ˜ Ω(k/ɛ), 2
Testing Boolean Function Isomorphism
"... Abstract. Two boolean functions f, g: {0, 1} n → {0, 1} are isomorphic if they are identical up to relabeling of the input variables. We consider the problem of testing whether two functions are isomorphic or far from being isomorphic with as few queries as possible. In the setting where one of the ..."
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Abstract. Two boolean functions f, g: {0, 1} n → {0, 1} are isomorphic if they are identical up to relabeling of the input variables. We consider the problem of testing whether two functions are isomorphic or far from being isomorphic with as few queries as possible. In the setting where one of the functions is known in advance, we show that the nonadaptive query complexity of the isomorphism testing problem is ˜ Θ(n). In fact, we show that the lower bound of Ω(n) queries for testing isomorphism to g holds for almost all functions g. In the setting where both functions are unknown to the testing algorithm, we show that the query complexity of the isomorphism testing problem is ˜ Θ(2 n/2). The bound in this result holds for both adaptive and nonadaptive testing algorithms. 1
DistributionFree Testing Lower Bounds for BasicBoolean Functions
"... Abstract. In the distributionfree property testing model, the distance betweenfunctions is measured with respect to an arbitrary and unknown probability distribution D over the input domain. We consider distributionfree testing of several basic Boolean function classes over { 0, 1}n, namely monot ..."
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Cited by 11 (1 self)
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Abstract. In the distributionfree property testing model, the distance betweenfunctions is measured with respect to an arbitrary and unknown probability distribution D over the input domain. We consider distributionfree testing of several basic Boolean function classes over { 0, 1}n, namely monotone conjunctions,general conjunctions, decision lists, and linear threshold functions. We prove that for each of these function classes, \Omega ((n / log n)1/5) oracle calls are required forany distributionfree testing algorithm. Since each of these function classes is known to be distributionfree properly learnable (and hence testable) using \Theta (n)oracle calls, our lower bounds are within a polynomial factor of the best possible. 1 Introduction The field of property testing deals with algorithms that decide whether an input objecthas a certain property or is far from having the property after reading only a small fraction of the object. Property testing was introduced in [21] and has evolved into a richfield of study (see [3, 7, 10, 19, 20] for some surveys). A standard approach in property testing is to view the input to the testing algorithm as a function over some finite domain;the testing algorithm is required to distinguish functions that have a certain property Pfrom functions that are fflfar from having property P. In the most commonly consideredproperty testing scenario, a function