Results 1  10
of
35
Bounded Independence Fools Halfspaces
 In Proc. 50th Annual Symposium on Foundations of Computer Science (FOCS), 2009
"... We show that any distribution on {−1, +1} n that is kwise independent fools any halfspace (a.k.a. linear threshold function) h: {−1, +1} n → {−1, +1}, i.e., any function of the form h(x) = sign ( ∑n i=1 wixi − θ) where the w1,..., wn, θ are arbitrary real numbers, with error ɛ for k = O(ɛ−2 log 2 ..."
Abstract

Cited by 43 (17 self)
 Add to MetaCart
(Show Context)
We show that any distribution on {−1, +1} n that is kwise independent fools any halfspace (a.k.a. linear threshold function) h: {−1, +1} n → {−1, +1}, i.e., any function of the form h(x) = sign ( ∑n i=1 wixi − θ) where the w1,..., wn, θ are arbitrary real numbers, with error ɛ for k = O(ɛ−2 log 2 (1/ɛ)). Our result is tight up to log(1/ɛ) factors. Using standard constructions of kwise independent distributions, we obtain the first explicit pseudorandom generators G: {−1, +1} s → {−1, +1} n that fool halfspaces. Specifically, we fool halfspaces with error ɛ and seed length s = k · log n = O(log n · ɛ−2 log 2 (1/ɛ)). Our approach combines classical tools from real approximation theory with structural results on halfspaces by Servedio (Comput. Complexity 2007).
Agnostic Learning of Monomials by Halfspaces is Hard
"... Abstract — We prove the following strong hardness result for learning: Given a distribution on labeled examples from the hypercube such that there exists a monomial (or conjunction) consistent with (1 − ϵ)fraction of the examples, it is NPhard to find a halfspace that is correct on ( 1 +ϵ)fractio ..."
Abstract

Cited by 26 (10 self)
 Add to MetaCart
(Show Context)
Abstract — We prove the following strong hardness result for learning: Given a distribution on labeled examples from the hypercube such that there exists a monomial (or conjunction) consistent with (1 − ϵ)fraction of the examples, it is NPhard to find a halfspace that is correct on ( 1 +ϵ)fraction of the examples, 2 for arbitrary constant ϵ> 0. In learning theory terms, weak agnostic learning of monomials by halfspaces is NPhard. This hardness result bridges between and subsumes two previous results which showed similar hardness results for the proper learning of monomials and halfspaces. As immediate corollaries of our result, we give the first optimal hardness results for weak agnostic learning of decision lists and majorities. Our techniques are quite different from previous hardness proofs for learning. We use an invariance principle and sparse approximation of halfspaces from recent work on fooling halfspaces to give a new natural list decoding of a halfspace in the context of dictatorship tests/label cover reductions. In addition, unlike previous invariance principle based proofs which are only known to give Unique Games hardness, we give a reduction from a smooth version of Label Cover that is known to be NPhard.
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 ..."
Abstract

Cited by 22 (3 self)
 Add to MetaCart
(Show Context)
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
Improved approximation of linear threshold functions
 In Proc. 24nd Annual IEEE Conference on Computational Complexity (CCC
, 2009
"... We prove two main results on how arbitrary linear threshold functions f(x) = sign(w · x − θ) over the ndimensional Boolean hypercube can be approximated by simple threshold functions. Our first result shows that every nvariable threshold function f is ɛclose to a threshold function depending only ..."
Abstract

Cited by 19 (12 self)
 Add to MetaCart
(Show Context)
We prove two main results on how arbitrary linear threshold functions f(x) = sign(w · x − θ) over the ndimensional Boolean hypercube can be approximated by simple threshold functions. Our first result shows that every nvariable threshold function f is ɛclose to a threshold function depending only on Inf(f) 2 · poly(1/ɛ) many variables, where Inf(f) denotes the total influence or average sensitivity of f. This is an exponential sharpening of Friedgut’s wellknown theorem [Fri98], which states that every Boolean function f is ɛclose to a function depending only on 2 O(Inf(f)/ɛ) many variables, for the case of threshold functions. We complement this upper bound by showing that Ω(Inf(f) 2 + 1/ɛ 2) many variables are required for ɛapproximating threshold functions. Our second result is a proof that every nvariable threshold function is ɛclose to a threshold function with integer weights at most poly(n) · 2 Õ(1/ɛ2/3). This is an improvement, in the dependence on the error parameter ɛ, on an earlier result of [Ser07] which gave a poly(n) · 2 Õ(1/ɛ2) bound. Our improvement is obtained via a new proof technique that uses strong anticoncentration bounds from probability theory. The new technique also gives a simple and modular proof of the original [Ser07] result, and extends to give lowweight approximators for threshold functions under a range of probability distributions other than the uniform distribution.
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 ..."
Abstract

Cited by 18 (6 self)
 Add to MetaCart
(Show Context)
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.
A regularity lemma, and lowweight approximators, for lowdegree polynomial threshold functions
, 2009
"... ..."
Lower bounds for testing function isomorphism
, 2009
"... We prove new lower bounds in the area of property testing of boolean functions. Specifically, we study the problem of testing whether a boolean function f is isomorphic to a fixed function g (i.e., is equal to g up to permutation of the input variables). The analogous problem for testing graphs was ..."
Abstract

Cited by 16 (6 self)
 Add to MetaCart
(Show Context)
We prove new lower bounds in the area of property testing of boolean functions. Specifically, we study the problem of testing whether a boolean function f is isomorphic to a fixed function g (i.e., is equal to g up to permutation of the input variables). The analogous problem for testing graphs was solved by Fischer in 2005. The setting of boolean functions, however, appears to be more difficult, and no progress has been made since the initial study of the problem by Fischer et al. in 2004. Our first result shows that any nonadaptive algorithm for testing isomorphism to a function that “strongly” depends on k variables requires log k − O(1) queries (assuming k/n is bounded away from 1). This lower bound affirms and strengthens a conjecture appearing in the 2004 work of Fischer et al. Its proof relies on total variation bounds between hypergeometric distributions which may be of independent interest. Our second result concerns the simplest interesting case not covered by our first result: nonadaptively testing isomorphism to the Majority function on k variables. Here we show that Ω(k 1/12) queries are necessary (again assuming k/n is bounded away from 1); this exponentially improves on a related lower bound of Matulef et al. from 2009. The proof of this result relies on recently developed multidimensional “invariance principle ” tools.
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 ..."
Abstract

Cited by 11 (5 self)
 Add to MetaCart
(Show Context)
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 ..."
Abstract

Cited by 11 (1 self)
 Add to MetaCart
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