Testing (Subclasses of) Halfspaces

Matulef, Kevin; O’Donnell, Ryan; Rubinfeld, Ronitt; Servedio, Rocco A.

doi:10.1007/978-3-642-16367-8_27

Cited by 3 publications

(3 citation statements)

References 21 publications

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“…The high level idea in the proof of Theorem 9 is to exploit the so-called "structure versus randomness" phenomenon for halfspaces which was introduced in the influential work of Servedio [43] and has subsequently played a crucial role in the recent developments in the complexity theoretic analysis of halfspaces [43,18,15,33,32] (we explain this phenomenon a little later). Results in this line of work have mostly looked at halfspaces of the form g(X 1 , .…”

Section: Theoremmentioning

confidence: 99%

Boolean function analysis meets stochastic optimization: An approximation scheme for stochastic knapsack

De¹

2018

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

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The stochastic knapsack problem is the stochastic variant of the classical knapsack problem in which the algorithm designer is given a a knapsack with a given capacity and a collection of items where each item is associated with a profit and a probability distribution on its size. The goal is to select a subset of items with maximum profit and violate the capacity constraint with probability at most p (referred to as the overflow probability).While several approximation algorithms [27,22,4,17,30] have been developed for this problem, most of these algorithms relax the capacity constraint of the knapsack. In this paper, we design efficient approximation schemes for this problem without relaxing the capacity constraint.(i) Our first result is in the case when item sizes are Bernoulli random variables. In this case, we design a (nearly) fully polynomial time approximation scheme (FPTAS) which only relaxes the overflow probability.(ii) Our second result generalizes the first result to the case when all the item sizes are supported on a (common) set of constant size. In this case, we obtain a quasi-FPTAS.(iii) Our third result is in the case when item sizes are socalled "hypercontractive" random variables i.e., random variables whose second and fourth moments are within constant factors of each other. In other words, the kurtosis of the random variable is upper bounded by a constant. This class has been widely studied in probability theory and most natural random variables are hypercontractive including well-known families such as Poisson, Gaussian, exponential and Laplace distributions. In this case, we design a polynomial time approximation scheme which relaxes both the overflow probability and maximum profit.Crucially, all of our algorithms meet the capacity constraint exactly, a result which was previously known only when the item sizes were Poisson or Gaussian random variables [22,24]. Our results rely on new connections between Boolean function analysis and stochastic optimization and are obtained by an adaption and extension of ideas such as (central) limit theorems, moment matching theorems and the influential critical index machinery of Servedio [43] developed in the context of complexity theoretic analysis of halfspaces. We believe that these ideas and techniques may prove to be useful in other stochastic optimization problems as well.

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Section: Theoremmentioning

confidence: 99%

Boolean function analysis meets stochastic optimization: An approximation scheme for stochastic knapsack

De¹

2018

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

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“…We say that an algorithm is a tester for a property P if, given query access to a measurable function f : R n → R, sampling access to the standard Gaussian, and ε > 0, it accepts with probability at least 2/3 when f satisfies P , and rejects with probability at least 2/3 when f is ε-far from P . Testability of a variety of properties has been considered, including surface area of a set [29,34], half spaces [31][32][33], linear separators [3], high-dimensional convexity [13], and linear k-junta [15]. Although the standard Gaussian is natural, it barely appears in practice.…”

Section: Introductionmentioning

confidence: 99%

Distribution-Free Testing of Linear Functions on R^n

Fleming¹,

Yoshida²

2019

Preprint

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We study the problem of testing whether a function f : R n → R is linear (i.e., both additive and homogeneous) in the distribution-free property testing model, where the distance between functions is measured with respect to an unknown probability distribution over R. We show that, given query access to f , sampling access to the unknown distribution as well as the standard Gaussian, and ε > 0, we can distinguish additive functions from functions that are ε-far from additive functions with O 1 ε log 1 ε queries, independent of n. Furthermore, under the assumption that f is a continuous function, the additivity tester can be extended to a distribution-free tester for linearity using the same number of queries. On the other hand, we show that if we are only allowed to get values of f on sampled points, then any distribution-free tester requires Ω(n) samples, even if the underlying distribution is the standard Gaussian.

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“…For Boolean functions, however, testability is less well understood. On one hand, there are a fair number of testing algorithms for specific classes of functions such as F 2 -linear functions [10,6], dictators [7,23], low-degree F 2 -polynomials [2,24], juntas [15,9], and halfspaces [22]. But there is not much by way of general characterizations of what makes a property of Boolean functions testable.…”

Section: Introductionmentioning

confidence: 99%

Testing Fourier Dimensionality and Sparsity

Gopalan

O’Donnell

Servedio

et al. 2009

Automata, Languages and Programming

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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 low-dimensional 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 onemust 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 sub-property 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].

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Testing (Subclasses of) Halfspaces

Cited by 3 publications

References 21 publications

Boolean function analysis meets stochastic optimization: An approximation scheme for stochastic knapsack

Boolean function analysis meets stochastic optimization: An approximation scheme for stochastic knapsack

Distribution-Free Testing of Linear Functions on R^n

Testing Fourier Dimensionality and Sparsity

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