An invariance principle for polytopes

Harsha, Prahladh; Klivans, Adam R.; Meka, Raghu

doi:10.1145/1806689.1806764

Cited by 17 publications

(84 citation statements)

References 30 publications

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“…Very recently, however, there has been a burst of progress on constructing PRGs for halfspaces [RS08, DGJ + 09, MZ09]. The present paper makes progress on this problem in several different directions, as do several concurrent and independent works [HKM09,DKN09,BELY09].…”

Section: Introductionmentioning

confidence: 90%

Fooling Functions of Halfspaces under Product Distributions

Gopalan

O’Donnell

et al. 2010

2010 IEEE 25th Annual Conference on Computational Complexity

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We construct pseudorandom generators that fool functions of halfspaces (threshold functions) under a very broad class of product distributions. This class includes not only familiar cases such as the uniform distribution on the discrete cube, the uniform distribution on the solid cube, and the multivariate Gaussian distribution, but also includes any product of discrete distributions with probabilities bounded away from 0.Our first main result shows that a recent pseudorandom generator construction of Meka and Zuckerman [MZ09], when suitably modified, can fool arbitrary functions of d halfspaces under product distributions where each coordinate has bounded fourth moment. To ǫ-fool any size-s, depth-d decision tree of halfspaces, our pseudorandom generator uses seed length O((d log(ds/ǫ) + log n) · log(ds/ǫ)). For monotone functions of d halfspaces, the seed length can be improved to O((d log(d/ǫ) + log n) · log(d/ǫ)). We get better bounds for larger ǫ; for example, to 1/polylog(n)-fool all monotone functions of (log n)/ log log n halfspaces, our generator requires a seed of length just O(log n).Our second main result generalizes the work of Diakonikolas et al. [DGJ + 09] to show that bounded independence suffices to fool functions of halfspaces under product distributions. Assuming each coordinate satisfies a certain stronger moment condition, we show that any function computable by a size-s, depth-d decision tree of halfspaces is ǫ-fooled byÕ(d 4 s 2 /ǫ 2 )-wise independence.Our technical contributions include: a new multidimensional version of the classical BerryEsseen theorem; a derandomization thereof; a generalization of Servedio [Ser07]'s regularity lemma for halfspaces which works under any product distribution with bounded fourth moments; an extension of this regularity lemma to functions of many halfspaces; and, new analysis of the sandwiching polynomials technique of Bazzi [Baz09] for arbitrary product distributions.

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Section: Introductionmentioning

confidence: 90%

Fooling Functions of Halfspaces under Product Distributions

Gopalan

O’Donnell

et al. 2010

2010 IEEE 25th Annual Conference on Computational Complexity

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“…Although this bound has been believed for some time, attempts to prove it have been unsuccessful. Perhaps the closest attempt thus far was by Harsha, Klivans and Meka who show in [7] that an intersection of k sufficiently regular halfspaces has noise sensitivity with parameter at most log(k) O(1) 1/6 . In this paper, we prove that the bound of O n log(k) is in face correct.…”

Section: Introductionmentioning

confidence: 99%

The average sensitivity of an intersection of half spaces

Kane¹

2014

Res Math Sci

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We prove new bounds on the average sensitivity of the indicator function of an intersection of k halfspaces. In particular, we prove the optimal bound of O n log(k) .This generalizes a result of Nazarov, who proved the analogous result in the Gaussian case, and improves upon a result of Harsha, Klivans and Meka. Furthermore, our result has implications for the runtime required to learn intersections of halfspaces.

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“…Using this framework, Harsha, Klivans and Meka [HKM13] proved an invariance principle for regular polytopes (i.e., when the coefficients in (all) the halfspaces are "regular"). The main novelty in their work was the poly-logarithmic (in the input parameters) error dependence.…”

Section: Prior Workmentioning

confidence: 99%

Positive spectrahedra: Invariance principles and Pseudorandom generators

Arunachalam¹,

Yao²

2021

Preprint

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In a recent work, O'Donnell, Servedio and Tan (STOC 2019) gave explicit pseudorandom generators (PRGs) for arbitrary m-facet polytopes in n variables with seed length poly-logarithmic in m, n, concluding a sequence of works in the last decade, that was started by Diakonikolas, Gopalan, Jaiswal, Servedio, Viola (SICOMP 2010) and Meka, Zuckerman (SICOMP 2013) for fooling linear and polynomial threshold functions, respectively. In this work, we consider a natural extension of PRGs for intersections of positive spectrahedrons. A positive spectrahedron is a Boolean function fwhere the A i s are k × k positive semidefinite matrices. We construct explicit PRGs that δ-fool "regular" width-M positive spectrahedrons (i.e., when none of the A i s are dominant) over the Boolean space with seed length poly(log k, log n, M, 1/δ).Our main technical contributions are the following: We first prove an invariance principle for positive spectrahedrons via the well-known Lindeberg method. As far as we are aware such a generalization of the Lindeberg method was unknown. Second, we prove various geometric properties of positive spectrahedrons such as their noise sensitivity, Gaussian surface area and a Littlewood-Offord theorem for positive spectrahedrons. Using these results, we give applications for constructing PRGs for positive spectrahedrons, learning theory, discrepancy sets for positive spectrahedrons (over the Boolean cube) and PRGs for intersections of structured polynomial threshold functions.

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An invariance principle for polytopes

Cited by 17 publications

References 30 publications

Fooling Functions of Halfspaces under Product Distributions

Fooling Functions of Halfspaces under Product Distributions

The average sensitivity of an intersection of half spaces

Positive spectrahedra: Invariance principles and Pseudorandom generators

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