Fooling Functions of Halfspaces under Product Distributions

Abstract

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.

New Algorithms and Lower Bounds for Monotonicity Testing

“…Their proof employed a D yes distribution which is the uniform distribution over all embedded majority functions of size n/2, and a D no distribution which is supported on certain monotone LTFs (which are far from embedded majority functions of size n/2). A key technical ingredient in the proofs of [BO10] is a multidimensional extension of the Berry-Esséen theorem (to independent sums of Ê q -valued random variables) which was essentially established in the work of [GOWZ10], building on ingredients from [Mos08]. Subsequently Ron and Servedio [RS13] adapted the arguments of [BO10] to give an improved analysis of the same D yes and D no distributions from [MORS09] and establish an Ω(n 1/12 )-query lower bound for non-adaptive algorithms that ε 0 -test whether f : {−1, 1} n → {−1, 1} is a signed majority function, thus exponentially improving over the [MORS09] lower bounds for this problem.…”

Pseudorandomness via the Discrete Fourier Transform

“…PRGs for (m, n)-Fourier shapes imply PRGs for generalized halfspaces. This in turn captures settings of fooling halfspaces with respect to the Gaussian distribution and the uniform distribution on the sphere [KRS12,MZ13,Kan14,KM15], and a large class of product distributions over R n [GOWZ10].…”

Bounded Independence Fools Halfspaces

“…Two natural questions are (1) to construct generators for halfspaces with a better dependence on in their seed length, ultimately achieving the information-theoretic optimum s = O(log(n/ )), and (2) to understand the degree of independence required to fool degree-d polynomial threshold functions (PTFs). After our work, there has been progress on the above (and other related) questions by several researchers [45,26,12,19,18,30] who improved and generalized our results in various directions. Meka and Zuckerman [45] constructed a generator for halfspaces with seed-length O(log(1/ ) · log n), and subsequently [26,30] gave generators for intersections of halfspaces.…”

“…After our work, there has been progress on the above (and other related) questions by several researchers [45,26,12,19,18,30] who improved and generalized our results in various directions. Meka and Zuckerman [45] constructed a generator for halfspaces with seed-length O(log(1/ ) · log n), and subsequently [26,30] gave generators for intersections of halfspaces. In [45] the authors also constructed generators for degree-d PTFs with seed-length 2…”