Bounded Independence Fools Halfspaces
Computational Complexity
2009-02-24 v1
Abstract
We show that any distribution on {-1,1}^n that is k-wise independent fools any halfspace h with error \eps for k = O(\log^2(1/\eps) /\eps^2). Up to logarithmic factors, our result matches a lower bound by Benjamini, Gurel-Gurevich, and Peled (2007) showing that k = \Omega(1/(\eps^2 \cdot \log(1/\eps))). Using standard constructions of k-wise 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 eps and seed length s = k \log n = O(\log n \cdot \log^2(1/\eps) /\eps^2). Our approach combines classical tools from real approximation theory with structural results on halfspaces by Servedio (Computational Complexity 2007).
Cite
@article{arxiv.0902.3757,
title = {Bounded Independence Fools Halfspaces},
author = {Ilias Diakonikolas and Parikshit Gopalan and Ragesh Jaiswal and Rocco Servedio and Emanuele Viola},
journal= {arXiv preprint arXiv:0902.3757},
year = {2009}
}