# Bounded Independence Fools Halfspaces

Ilias Diakonikolas, Parikshit Gopalan, Ragesh Jaiswal, Rocco A. Servedio, Emanuele Viola

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## 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)
Original language English 16 1 Electronic Colloquium on Computational Complexity (ECCC) 16 Published - 2009