Bounded Independence Fools Halfspaces

TL;DR

This paper proves that k-wise independent distributions can effectively fool halfspaces with small error, providing explicit pseudorandom generators with near-optimal seed length, by combining approximation theory and structural analysis.

Contribution

It establishes near-optimal bounds for fooling halfspaces with k-wise independence and constructs explicit pseudorandom generators with efficient seed length.

Findings

k-wise independence fools halfspaces with error psilon for k = O(\u03bclog^2(1/psilon)/psilon^2)

First explicit pseudorandom generators fooling halfspaces with seed length O(\u0003bclog n d7 bclog^2(1/psilon)/psilon^2)

Combines real approximation theory with structural results to achieve these bounds

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).