Pseudorandom Generators for Polynomial Threshold Functions
Abstract
We study the natural question of constructing pseudorandom generators (PRGs) for low-degree polynomial threshold functions (PTFs). We give a PRG with seed-length log n/eps^{O(d)} fooling degree d PTFs with error at most eps. Previously, no nontrivial constructions were known even for quadratic threshold functions and constant error eps. For the class of degree 1 threshold functions or halfspaces, we construct PRGs with much better dependence on the error parameter eps and obtain a PRG with seed-length O(log n + log^2(1/eps)). Previously, only PRGs with seed length O(log n log^2(1/eps)/eps^2) were known for halfspaces. We also obtain PRGs with similar seed lengths for fooling halfspaces over the n-dimensional unit sphere. The main theme of our constructions and analysis is the use of invariance principles to construct pseudorandom generators. We also introduce the notion of monotone read-once branching programs, which is key to improving the dependence on the error rate eps for halfspaces. These techniques may be of independent interest.
Keywords
Cite
@article{arxiv.0910.4122,
title = {Pseudorandom Generators for Polynomial Threshold Functions},
author = {Raghu Meka and David Zuckerman},
journal= {arXiv preprint arXiv:0910.4122},
year = {2015}
}
Comments
Revision 5: Updated to the journal version to appear in SICOMP. Revision 4: Improves seed-length for halfspaces to O(log n + log^2(1/eps)) (the change in analysis is minor: use INW PRG instead of Nisan's PRG). Revision 3: Fixed some more minor errors (mainly in proof of Theorem 4.3). Revision 2: Corrected the non-explicit bound to O(d log n + log(1/eps)) and some minor typos