Pseudorandom Generators for Polynomial Threshold Functions (0910.4122v5)

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.