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 $\log n/\epsilon^{O(d)}$ fooling degree $d$ PTFs with error at most $\epsilon$. Previously, no nontrivial constructions were known even for quadratic threshold functions and constant error $\epsilon$. For the class of degree $1$ threshold functions or halfspaces, previously only PRGs with seed length $O(\log n \log^2(1/\epsilon)/\epsilon^2)$ were known. We improve this dependence on the error parameter and construct PRGs with seed length $O(\log n + \log^2 (1/\epsilon))$ that $\epsilon$-fool 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 PRGs. We also introduce the notion of monotone read-once branching programs, which is key to improving the dependence on the error rate $\epsilo...