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/eO(d) fooling degree d PTFs with error at most e. Previously, no nontrivial constructions were known even for quadratic threshold functions and constant error e. For the class of degree 1 threshold functions or halfspaces, we construct PRGs with much better dependence on the error parameter e and obtain the following results. A PRG with seed length O(log n log(1/e)) for error e ≥ 1/poly(n). A PRG with seed length O(log n) for e ≥ 1/poly(log n). Previously, only PRGs with seed length O(log n log2(1/e)/ e2) 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 e for halfspaces. These techniques may be of independent interest.