Pseudorandom Generators for Polynomial Threshold Functions

TL;DR

This paper develops new pseudorandom generators for polynomial threshold functions, achieving better seed-lengths and error dependence, especially for low-degree and halfspace functions, using invariance principles and monotone read-once branching programs.

Contribution

It introduces the first nontrivial PRGs for low-degree PTFs and improves seed-length bounds for halfspaces using novel invariance and branching program techniques.

Findings

PRGs for degree d PTFs with seed-length log n/eps^{O(d)}

Improved seed-length O(log n + log^2(1/eps)) for halfspaces

Techniques applicable to fooling halfspaces over the unit sphere

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.