Pseudorandom generators for polynomial threshold functions

  • Authors:

  • Raghu Meka;David Zuckerman

  • Affiliations:

  • University of Texas, Austin, TX, USA;University of Texas, Austin, TX, USA

  • Venue:
  • Proceedings of the forty-second ACM symposium on Theory of computing
  • Year:
  • 2010

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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/εO(d) fooling degree d PTFs with error at most ε. Previously, no nontrivial constructions were known even for quadratic threshold functions and constant error ε. For the class of degree 1 threshold functions or halfspaces, we construct PRGs with much better dependence on the error parameter ε and obtain the following results. A PRG with seed length O(log n log(1/ε)) for error ε ≥ 1/poly(n). A PRG with seed length O(log n) for ε ≥ 1/poly(log n). Previously, only PRGs with seed length O(log n log2(1/ε)/ ε2) 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 ε for halfspaces. These techniques may be of independent interest.