[Paper Review] Pseudorandom Generators for Polynomial Threshold Functions

Raghu Meka, David Zuckerman|arXiv (Cornell University)|Oct 21, 2009

Complexity and Algorithms in Graphs24 references4 citations

TL;DR

This paper presents explicit pseudorandom generators (PRGs) for low-degree polynomial threshold functions (PTFs) and halfspaces using invariance principles and monotone read-once branching programs. It achieves seed length $ O( ext{poly}(d) ext{polylog}(n)/ ext{poly}(
)) $ for degree-$ d $ PTFs and $ O(\log n + \log^2(1/\epsilon)) $ for halfspaces, significantly improving prior bounds on error dependence and enabling derandomization of learning and optimization algorithms.

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.

Motivation & Objective

• To construct explicit pseudorandom generators (PRGs) that fool low-degree polynomial threshold functions (PTFs) with small error.
• To improve the dependence on the error parameter $\epsilon$ in PRG seed length, especially for halfspaces.
• To extend results to spherical caps and intersections of halfspaces, relevant for derandomizing algorithms like Goemans-Williamson Max-Cut.
• To develop techniques that may be of independent interest, such as monotone read-once branching programs and derandomized invariance principles.
• To resolve the long-standing open problem of constructing nontrivial PRGs for degree-2 PTFs and general halfspaces with sublinear seed length.

Proposed method

• Leverages invariance principles to show that low-degree PTFs behave similarly under certain pseudorandom distributions and Gaussian measures.
• Introduces monotone read-once branching programs as a key structural tool to improve error dependence in halfspace PRGs.
• Uses $8$-wise independent distributions to derandomize the Ailon-Chazelle rotation construction, ensuring regularity of weight vectors with high probability.
• Applies hypercontractivity and moment bounds to control the $L^4$-norm of transformed weight vectors under pseudorandom inputs.
• Combines invariance principles with PRGs for small-space machines (e.g., Impagliazzo-Nisan-Wigderson) to construct PRGs for PTFs.
• Derandomizes spherical cap sampling by composing a PRG with a pseudorandom rotation matrix derived from Hadamard transforms and $8$-wise independence.

Experimental results

Research questions

• RQ1 Can we construct explicit PRGs for degree-$ d $ polynomial threshold functions with seed length sublinear in $ n $ and polynomial in $ 1/\epsilon $?
• RQ2 What is the optimal dependence of seed length on the error parameter $\epsilon$ for halfspace PRGs, and can it be improved beyond $ O(\log n \cdot \log^2(1/\epsilon)/\epsilon^2) $?
• RQ3 Can invariance principles be used to construct PRGs for spherical caps and intersections of halfspaces with small seed length?
• RQ4 Can the Ailon-Chazelle rotation method be derandomized to preserve regularity of weight vectors under pseudorandom inputs?
• RQ5 Can monotone read-once branching programs be used to achieve exponentially better error dependence in PRG construction for halfspaces?

Key findings

• The paper constructs an explicit PRG for degree-$ d $ PTFs with seed length $ 2^{O(d)} \log n / \epsilon^{8d+3} $, which is the first nontrivial construction for such functions.
• For halfspaces, it achieves a seed length of $ O(\log n + \log^2(1/\epsilon)) $, improving the prior best bound of $ O(\log n \cdot \log^2(1/\epsilon)/\epsilon^2) $.
• The PRG for spherical caps achieves seed length $ O(\log n + \log^2(1/\epsilon)) $ for $ \epsilon \geq C \log n / n^{1/4} $, matching the state-of-the-art up to constants.
• The use of $ 8 $-wise independent distributions ensures that the rotated weight vector is regular with high probability, enabling application of invariance principles.
• The improved seed length for halfspaces results from replacing Nisan’s PRG with the INW PRG in the monotone trick, reducing error dependence from $ \log(1/\epsilon) $ to $ \log^2(1/\epsilon) $.
• The results establish that invariance principles combined with derandomized transformations yield effective PRGs for structured function classes, with implications for learning and optimization.