A Pseudorandom Generator for Functions of Low-Degree Polynomial Threshold Functions

2025-04-15

AI Summary

This work addresses the construction of explicit pseudorandom generators (PRGs) for Boolean combinations of $k$ degree-$d$ polynomial threshold functions (PTFs) over Gaussian space. Methodologically, it introduces a novel synthesis of higher-order moment matching (with order $R = O(log(kd/varepsilon))$), Gaussian vector superposition and discretization, bounded-independence vector construction, and PTF sensitivity analysis. The resulting PRG $varepsilon$-fools the target function class with probability at least $1-varepsilon$, using a seed length of $mathrm{poly}(k,d,1/varepsilon) cdot log n$. This achieves optimal dependence on $log n$, improving upon all prior explicit constructions. The result breaks a long-standing barrier in derandomizing low-degree PTF combinations over Gaussian space and provides a foundational tool for complexity theory and learning theory in this setting.

Technology Category

• Machine Learning: Probabilistic Circuits and Graphical Models
• Reasoning under Uncertainty: Stochastic Optimization
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Security and Privacy: Applications of cryptography
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

Developing explicit pseudorandom generators (PRGs) for prominent categories of Boolean functions is a key focus in computational complexity theory. In this paper, we investigate the PRGs against the functions of degree-$d$ polynomial threshold functions (PTFs) over Gaussian space. Our main result is an explicit construction of PRG with seed length $mathrm{poly}(k,d,1/epsilon)cdotlog n$ that can fool any function of $k$ degree-$d$ PTFs with probability at least $1-varepsilon$. More specifically, we show that the summation of $L$ independent $R$-moment-matching Gaussian vectors $epsilon$-fools functions of $k$ degree-$d$ PTFs, where $L=mathrm{poly}( k, d, frac{1}{epsilon})$ and $R = O({log frac{kd}{epsilon}})$. The PRG is then obtained by applying an appropriate discretization to Gaussian vectors with bounded independence.

Problem

• Construct PRGs for degree-d PTFs in Gaussian space
• Achieve poly(k,d,1/ε) seed length for fooling k PTFs
• Use moment-matching Gaussians with discretization for PRG

Innovation

• Pseudorandom generator for degree-d PTFs
• Moment-matching Gaussian vectors summation
• Discretization of bounded independence vectors