Fooling intersections of low-weight halfspaces

TL;DR

This paper presents an explicit pseudorandom generator that effectively fools intersections of low-weight halfspaces, significantly advancing derandomization techniques for complex Boolean functions.

Contribution

It introduces a novel PRG that fools intersections of weight-$t$ halfspaces with polylogarithmic seed length, combining techniques from multiple prior approaches.

Findings

Fools intersections of $k$ weight-$t$ halfspaces with seed length poly$( ext{log } n, ext{log } k, t, 1/ extdelta)$.

Achieves fooling of quasipoly$(n)$ halfspaces with seed length poly$ ext{log } n$.

No prior explicit PRG with non-trivial seed length was known for these functions.

Abstract

A weight-$t$ halfspace is a Boolean function $f(x)=$sign$(w_1 x_1 + \cdots + w_n x_n - \theta)$ where each $w_i$ is an integer in $\{-t,\dots,t\}.$ We give an explicit pseudorandom generator that $\delta$-fools any intersection of $k$ weight-$t$ halfspaces with seed length poly$(\log n, \log k,t,1/\delta)$. In particular, our result gives an explicit PRG that fools any intersection of any quasipoly$(n)$ number of halfspaces of any poly$\log(n)$ weight to any $1/$poly$\log(n)$ accuracy using seed length poly$\log(n).$ Prior to this work no explicit PRG with non-trivial seed length was known even for fooling intersections of $n$ weight-1 halfspaces to constant accuracy. The analysis of our PRG fuses techniques from two different lines of work on unconditional pseudorandomness for different kinds of Boolean functions. We extend the approach of Harsha, Klivans and Meka \cite{HKM12} for fooling intersections of regular halfspaces, and combine this approach with results of Bazzi \cite{Bazzi:07} and Razborov \cite{Razborov:09} on bounded independence fooling CNF formulas. Our analysis introduces new coupling-based ingredients into the standard Lindeberg method for establishing quantitative central limit theorems and associated pseudorandomness results.