Fooling Intersections of Low-Weight Halfspaces

A weight-t halfspace} is a Boolean function f(x)=\sign(w_1 x_1 + … + w_n x_n - θ) where each w_i is an integer in \{-t,\dots,t\}. We give an explicit pseudorandom generator that δ-fools any intersection of k weight-t halfspaces with seed length \poly(\log n, \log k,t,1/δ). In particular, our result gives an explicit PRG that fools any intersection of any quasi\poly(n) number of halfspaces of any \polylog(n) weight to any 1/\polylog(n) accuracy using seed length \polylog(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.