Fooling intersections of low-weight halfspaces

A weight-$t$ halfspace is a Boolean function $f(x)=\mathrm{sign}(w_1 x_1 + \cdots + w_n x_n - \theta)$ 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 polylog$(n)$ weight to any $1/$polylog$(n)$ accuracy using seed length polylog$(n).$

Prior to this work no explicit PRG with seed length $o(n)$ was known even for fooling intersections of $n$ weight-1 halfspaces to constant accuracy.

Our analysis introduces new coupling-based ingredients into the standard Lindeberg method for establishing quantitative central limit theorems and associated pseudorandomness results.

Joint work with Li-Yang Tan.