Fooling intersections of low-weight halfspaces

Computer Science/Discrete Mathematics Seminar I
Topic:Fooling intersections of low-weight halfspaces
Speaker:Rocco Servedio
Affiliation:Columbia University
Date:Monday, October 30
Time/Room:11:00am - 12:15pm/S-101
Video Link:https://video.ias.edu/csdm/2017/1030-RoccoServedio

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.