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A weight-t halfspace is a Boolean function f(x) = sign(w1x1 + &#x22EF; + wnxn - &#x03B8;) where each wi is an integer in {-t, . . . , t}. We give an explicit pseudorandom generator that &#x03B4;-fools any intersection of k weight-t halfspaces with seed length poly(log n, log k, t, 1/&#x03B4;). 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 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 [HKM12] for fooling intersections of regular halfspaces, and combine this approach with results of Bazzi [Baz07] and Razborov [Raz09] on bounded independence fooling CNF formulas. Our analysis introduces new couplingbased ingredients into the standard Lindeberg method for establishing quantitative central limit theorems and associated pseudorandomness results.

Fooling Intersections of Low-Weight Halfspaces