Bounded Independence Fools Halfspaces

We show that any distribution on {-1,+1}
ⁿ
that is k-wise independent fools any halfspace (a.k.a. threshold) h : {-1,+1}
ⁿ
¿ {-1,+1}, i.e., any function of the form h(x) = sign(¿
_i=1

ⁿ
w
_i
X
_i
- ¿) where the w
₁
,..., w
_n
, ¿ are arbitrary real numbers, with error ¿ for k = O(¿
^-2
log
²
(1/¿)). Our result is tight up to log(1/¿) factors. Using standard constructions of k-wise independent distributions, we obtain the first explicit pseudorandom generators G : {-1,+1}
^s
¿ {-1,+1}
ⁿ
that fool halfspaces. Specifically, we fool halfspaces with error e and seed length s = k · log n = O(log n · ¿
^-2
log
²
(1/¿)). Our approach combines classical tools from real approximation theory with structural results on halfspaces by Servedio (Comput. Complexity 2007).