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Bounded Independence Fools Halfspaces

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5 Author(s)
Diakonikolas, I. ; Dept. of Comput. Sci., Columbia Univ., New York, NY, USA ; Gopalan, P. ; Jaiswal, R. ; Servedio, R.A.
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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 wiXi - ¿) where the w1,..., wn, ¿ are arbitrary real numbers, with error ¿ for k = O(¿-2 log2(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 log2(1/¿)). Our approach combines classical tools from real approximation theory with structural results on halfspaces by Servedio (Comput. Complexity 2007).

Published in:

Foundations of Computer Science, 2009. FOCS '09. 50th Annual IEEE Symposium on

Date of Conference:

25-27 Oct. 2009