Bounded Independence Fools Halfspaces

Ilias Diakonikolas, Parikshit Gopalan, Ragesh Jaiswal, Rocco A. Servedio, Emanuele Viola

Abstract: We show that any distribution on {-1,+1} <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> that is k-wise independent fools any halfspace (a.k.a. threshold) h : {-1,+1} <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> ¿ {-1,+1}, i.e., any function of the form h(x) = sign(¿ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i=1</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> w <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> X <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> - ¿) where the w <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> ,..., w <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> , ¿ are arbitrary real numbers, with error ¿ for k = O(¿ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-2</sup> log <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> (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} <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">s</sup> ¿ {-1,+1} <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> that fool halfspaces. Specifically, we fool halfspaces with error e and seed length s = k · log n = O(log n · ¿ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-2</sup> log <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> (1/¿)). Our approach combines classical tools from real approximation theory with structural results on halfspaces by Servedio (Comput. Complexity 2007).