Title: Bounded Independence Fools Halfspaces
Speaker: Ilias Diakonikolas, Columbia
We show that any distribution on {-1,1}^n that is k-wise independent fools any halfspace h with error \eps for k = O(\log2(1/\eps) /\eps2). Up to logarithmic factors, our result matches a lower bound by Benjamini, Gurel-Gurevich, and Peled (2007) showing that k = \Omega(1/(\eps2 \cdot \log(1/\eps))). 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 \eps and seed length s = k \log n = O(\log n \cdot \log2(1/\eps) /\eps2).
Our approach combines classical tools from real approximation theory with structural results on halfspaces by Servedio (Computational Complexity 2007).
Joint work with Parikshit Gopalan (MSR SV), Ragesh Jaiswal (Columbia U.), Rocco Servedio (Columbia U.) and Emanuele Viola (Northeastern U.).