Bounded Independence Fools Halfspaces
Ilias Diakonikolas, Parikshit Gopalan, Ragesh Jaiswal, Rocco A. Servedio, Emanuele Viola · SIAM Journal on Computing · 2010
We show that any distribution on $\{-1,+1\}^n$ that is k-wise independent fools any halfspace (or linear threshold function) $h:\{-1,+1\}^n\to\{-1,+1\}$, i.e., any function of the form $h(x)=\operatorname{sign}(\sum_{i=1}^{n}w_{i}x_{i}-\theta)$, where the $w_1,\dots,w_n$ and $\theta$ are arbitrary real numbers, with error $\epsilon$ for $k=O(\epsilon^{-2}\log^2(1/\epsilon))$. Our result is tight up to $\log(1/\epsilon)$ factors. Using standard constructions of k-wise independent distributions, we obtain the first explicit pseudorandom generators $G:\{-1,+1\}^s\to\{-1,+1\}^n$ that fool halfspaces. Specifically, we fool halfspaces with error $\epsilon$ and seed length $s=k\cdot\log n=O(\log n\cdot\epsilon^{-2}\log^2(1/\epsilon))$. Our approach combines classical tools from real approximation theory with structural results on halfspaces by Servedio [Comput. Complexity, 16 (2007), pp. 180–209].