Bounded Independence Fools Degree-2 Threshold Functions

Abstract

For an $n$-variate degree–$2$ real polynomial $p$, we prove that $\E_{x\sim \mathcal{D}}[\sgn(p(x))]$ is determined up to an additive $\eps$ as long as $\mathcal{D}$ is a $k$-wise independent distribution over $\bits^n$ for $k = \poly(1/\eps)$. This gives a broad class of explicit pseudorandom generators against degree-$2$ boolean threshold functions, and answers an open question of Diakonikolas et al. (FOCS 2009).