Testing Halfspaces over Rotation-Invariant Distributions

We present an algorithm for testing halfspaces over arbitrary, unknown rotation-invariant distributions. Using $\tilde O(\sqrt{n}ε^{-7})$ random examples of an unknown function $f$, the algorithm determines with high probability whether $f$ is of the form $f(x) = sign(\sum_i w_ix_i-t)$ or is $ε$-far from all such functions. This sample size is significantly smaller than the well-known requirement of $Ω(n)$ samples for learning halfspaces, and known lower bounds imply that our sample size is optimal (in its dependence on $n$) up to logarithmic factors. The algorithm is distribution-free in the sense that it requires no knowledge of the distribution aside from the promise of rotation invariance. To prove the correctness of this algorithm we present a theorem relating the distance between a function and a halfspace to the distance between their centers of mass, that applies to arbitrary distributions.