Testing +/- 1-Weight Halfspaces

We consider the problem of testing whether a Boolean function f : {−1, 1}n → {−1, 1} is a ±1-weight halfspace, i.e. a function of the form f(x) = sgn(w1x1 + w2x2 + · · · + wnxn) where the weights wi take values in {−1, 1}. We show that the complexity of this problem is markedly different from the problem of testing whether f is a general halfspace with arbitrary weights. While the latter can be done with a number of queries that is independent of n [7], to distinguish whether f is a ±1-weight halfspace versus -far from all such halfspaces we prove that nonadaptive algorithms must make Ω(log n) queries. We complement this lower bound with a sublinear upper bound showing that O( √ n·poly( 1 )) queries suffice.