Testing ±1-weight halfspace

Abstract

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(w 1 x 1 + w 2 x 2 + ⋯ + w n x n) where the weights w i 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 ±-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(poly queries suffice. © 2009 Springer.