Halfspaces are hard to test with relative error

Several recent works [CDH + 25, CPPS25a, CPPS25b, CPPS26] have studied a model of property testing of Boolean functions under a relative-error criterion. In this model, the distance from a target function f : 0, 1 n → 0, 1 that is being tested to a function g is defined relative to the number of inputs x for which f (x) = 1; moreover, testing algorithms in this model have access both to a black-box oracle for f and to independent uniform satisfying assignments of f . The motivation for this model is that it provides a natural framework for testing sparse Boolean functions that have few satisfying assignments, analogous to well-studied models for property testing of sparse graphs.

The main result of this paper is a lower bound for testing halfspaces (i.e., linear threshold functions) in the relative error model: we show that Ω(log n) oracle calls are required for any relative-error halfspace testing algorithm over the Boolean hypercube 0, 1 n . This stands in sharp contrast both with the constant-query testability (independent of n) of halfspaces in the standard model [MORS10], and with the positive results for relative-error testing of many other classes given in [CDH + 25, CPPS25a, CPPS25b, CPPS26]. Our lower bound for halfspaces gives the first example of a well-studied class of functions for which relative-error testing is provably more difficult than standard-model testing.