Halfspaces are hard to test with relative error

TL;DR

This paper establishes a logarithmic lower bound on the number of oracle calls needed to test halfspaces under a relative-error model, highlighting increased complexity compared to standard testing.

Contribution

It provides the first proven lower bound showing that relative-error testing of halfspaces requires logarithmic queries, unlike constant-query testing in the standard model.

Findings

Halfspaces require  ( \, ext{log} \, n) oracle calls for testing.

Relative-error testing can be significantly harder than standard testing for certain classes.

This is the first example of a class with provably increased difficulty in the relative-error testing model.

Abstract

Several recent works [DHLNSY25, CPPS25a, CPPS25b] have studied a model of property testing of Boolean functions under a \emph{relative-error} criterion. In this model, the distance from a target function $f: \{0,1\}^n \to \{0,1\}$ 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 \emph{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 \emph{halfspaces} (i.e., linear threshold functions) in the relative error model: we show that $\tilde{\Omega}(\log n)$ 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 [DHLNSY25, CPPS25a, CPPS25b]. 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.