Halfspaces are hard to test with relative error

Xi Chen, Anindya De, Yizhi Huang, Shivam Nadimpalli, Rocco A. Servedio, Tianqi Yang

Abstract: Several recent works (Chen et al., SODA 2025; Chen et al., ICALP 2025; Chen et al., COLT 2025; Chen et al., manuscript) have studied a model of property testing of Boolean functions under a relative-error criterion. In this model, the distance from a target function \(f : \{0, 1\}^n \rightarrow \{0, 1\}\) 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 \(\tilde \Omega(\log n)\) 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 (Mossel et al., SICOMP, 2010) and with the positive results for relative-error testing of many other classes given in (Chen et al., SODA 2025; Chen et al., ICALP 2025; Chen et al., COLT 2025; Chen et al., manuscript). 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.

External IDs:dblp:conf/soda/0001DHNSY26