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 $\tildeΩ(\log n)$ 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.

翻译：近期多项研究[DHLNSY25, CPPS25a, CPPS25b]探讨了基于相对误差准则的布尔函数性质测试模型。在该模型中，被测目标函数 $f: \{0,1\}^n \to \{0,1\}$ 与函数 $g$ 之间的距离定义为相对于使 $f(x)=1$ 的输入 $x$ 数量的误差；此外，该模型下的测试算法可同时访问 $f$ 的黑箱预言机和独立均匀满足赋值。该模型的动机在于，它为测试具有少量满足赋值的稀疏布尔函数提供了自然框架，类似于稀疏图性质测试中已被充分研究的模型。本文的主要结果是在相对误差模型下针对半空间（即线性阈值函数）测试的下界：我们证明在布尔超立方体 $\{0,1\}^n$ 上，任何相对误差半空间测试算法至少需要 $\tildeΩ(\log n)$ 次预言机调用。这一结果与标准模型下半空间的常数查询可测试性（不依赖于 $n$）[MORS10]，以及[DHLNSY25, CPPS25a, CPPS25b]中其他多类函数的相对误差测试正向结果形成鲜明对比。我们的半空间下界首次证明：对于一个已被充分研究的函数类，相对误差测试的难度确实高于标准模型测试。