Halfspaces are hard to test with relative error
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911533348421632 |
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| author | Chen, Xi De, Anindya Huang, Yizhi Nadimpalli, Shivam Servedio, Rocco A. Yang, Tianqi |
| author_facet | Chen, Xi De, Anindya Huang, Yizhi Nadimpalli, Shivam Servedio, Rocco A. Yang, Tianqi |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_06171 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Halfspaces are hard to test with relative error Chen, Xi De, Anindya Huang, Yizhi Nadimpalli, Shivam Servedio, Rocco A. Yang, Tianqi Computational Complexity Data Structures and Algorithms 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. |
| title | Halfspaces are hard to test with relative error |
| topic | Computational Complexity Data Structures and Algorithms |
| url | https://arxiv.org/abs/2511.06171 |