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Sublinear-Query Relative-Error Testing of Halfspaces

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

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ACM Subject Classification
  • Theory of computation → Streaming, sublinear and near linear time algorithms
Keywords
  • Property testing
  • relative-error testing
  • halfspaces
  • Gaussian space

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Abstract

The relative-error property testing model was introduced in [Chen et al., 2024] to facilitate the study of property testing for "sparse" Boolean-valued functions, i.e. ones for which only a small fraction of all input assignments satisfy the function. In this framework, the distance from the unknown target function f that is being tested to a function g is defined as Vol(f△g)/Vol(f), where the numerator is the fraction of inputs on which f and g disagree and the denominator is the fraction of inputs that satisfy f. 
Recent work [Chen et al., 2026] has shown that over the Boolean domain {0,1}ⁿ, any relative-error testing algorithm for the fundamental class of {halfspaces} (i.e. linear threshold functions) must make Ω(log n) oracle calls. In this paper we complement the [Chen et al., 2026] lower bound by showing that halfspaces can be relative-error tested over ℝⁿ under the standard N(0,I_n) Gaussian distribution using a sublinear number of oracle calls - in particular, substantially fewer than would be required for learning. Our results use a wide range of tools including Hermite analysis, Gaussian isoperimetric inequalities, and geometric results on noise sensitivity and surface area.

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Xi Chen, Anindya De, Yizhi Huang, Shivam Nadimpalli, Rocco A. Servedio, and Tianqi Yang. Sublinear-Query Relative-Error Testing of Halfspaces. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 61:1-61:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.ICALP.2026.61

Author Details

Xi Chen
  • Columbia University, New York, NY, USA
Anindya De
  • University of Pennsylvania, Philadelphia, PA, USA
Yizhi Huang
  • Columbia University, New York, NY, USA
Shivam Nadimpalli
  • MIT, Cambridge, MA, USA
Rocco A. Servedio
  • Columbia University, New York, NY, USA
Tianqi Yang
  • Columbia University, New York, NY, USA

Funding

This work was awarded a grant by the AI Security Institute (AISI) via the Alignment Project ("Discovering rare harmful behaviors exhibited by high-dimensional AI systems"). This work is also funded by OpenAI. T.Y. and Y.H. are also supported by an Amazon Research Award and a Google CyberNYC award.
  • Chen, Xi: NSF grants CCF-2106429 and CCF-2107187.
  • De, Anindya: NSF grant CCF 2045128.
  • Huang, Yizhi: NSF grants CCF-2211238, CCF-2312242, CCF-2106429, and CCF-2238221.
  • Servedio, Rocco A.: NSF grants CCF-2211238 and CCF-2106429.
  • Yang, Tianqi: NSF grants CCF-2211238, CCF-2106429, CCF-2312242, and AF-Medium 2212136.

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