Year: 2026
Type: preprint
Abstract: The relative-error property testing model was introduced in [CDHLNSY24] 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 $\mathrm{Vol}(f \mathop{\triangle} g)/\mathrm{Vol}(f)$, 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 [CDHNSY26] 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 [CDHNSY26] lower bound by showing that halfspaces can be relative-error tested over $\mathbb{R}^n$ 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.
Source: arXiv (Cornell University)
Authors: Chen, Xi, De, Anindya, Huang, Yizhi, Nadimpalli, Shivam, Servedio, Rocco A., Yang, Tianqi
Institution: Columbia University
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
Funder: National Science Foundation
Awards: Collaborative Research: AF: Medium: Continuous Concrete Complexity, AF: Medium: The Trace Reconstruction Problem, Collaborative Research: AF:Medium: Advancing the Lower Bound Frontier, CAREER: Complexity of Matrix Operations, Collaborative Research: AF: Medium: Foundations of Anonymous Communication in Large-Scale Networks, CAREER: Learning and property testing -- a complexity theoretic perspective, AF: Medium: Smoothed Analysis for Optimization and Games
OpenAlex ID: W7149873989
Cites:
Cited by:
Related to: