Sublinear-query relative-error testing of halfspaces

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.

Theorems & Definitions (47)

• Remark 1
• Theorem 1: Gaussian halfspace testing for known $p$
• Theorem 2: Sample-based Gaussian halfspace testing for known $p$
• Theorem 3: Gaussian halfspace testing for unknown $p$
• Proposition 1: Proposition 2.1.2 of vershynin2018high or Exercise 2.2 of Wainwright19book
• Theorem 4: Theorem 26 of MORS10
• Theorem 5: Theorem 1.1 of Harms19
• Definition 1
• Theorem 6
• Theorem 7: Theorem 1.2, Theorem 2.1 of Neeman14