Sublinear-query relative-error testing of halfspaces

TL;DR

This paper demonstrates that halfspaces over Gaussian distributions can be tested with sublinear oracle calls in a relative-error model, contrasting prior lower bounds for Boolean domains.

Contribution

It introduces a sublinear-query relative-error testing algorithm for halfspaces over Gaussian distributions, surpassing known lower bounds for Boolean domains.

Findings

Halfspaces can be tested with fewer than linear oracle calls under Gaussian distribution.

The approach uses Hermite analysis, Gaussian isoperimetric inequalities, and geometric noise sensitivity results.

The testing complexity is substantially less than the learning complexity for 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 $\Omega(\log n)$ 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.