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 ${0,1}<sup>n$, 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 $\mathbb{R}<sup>n$ 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.

• The paper presents novel sublinear-query testers for halfspaces under Gaussian measure that leverage relative-error property testing.
• It introduces three algorithmic frameworks, including a Gaussian surface area tester and a sample-only Hermite-based tester, achieving sublinear complexities.
• The work establishes a testing versus learning separation and offers efficient protocols for high-dimensional, sparse concept verification.

Sublinear-Query Relative-Error Testing of Halfspaces

Introduction and Motivation

The paper "Sublinear-query relative-error testing of halfspaces" (2604.01557) addresses the query complexity of property testing for halfspaces—i.e., linear threshold functions—within the relative-error testing (RET) model. Classical property testing considers the absolute distance to a property, often unsuitable for sparse Boolean functions due to their insensitivity to small, structured deviations. RET, as introduced by [CDHLNSY24], instead normalizes the proximity criterion to the measure (often very small) of the "positive" set of the function under test, making it relevant for high-dimensional properties with small acceptance volumes.

Recent work [CDHNSY26soda] established an $\Omega(\log n)$ lower bound for RET of halfspaces under the uniform discrete setting, in contrast to the standard (absolute-error) property testing model, where halfspaces are testable with a query complexity independent of $n$. This brings forth a natural question: can one achieve sublinear-in-$n$ RET for halfspaces in non-discrete settings, e.g., over $\mathbb{R}^n$ under the standard Gaussian? This paper delivers affirmative results, presenting the first sublinear-query testers for halfspaces in the RET model under Gaussian measure, and elucidating the technical landscape enabling such improvements.

Problem Formulation and Relative-Error Model

Given an unknown $f: \mathbb{R}^n \rightarrow \{0,1\}$ accessible via two oracles—a pointwise (membership) query and a positive example sampler ($SAMP(f)$, sampling from the preimage of 1 under $f$ according to conditioned Gaussian measure)—the challenge is to decide, with high confidence using as few queries and samples as possible, whether $f$ is a halfspace (i.e., for some $w, \theta$, $f(x) = \mathbb{I}[w \cdot x \ge \theta]$), or $n$0 is $n$1-far (in the relative sense) from all halfspaces.

Relative error here is defined as

$n$2

where the denominator is typically small for sparse functions, ensuring the distance is meaningful for such problems.

Main Contributions and Algorithmic Frameworks

Sublinear RET Algorithms

The central contributions are three algorithms, each providing RET of halfspaces with query complexity sublinear in the ambient dimension:

1. Gaussian Surface Area Tester (With Known Volume): If the measure $n$3 is known (up to a fine estimate), there exists an RET of halfspaces using $n$4 queries/samples. This tester is based on robust, dimension-free isoperimetric inequalities for Gaussian surface area, and leverages a sophisticated surface area tester due to Neeman.
2. Sample-Only Hermite-Based Tester: When only access to $n$5 (positive samples from $n$6) is given and $n$7 is known, one can achieve RET using $n$8 sample complexity. The algorithm exploits the Hermite spectrum and sharp analytic characterizations of halfspaces in Gaussian space, comparing empirical estimates of level-1 Hermite moments to their extremal values.
3. Unknown-Volume (Volume-Adaptive) Tester: If only a lower bound $n$9 on $n$0 is known, one can combine the aforementioned techniques to achieve RET with $n$1 complexity, plus lower-order polynomial and logarithmic terms.

Technical Highlights

• Dimension-Free Robust Gaussian Isoperimetry: The crux for (1) is the quantitative stability theorem for Gaussian surface area minimization. Results of Barchiesi et al. and Mossel–Neeman are used to deduce that if a set $n$2 has (relative to its measure $n$3) nearly minimal Gaussian surface area, then $n$4 is close to a halfspace in symmetric difference. This stability underpins the correctness and soundness of the Gaussian surface area approach.
• Testing via Surface Area: Building on an algorithm of Neeman for surface area estimation, the tester estimates the normalized noise sensitivity, which corresponds to surface area in the small-noise limit, and compares it to the (explicit) minimal value achieved by halfspaces.
• Hermite Analysis and Level-1 Weight Characterization: For sample-only testing, the key observation is that for an LTF, the level-1 Hermite weight is uniquely determined by the function's mean. Deviations of a function's level-1 Hermite spectrum from that of any halfspace with matching measure indicate RET. Estimation is performed using an unbiased quadratic statistic computed from positive samples, with variance controlled by hypercontractive Gaussian inequalities.
• One-Sided Volume-Immune Reduction: The adaptive volume tester forms a "guess" for the measure via inversion of the analytical mapping from surface area/Hermite statistic to $n$5, and applies the robust surface-area tester on this $n$6. If $n$7 is far from all halfspaces, this guess will necessarily be biased in a one-sided fashion (overestimating $n$8), so rejection is sound.
• Sample Complexity Separation: The frameworks utilized scale strictly sublinearly with $n$9; for sample-only testers, the $\mathbb{R}^n$0 dependency is information-theoretically optimal up to logarithmic factors due to lower bounds on high-dimensional mean estimation.

Numerical and Strength Results

• Query Complexity: The Gaussian surface area tester operates with $\mathbb{R}^n$1 queries, which is exponentially better for sparse functions than the naive $\mathbb{R}^n$2 bound obtainable by reduction to standard-model testers, and sublinear in $\mathbb{R}^n$3 for all super-polynomially small $\mathbb{R}^n$4.
• Sample-Only Complexity: The Hermite-based sample-only tester achieves sample complexity scaling as $\mathbb{R}^n$5, a significant improvement over previous learning-based approaches. Moreover, with only a lower bound for $\mathbb{R}^n$6, the adaptive procedure loses only logarithmic factors.
• Structural Sharpness: The use of robust, quantitative isoperimetry yields tight relations between surface area/noise sensitivity and the proximity to halfspaces. The analytic arguments for Hermite weight extremality for LTFs further ensure that testers are both sample efficient and sharp.
• Volume Independence: For $\mathbb{R}^n$7 larger than an explicit threshold, the testers' sample complexity becomes independent of $\mathbb{R}^n$8, matching the best known for standard property testing and surpassing known distribution-free lower bounds.

Theoretical and Practical Implications

• Testing vs. Learning Separation: These results reinforce and quantify the principle that property testing can be strictly easier than learning, even in high-dimensional and sparse settings. Unlike learning halfspaces, which requires $\mathbb{R}^n$9 samples, the RET testers here demonstrate that property testing can circumvent this complexity, exploiting structural optimality criteria.
• Unified Analysis under Gaussian Measure: The presented techniques generalize and unify several analytic approaches to property testing, providing a template for other RET problems in continuous measure spaces.
• Practical Sampling Regimes: For problems in AI safety, high-dimensional optimization, and rare event detection (motivations referenced in the paper's funding acknowledgments), these RET testers provide actionable, sample-efficient protocols for verifying linear threshold structure in high dimensions, even when the "positive" region is vanishingly small.
• Future Directions: The dimension-agnostic analytic underpinnings invite extension to other geometric concept classes (e.g., convex bodies, polynomial threshold functions of degree $f: \mathbb{R}^n \rightarrow \{0,1\}$0). The explicit utility of positive-sample-only frameworks opens up avenues for property testing in settings with asymmetric or rare-positive feedback (e.g., anomaly detection, statistical auditing of AI models).

Conclusion

This work delivers the first sublinear-in-$f: \mathbb{R}^n \rightarrow \{0,1\}$1 relative-error testers for halfspaces under the Gaussian measure, leveraging robust isoperimetric, Hermite-analytic, and noise sensitivity tools. The results represent a sharp improvement over both naive bounds and existing lower bounds for related models, and solidify the structural and algorithmic separation between testing and learning in high-dimensional, sparse-concept regimes. The methods developed have considerable ramifications for both the theory of property testing and practical, high-dimensional data analysis, offering efficient solutions in domains previously limited by sample complexity.