Halfspaces are hard to test with relative error

2025-11-09

AI Summary

This work investigates the property testing complexity of halfspaces (linear threshold functions) under the relative error model, where a tester must distinguish functions fully satisfying the property from those satisfied by only a $(1-varepsilon)$-fraction of inputs—contrasting with the standard absolute distance criterion. The authors establish the first nontrivial query lower bound for this setting: reliably testing $n$-dimensional sparse Boolean halfspaces requires $widetilde{Omega}(log n)$ black-box queries, markedly exceeding the constant query complexity achievable in the standard model. Technically, the proof integrates uniform random assignment access, probabilistic analysis over the Boolean hypercube, and information-theoretic lower bound techniques. This result reveals the intrinsic hardness of property testing under relative error, providing a foundational theoretical distinction in testability across different error metrics and highlighting a fundamental limitation of relative-error-based testing for structured function classes.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Relational Probabilistic Models
• Machine Learning: Probabilistic Circuits and Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets

Abstract

Several recent works [DHLNSY25, CPPS25a, CPPS25b] have studied a model of property testing of Boolean functions under a emph{relative-error} criterion. In this model, the distance from a target function $f: {0,1}^n o {0,1}$ that is being tested to a function $g$ is defined relative to the number of inputs $x$ for which $f(x)=1$; moreover, testing algorithms in this model have access both to a black-box oracle for $f$ and to independent uniform satisfying assignments of $f$. The motivation for this model is that it provides a natural framework for testing emph{sparse} Boolean functions that have few satisfying assignments, analogous to well-studied models for property testing of sparse graphs. The main result of this paper is a lower bound for testing emph{halfspaces} (i.e., linear threshold functions) in the relative error model: we show that $ ilde{Omega}(log n)$ oracle calls are required for any relative-error halfspace testing algorithm over the Boolean hypercube ${0,1}^n$. This stands in sharp contrast both with the constant-query testability (independent of $n$) of halfspaces in the standard model [MORS10], and with the positive results for relative-error testing of many other classes given in [DHLNSY25, CPPS25a, CPPS25b]. Our lower bound for halfspaces gives the first example of a well-studied class of functions for which relative-error testing is provably more difficult than standard-model testing.

Problem

• Testing halfspaces requires logarithmic queries under relative error
• Relative error testing is harder for halfspaces than standard testing
• Lower bound shows sparse function testing challenges for halfspaces

Innovation

• Relative-error testing model for sparse functions
• Lower bound for halfspace testing complexity
• Oracle calls required increase with input size