Classes Testable with $O(1/ε)$ Queries for Small $ε$ Independent of the Number of Variables

2026-04-07

AI Summary

This work investigates the design of efficient property testing algorithms for Boolean functions whose query complexity is independent of the total number of variables $n$. Focusing on important function classes such as $k$-juntas, functions with Fourier degree at most $d$, and sparse polynomials, it introduces a unified framework that extends known $O(1/\varepsilon)$-query testability results—previously established for term-wise and XOR functions—to a broader range of classes. The core contribution establishes that when the error parameter satisfies $\varepsilon \leq 1/\psi$, where $\psi$ depends only on the structural properties of the function class, there exists a tester with query complexity $O(1/\varepsilon)$, matching the theoretical lower bound $\Omega(1/\varepsilon)$. This result also reveals a profound connection between learnability and testability.

Technology Category

• Machine Learning: Learning Theory
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

In this paper, we study classes of Boolean functions that are testable with $O(ψ+1/ε)$ queries, where $ψ$ depends on the parameters of the class (e.g., the number of terms, the number of relevant variables, etc.) but not on the total number of variables $n$. In particular, when $ε\le 1/ψ$, the query complexity is $O(1/ε)$, matching the known tight bound $Ω(1/ε)$. This result was previously known for classes of terms of size at most $k$ and exclusive OR functions of at most $k$ variables. In this paper, we extend this list to include the classes: $k$-junta, functions with Fourier degree at most $d$, $s$-sparse polynomials of degree at most $d$, and $s$-sparse polynomials. Additionally, we show that for any class $C$ of Boolean functions that depend on at most $k$ variables, if $C$ is properly exactly learnable, then it is testable with $O(1/ε)$ queries for $ε<1/ψ$, where $ψ$ depends on $k$ and independent of the total number of variables $n$.

Problem

• Boolean functions
• property testing
• query complexity
• junta
• Fourier degree

Innovation

• property testing
• Boolean functions
• query complexity
• junta
• Fourier degree