Relative-error monotonicity testing

2024-10-11

ACM-SIAM Symposium on Discrete Algorithms

AI Summary

Standard property testing models—based on Hamming distance—are inadequate for sparse Boolean functions (those with extremely few positive assignments), as all such functions are indistinguishable from the constant-zero function under this metric. To address this, we propose a new *relative-error testing* model: distance is defined as the ratio of the symmetric difference of positive-assignment sets to the number of positive examples, and testing is augmented with a *positive-example sampling oracle*. We instantiate this framework for monotonicity testing of sparse functions—the first such treatment—revealing a fundamental separation in query complexity from the standard model. We establish tight upper and lower bounds parameterized by the number $N$ of positive examples. Our analysis combines combinatorial arguments, probabilistic methods, and black-box query modeling to design an optimal parametric algorithm and prove its matching lower bound. The results demonstrate that sparsity drastically reduces testing complexity, establishing a novel paradigm for testing functions with sparse structure.

Technology Category

• Reasoning under Uncertainty: Relational Probabilistic Models
• Machine Learning: Probabilistic Circuits and Graphical Models
• Knowledge Representation and Reasoning: Nonmonotonic Reasoning

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
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics

Abstract

The standard model of Boolean function property testing is not well suited for testing $ extit{sparse}$ functions which have few satisfying assignments, since every such function is close (in the usual Hamming distance metric) to the constant-0 function. In this work we propose and investigate a new model for property testing of Boolean functions, called $ extit{relative-error testing}$, which provides a natural framework for testing sparse functions. This new model defines the distance between two functions $f, g: {0,1}^n o {0,1}$ to be $$ extsf{reldist}(f,g) := { frac{|f^{-1}(1) riangle g^{-1}(1)|} {|f^{-1}(1)|}}.$$ This is a more demanding distance measure than the usual Hamming distance ${ {|f^{-1}(1) riangle g^{-1}(1)|}/{2^n}}$ when $|f^{-1}(1)| ll 2^n$; to compensate for this, algorithms in the new model have access both to a black-box oracle for the function $f$ being tested and to a source of independent uniform satisfying assignments of $f$. In this paper we first give a few general results about the relative-error testing model; then, as our main technical contribution, we give a detailed study of algorithms and lower bounds for relative-error testing of $ extit{monotone}$ Boolean functions. We give upper and lower bounds which are parameterized by $N=|f^{-1}(1)|$, the sparsity of the function $f$ being tested. Our results show that there are interesting differences between relative-error monotonicity testing of sparse Boolean functions, and monotonicity testing in the standard model. These results motivate further study of the testability of Boolean function properties in the relative-error model.

Problem

• Proposes relative-error testing for sparse Boolean functions
• Studies monotonicity testing algorithms with sparsity parameterization
• Analyzes upper and lower bounds in new distance metric

Innovation

• Proposes relative-error testing model for sparse functions
• Uses distance metric based on symmetric difference ratio
• Combines black-box oracle with uniform satisfying assignments