Distribution-Free Testing Lower Bounds for Basic Boolean Functions

TitleDistribution-Free Testing Lower Bounds for Basic Boolean Functions
Publication TypeBook Chapters
Year of Publication2007
AuthorsDachman-Soled D, Servedio RA
EditorCharikar M, Jansen K, Reingold O, Rolim JDP
Book TitleApproximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques
Series TitleLecture Notes in Computer Science
Pagination494 - 508
PublisherSpringer Berlin Heidelberg
ISBN Number978-3-540-74207-4, 978-3-540-74208-1
KeywordsAlgorithm Analysis and Problem Complexity, Discrete Mathematics in Computer Science, Numeric Computing

In the distribution-free property testing model, the distance between functions is measured with respect to an arbitrary and unknown probability distribution \mathcal{D} over the input domain. We consider distribution-free testing of several basic Boolean function classes over {0,1} n , namely monotone conjunctions, general conjunctions, decision lists, and linear threshold functions. We prove that for each of these function classes, Ω((n/logn)1/5) oracle calls are required for any distribution-free testing algorithm. Since each of these function classes is known to be distribution-free properly learnable (and hence testable) using Θ(n) oracle calls, our lower bounds are within a polynomial factor of the best possible.