Efficiently Testing Sparse GF(2) Polynomials

Ilias Diakonikolas, HominK. Lee, Kevin Matulef, RoccoA. Servedio, Andrew Wan

Research output: Chapter in Book/Report/Conference proceedingConference contribution


We give the first algorithm that is both query-efficient and time-efficient for testing whether an unknown function f: {0,1} n →{0,1} is an s-sparse GF(2) polynomial versus ε-far from every such polynomial. Our algorithm makes poly(s,1/ε) black-box queries to f and runs in time n ·poly(s,1/ε). The only previous algorithm for this testing problem [DLM + 07] used poly(s,1/ε) queries, but had running time exponential in s and super-polynomial in 1/ε.
Our approach significantly extends the “testing by implicit learning” methodology of [DLM + 07]. The learning component of that earlier work was a brute-force exhaustive search over a concept class to find a hypothesis consistent with a sample of random examples. In this work, the learning component is a sophisticated exact learning algorithm for sparse GF(2) polynomials due to Schapire and Sellie [SS96]. A crucial element of this work, which enables us to simulate the membership queries required by [SS96], is an analysis establishing new properties of how sparse GF(2) polynomials simplify under certain restrictions of “low-influence” sets of variables.
Original languageEnglish
Title of host publicationAutomata, Languages and Programming
EditorsLuca Aceto, Ivan Damgård, LeslieAnn Goldberg, MagnúsM. Halldórsson, Anna Ingólfsdóttir, Igor Walukiewicz
PublisherSpringer Berlin Heidelberg
Number of pages13
ISBN (Electronic)978-3-540-70575-8
ISBN (Print)978-3-540-70574-1
Publication statusPublished - 2008

Publication series

NameLecture Notes in Computer Science
PublisherSpringer Berlin Heidelberg


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