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

Abstract

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
Pages502-514
Number of pages13
Volume5125
ISBN (Electronic)978-3-540-70575-8
ISBN (Print)978-3-540-70574-1
DOIs
Publication statusPublished - 2008

Publication series

NameLecture Notes in Computer Science
PublisherSpringer Berlin Heidelberg
Volume5125

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