AND testing and robust judgement aggregation

Yuval Filmus, Noam Lifshitz, Dor Minzer, Elchanan Mossel

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

5 Scopus citations

Abstract

A function fg¶{0,1}n→ {0,1} is called an approximate AND-homomorphism if choosing x,ygn uniformly at random, we have that f(xg§ y) = f(x)g§ f(y) with probability at least 1-ϵ, where xg§ y = (x1g§ y1,...,xng§ yn). We prove that if fg¶ {0,1}n → {0,1} is an approximate AND-homomorphism, then f is -close to either a constant function or an AND function, where (ϵ) → 0 as ϵ→ 0. This improves on a result of Nehama, who proved a similar statement in which δdepends on n. Our theorem implies a strong result on judgement aggregation in computational social choice. In the language of social choice, our result shows that if f is ϵ-close to satisfying judgement aggregation, then it is (ϵ)-close to an oligarchy (the name for the AND function in social choice theory). This improves on Nehama's result, in which δdecays polynomially with n. Our result follows from a more general one, in which we characterize approximate solutions to the eigenvalue equation f = λ g, where is the downwards noise operator f(x) = y[f(x g§ y)], f is [0,1]-valued, and g is {0,1}-valued. We identify all exact solutions to this equation, and show that any approximate solution in which f and λ g are close is close to an exact solution.

Original languageEnglish
Title of host publicationSTOC 2020 - Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing
EditorsKonstantin Makarychev, Yury Makarychev, Madhur Tulsiani, Gautam Kamath, Julia Chuzhoy
PublisherAssociation for Computing Machinery
Pages222-233
Number of pages12
ISBN (Electronic)9781450369794
DOIs
StatePublished - 8 Jun 2020
Externally publishedYes
Event52nd Annual ACM SIGACT Symposium on Theory of Computing, STOC 2020 - Chicago, United States
Duration: 22 Jun 202026 Jun 2020

Publication series

NameProceedings of the Annual ACM Symposium on Theory of Computing
ISSN (Print)0737-8017

Conference

Conference52nd Annual ACM SIGACT Symposium on Theory of Computing, STOC 2020
Country/TerritoryUnited States
CityChicago
Period22/06/2026/06/20

Bibliographical note

Publisher Copyright:
© 2020 ACM.

Funding

∗This project has received funding from the European Union’s Horizon 2020 research and innovation programme under grant agreement No 802020-ERC-HARMONIC. †Supported partially by NSF grant CCF-1412958 and Rothschild Fellowship. ‡Supported by NSF Award DMS-1737944, by AROMURI W911NF1910217 and by a Simons Investigator Award in Mathematics (622132).

FundersFunder number
AROMURIW911NF1910217, 622132
National Science FoundationDMS-1737944, CCF-1412958, 1737944, 1412958, 802020
Horizon 2020 Framework Programme

    Keywords

    • Analysis of Boolean Functions
    • Property Testing

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