Learning pseudo-Boolean k-DNF and submodular functions

Sofya Raskhodnikova, Grigory Yaroslavtsev

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

12 Scopus citations

Abstract

We prove that any submodular function f : {0, 1}n → {0, 1, ..., k} can be represented as a pseudo-Boolean 2k-DNF formula. Pseudo-Boolean DNFs are a natural generalization of DNF representation for functions with integer range. Each term in such a formula has an associated integral constant. We show that an analog of Håstad's switching lemma holds for pseudo-Boolean k-DNFs if all constants associated with the terms of the formula are bounded. This allows us to generalize Mansour's PAC-learning algorithm for k-DNFs to pseudo-Boolean k-DNFs, and hence gives a PAC-learning algorithm with membership queries under the uniform distribution for submodular functions of the form f : {0, 1}n → {0, 1, ..., k}. Our algorithm runs in time polynomial in n, kO(k log k/∈) and log(1/δ) and works even in the agnostic setting. The line of previous work on learning submodular functions [Balcan, Harvey (STOC '11), Gupta, Hardt, Roth, Ullman (STOC '11), Cheraghchi, Klivans, Kothari, Lee (SODA '12)] implies only nO(k) query complexity for learning submodular functions in this setting, for fixed ∈ and δ. Our learning algorithm implies a property tester for submodularity of functions f : {0, 1}n → {0, . . . , k} with query complexity polynomial in n for k = O((log n/log log n)1/2) and constant proximity parameter ∈.

Original languageEnglish (US)
Title of host publicationProceedings of the 24th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2013
PublisherAssociation for Computing Machinery
Pages1356-1368
Number of pages13
ISBN (Print)9781611972511
DOIs
StatePublished - 2013
Event24th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2013 - New Orleans, LA, United States
Duration: Jan 6 2013Jan 8 2013

Publication series

NameProceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms

Other

Other24th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2013
Country/TerritoryUnited States
CityNew Orleans, LA
Period1/6/131/8/13

All Science Journal Classification (ASJC) codes

  • Software
  • Mathematics(all)

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