On the minimal fourier degree of symmetric Boolean functions

Amir Shpilka*, Avishay Tal

*Corresponding author for this work

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

Abstract

In this paper we give a new upper bound on the minimal degree of a nonzero Fourier coefficient in any nonlinear symmetric Boolean function. Specifically, we prove that for every non-linear and symmetric f: {0,1}k →{0,1} there exists a set θ ≠ S ⊂ [k] such that |S| = O(Λ(k) + √k), and f̂(S) ≠ 0, where Λ(m) ≤ m 0.525 is the largest gap between consecutive prime numbers in {1,⋯,m}. As an application we obtain a new analysis of the PAC learning algorithm for symmetric juntas, under the uniform distribution, of Mossel et al. [JCSS, 2004]. Namely, we show that the running time of their algorithm is at most nO(k0.525) · poly(n,2k,log(1/δ)) where n is the number of variables, k is the size of the junta (i.e. number of relevant variables) and δ is the error probability. In particular, for k ≥ log(n)1/(1-0.525) ≈ log(n)2.1 our analysis matches the lower bound 2k (up to polynomial factors). Our bound on the degree greatly improves the previous result of Kolountzakis et al. [Combinatorica, 2009] who proved that |S| = O(k/logk).

Original languageEnglish
Title of host publicationProceedings - 26th Annual IEEE Conference on Computational Complexity, CCC 2011
Pages200-209
Number of pages10
DOIs
StatePublished - 2011
Externally publishedYes
Event26th Annual IEEE Conference on Computational Complexity, CCC 2011 - San Jose, CA, United States
Duration: 8 Jun 201110 Jun 2011

Publication series

NameProceedings of the Annual IEEE Conference on Computational Complexity
ISSN (Print)1093-0159

Conference

Conference26th Annual IEEE Conference on Computational Complexity, CCC 2011
Country/TerritoryUnited States
CitySan Jose, CA
Period8/06/1110/06/11

Keywords

  • Fourier spectrum
  • Learning juntas
  • Symmetric functions

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