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San Jose, California USA
June 8, 2011 to June 11, 2011
ISBN: 978-0-7695-4411-3
pp: 200-209
ABSTRACT
In this paper we give a new upper bound on the minimal degree of a nonzero Fouriercoefficient in any non-linear symmetric Boolean function. Specifically, we prove that forevery non-linear and symmetric $f:\B^{k}\to\B$ there exists a set $\emptyset\neqS\subset[k]$ such that $|S|=O(\Gamma(k)+\sqrt{k})$, and $\hat{f}(S)\neq0$, where$\Gamma(m)\leq m^{0.525}$ is the largest gap between consecutive prime numbers in$\{1,\ldots,m\}$. As an application we obtain a new analysis of the PAC learningalgorithm 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$n^{O(k^{0.525})}\cdot\poly(n,2^{k},\log(1/\delta))$ where $n$ is the number ofvariables, $k$ is the size of the junta (i.e. number of relevant variables) and $\delta$is the error probability. In particular, for $k\geq\log(n)^{1/(1-0.525)}\approx\log(n)^{2.1}$ our analysis matches the lower bound $2^k$ (up to polynomial factors).\sloppy Our bound on the degree greatly improves the previous result of Kolountzakis etal. [Combinatorica, 2009] who proved that $|S|=O(k/\log k)$.
INDEX TERMS
Fourier spectrum, symmetric functions, learning juntas
CITATION
Amir Shpilka, Avishay Tal, "On the Minimal Fourier Degree of Symmetric Boolean Functions", CCC, 2011, 2012 IEEE 27th Conference on Computational Complexity, 2012 IEEE 27th Conference on Computational Complexity 2011, pp. 200-209, doi:10.1109/CCC.2011.16
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