Technical Report MSC-2012-04

Title: On The Minimal Fourier Degree of Symmetric Boolean Functions
Authors: Avishay Tal
Supervisors: Amir Shpilka
Abstract: In this thesis we give a new upper bound on the minimal degree of a nonzero Fourier coefficient in any non-linear symmetric Boolean function. Specifically, we prove that for every non-linear and symmetric f:{0,1}^k → {0,1} there exists a non-empty subset S of {1,...,n} such that |S| = O(Γ(k) + sqrt(k)), and the S-fourier coefficient of f is non-zero, where Γ(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. [MOS04]. Namely, we show that the running time of their algorithm is at most n^(O(Γ(k) + sqrt(k))) * poly(n, 2^k, 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 2^k (up to polynomial factors).

Our bound on the degree greatly improves the previous result of Kolountzakis et al. [KLMMV09] who proved that |S| = O(k / log(k)).

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