TY - GEN

T1 - On the minimal fourier degree of symmetric Boolean functions

AU - Shpilka, Amir

AU - Tal, Avishay

PY - 2011

Y1 - 2011

N2 - 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).

AB - 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).

KW - Fourier spectrum

KW - Learning juntas

KW - Symmetric functions

UR - http://www.scopus.com/inward/record.url?scp=80052001791&partnerID=8YFLogxK

U2 - 10.1109/CCC.2011.16

DO - 10.1109/CCC.2011.16

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AN - SCOPUS:80052001791

SN - 9780769544113

T3 - Proceedings of the Annual IEEE Conference on Computational Complexity

SP - 200

EP - 209

BT - Proceedings - 26th Annual IEEE Conference on Computational Complexity, CCC 2011

T2 - 26th Annual IEEE Conference on Computational Complexity, CCC 2011

Y2 - 8 June 2011 through 10 June 2011

ER -