On the Fourier Spectrum of Symmetric Boolean Functions with Applications to Learning Symmetric Juntas
R.J. Lipton, Evangelos Markakis, Aranyak Mehta, Nisheeth K. Vishnoi · 2005
We study the following question: What is the smallest t such that every symmetric boolean function on k variables (which is not a constant or a parity function), has a non-zero Fourier coefficient of order at least 1 and at most t? We exclude the constant functions for which there is no such t and the parity functions for which t has to be k. Let τ(k) be the smallest such t. The main contribution of this paper is a proof of the following self similar nature of this question: If τ(l) ≤ s, then for any ɛ> 0 and � for k ≥ k0(l, ɛ), τ(k) ≤ k s+1 l+1 + ɛ Coupling this result with a computer based search which establishes τ(30) = 2, one obtains that for large enough k, τ(k) ≤ 3k/31. The motivation for our work is to understand the complexity of learning symmetric juntas. A k-junta is a boolean function of n variables that depends only on an unknown subset of k variables. If f is symmetric in the variables it depends on, it is called a symmetric k-junta. Our results imply an algorithm to learn the class of symmetric k-juntas, in the uniform PAC learning model, in time approximately n 3k 31. This improves on a result of Mossel, O’Donnell and Servedio in [11], who show that symmetric k-juntas can be ∗ Research supported by NSF grants CCR-0002299 and CCF-0431023.