Learning symmetric k-juntas in time n^o(k)

TL;DR

This paper presents a new algorithm for learning symmetric k-juntas in the PAC model with a runtime of n^{O(k/ ext{log} k)}, improving previous bounds and leveraging Fourier analysis of symmetric boolean functions.

Contribution

The paper introduces an algorithm with sub-exponential runtime for learning symmetric k-juntas and proves a new Fourier coefficient bound for symmetric boolean functions.

Findings

Achieved an n^{O(k/ ext{log} k)} time algorithm for learning symmetric k-juntas.

Proved that all symmetric boolean functions, except parity and constant functions, have a significant Fourier coefficient.

Improved the Fourier coefficient bound from (3/31)k to O(k/ ext{log} k).

Abstract

We give an algorithm for learning symmetric k-juntas (boolean functions of $n$ boolean variables which depend only on an unknown set of $k$ of these variables) in the PAC model under the uniform distribution, which runs in time n^{O(k/\log k)}. Our bound is obtained by proving the following result: Every symmetric boolean function on k variables, except for the parity and the constant functions, has a non-zero Fourier coefficient of order at least 1 and at most O(k/\log k). This improves the previously best known bound of (3/31)k, and provides the first n^{o(k)} time algorithm for learning symmetric juntas.