Learning Boolean Functions in AC ⁰ on Attribute and Classification Noise

Abstract

Linial, Mansour and Nisan introduced a learning algorithm of Boolean functions in AC ⁰ under the uniform distribution. Following their work, Bshouty, Jackson and Tamon, also Ohtsuki and Tomita extended the algorithm to one that is robust for noisy data. The noise process we are concerned here is as follows: for an example $\langle x,f(x) \rangle (x \in \{ 0,1 \}^n,\ f(x) \in \{-1,1\})$, each bit of the vector x and the f(x) are obliged to change independently with probability η during a learning process. Before our present paper, we were on the assumption that the correct noise rate η or its upper bound is known in advance. In this paper, we eliminate the assumption. We estimate an upper bound of the noise rate by evaluating noisy power spectrum on the frequency domain by using a sampling trick. Thereby, by combining this procedure with Ohtsuki et al.’s algorithm, we obtain a quasipolynomial time learning algorithm that can cope with noise without knowing any information about noise in advance.

This work has been supported in part by Grants-in-Aid for Scientific Research Nos. 13680435 and 16300001 from the Ministry of Education, Culture, Sports, Science and Technology, Japan.