Learning DNF Expressions from Fourier Spectrum
Abstract
Since its introduction by Valiant in 1984, PAC learning of DNF expressions remains one of the central problems in learning theory. We consider this problem in the setting where the underlying distribution is uniform, or more generally, a product distribution. Kalai, Samorodnitsky and Teng (2009) showed that in this setting a DNF expression can be efficiently approximated from its "heavy" lowdegree Fourier coefficients alone. This is in contrast to previous approaches where boosting was used and thus Fourier coefficients of the target function modified by various distributions were needed. This property is crucial for learning of DNF expressions over smoothed product distributions, a learning model introduced by Kalai et al. (2009) and inspired by the seminal smoothed analysis model of Spielman and Teng (2001). We introduce a new approach to learning (or approximating) a polynomial threshold functions which is based on creating a function with range [1,1] that approximately agrees with the unknown function on lowdegree Fourier coefficients. We then describe conditions under which this is sufficient for learning polynomial threshold functions. Our approach yields a new, simple algorithm for approximating any polynomialsize DNF expression from its "heavy" lowdegree Fourier coefficients alone. Our algorithm greatly simplifies the proof of learnability of DNF expressions over smoothed product distributions. We also describe an application of our algorithm to learning monotone DNF expressions over product distributions. Building on the work of Servedio (2001), we give an algorithm that runs in time $\poly((s \cdot \log{(s/\eps)})^{\log{(s/\eps)}}, n)$, where $s$ is the size of the target DNF expression and $\eps$ is the accuracy. This improves on $\poly((s \cdot \log{(ns/\eps)})^{\log{(s/\eps)} \cdot \log{(1/\eps)}}, n)$ bound of Servedio (2001).
 Publication:

arXiv eprints
 Pub Date:
 March 2012
 arXiv:
 arXiv:1203.0594
 Bibcode:
 2012arXiv1203.0594F
 Keywords:

 Computer Science  Machine Learning;
 Computer Science  Computational Complexity;
 Computer Science  Data Structures and Algorithms
 EPrint:
 Appears in Conference on Learning Theory (COLT) 2012