Learning DNF Expressions from Fourier Spectrum

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 (2009b) showed that in this setting a DNF expression can be efficiently approximated from its “heavy” low-degree 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. (2009b) and inspired by the seminal smoothed analysis model of Spielman and Teng (2004). 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 low-degree Fourier coefficients. We then describe conditions under which this is sufficient for learning polynomial threshold functions. As an application of our approach, we give a new, simple algorithm for approximating any polynomial-size DNF expression from its “heavy” low-degree Fourier coefficients alone. Our algorithm greatly simplifies the proof of learnability of DNF expressions over smoothed product distributions and is simpler than all previous algorithm for PAC learning of DNF expression using membership queries. We also describe an application of our algorithm to learning monotone DNF expressions over product distributions. Building on the work of Servedio (2004), we give an algorithm that runs in time poly((s · log (s/ǫ)) , n), where s is the size of the DNF expression and ǫ is the accuracy. This improves on poly((s · log (ns/ǫ)) (s/ǫ)·log , n) bound of Servedio (2004). Another advantage of our algorithm is that it can be applied to a large class of polynomial threshold functions whereas previous algorithms for both applications relied on the function being a polynomial-size DNF expression.