Uniform-Distribution Learnability of Noisy Linear Threshold Functions with Restricted Focus of Attention

Abstract

Recently, Kalai et al. [1] have shown (among other things) that linear threshold functions over the Boolean cube and unit sphere are agnostically learnable with respect to the uniform distribution using the hypothesis class of polynomial threshold functions. Their primary algorithm computes monomials of large constant degree, although they also analyze a low-degree algorithm for learning origin-centered halfspaces over the unit sphere. This paper explores noise-tolerant learnability of linear thresholds over the cube when the learner sees a very limited portion of each instance. Uniform-distribution weak learnability results are derived for the agnostic, unknown attribute noise, and malicious noise models. The noise rates that can be tolerated vary: the rate is essentially optimal for attribute noise, constant (roughly 1/8) for agnostic learning, and non-trivial ($\Omega(1/\sqrt{n})$) for malicious noise. In addition, a new model that lies between the product attribute and malicious noise models is introduced, and in this stronger model results similar to those for the standard attribute noise model are obtained for learning homogeneous linear thresholds with respect to the uniform distribution over the cube. The learning algorithms presented are simple and have small-polynomial running times.