# Reliably Learning the ReLU in Polynomial Time

S. Goel, V. Kanade, A. Klivans, and J. Thaler

Abstract:

We give the first dimension-efficient algorithms for learning Rectified Linear Units (ReLUs), which are functions of the form x \mapsto max(0,w \cdot x) with w in \mathbb{S}^{n-1}. Our algorithm works in the challenging Reliable Agnostic learning model of Kalai, Kanade, and Mansour (2009) where the learner is given access to a distribution D on labeled examples but the labeling may be arbitrary. We construct a hypothesis that simultaneously minimizes the false-positive rate and the loss on inputs given positive labels by D, for any convex, bounded, and Lipschitz loss function.

The algorithm runs in polynomial-time (in n) with respect to any distribution on \mathbb{S}^{n-1} (the unit sphere in n dimensions) and for any error parameter epsilon=Omega(1/log n) (this yields a PTAS for a question raised by F. Bach on the complexity of maximizing ReLUs). These results are in contrast to known efficient algorithms for reliably learning linear threshold functions, where epsilon must be Omega(1) and strong assumptions are required on the marginal distribution. We can compose our results to obtain the first set of efficient algorithms for learning constant-depth networks of ReLUs.

Our techniques combine kernel methods and polynomial approximations with a "dual-loss" approach to convex programming. As a byproduct we obtain a number of applications including the first set of efficient algorithms for "convex piecewise-linear fitting" and the first efficient algorithms for noisy polynomial reconstruction of low-weight polynomials on the unit sphere.

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