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# Reliably Learning the ReLU in Polynomial Time

*Proceedings of the 2017 Conference on Learning Theory*, PMLR 65:1004-1042, 2017.

#### Abstract

We give the first dimension-efficient algorithms for learning Rectified Linear Units (ReLUs), which are functions of the form $\mathbf{x} \mapsto \mathsf{max}(0, \mathbf{w} ⋅\mathbf{x})$ with $\mathbf{w} ∈\mathbb{S}^n-1$. Our algorithm works in the challenging Reliable Agnostic learning model of Kalai, Kanade and Mansour (2012) where the learner is given access to a distribution $\mathcal{D}$ 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 $\mathcal{D}$, for any convex, bounded, and Lipschitz loss function. The algorithm runs in polynomial-time (in $n$) with respect to \em any distribution on $\mathbb{S}^n-1$ (the unit sphere in $n$ dimensions) and for any error parameter $ε= Ω(1 / \log n)$ (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 $ε$ must be $Ω(1)$ 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 ReLU with fixed polynomial-dependence in the dimension. For depth-2 networks of sigmoids, we obtain the first algorithms that have a polynomial dependency in \em all parameters. 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.