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arXiv · 1511.07948

Learning Halfspaces and Neural Networks with Random Initialization

Yuchen Zhang·Jason D. Lee·Martin J. Wainwright·Michael I. Jordan
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Abstract

We study non-convex empirical risk minimization for learning halfspaces and neural networks. For loss functions that are $L$-Lipschitz continuous, we present algorithms to learn halfspaces and multi-layer neural networks that achieve arbitrarily small excess risk $ε>0$. The time complexity is polynomial in the input dimension $d$ and the sample size $n$, but exponential in the quantity $(L/ε^2)\log(L/ε)$. These algorithms run multiple rounds of random initialization followed by arbitrary optimization steps. We further show that if the data is separable by some neural network with constant margin $γ>0$, then there is a polynomial-time algorithm for learning a neural network that separates the training data with margin $Ω(γ)$. As a consequence, the algorithm achieves arbitrary generalization error $ε>0$ with ${\rm poly}(d,1/ε)$ sample and time complexity. We establish the same learnability result when the labels are randomly flipped with probability $η<1/2$.

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Yuchen Zhang, Jason D. Lee, Martin J. Wainwright, Michael I. Jordan. 2015-11-25. Learning Halfspaces and Neural Networks with Random Initialization. https://arxiv.org/abs/1511.07948

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