Recovery Guarantees for One-hidden-layer Neural Networks

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Kai Zhong, Zhao Song, Prateek Jain, Peter L. Bartlett, Inderjit S. Dhillon ;
Proceedings of the 34th International Conference on Machine Learning, PMLR 70:4140-4149, 2017.

Abstract

In this paper, we consider regression problems with one-hidden-layer neural networks (1NNs). We distill some properties of activation functions that lead to local strong convexity in the neighborhood of the ground-truth parameters for the 1NN squared-loss objective and most popular nonlinear activation functions satisfy the distilled properties, including rectified linear units (ReLUs), leaky ReLUs, squared ReLUs and sigmoids. For activation functions that are also smooth, we show local linear convergence guarantees of gradient descent under a resampling rule. For homogeneous activations, we show tensor methods are able to initialize the parameters to fall into the local strong convexity region. As a result, tensor initialization followed by gradient descent is guaranteed to recover the ground truth with sample complexity $ d \cdot \log(1/\epsilon) \cdot \mathrm{poly}(k,\lambda )$ and computational complexity $n\cdot d \cdot \mathrm{poly}(k,\lambda) $ for smooth homogeneous activations with high probability, where $d$ is the dimension of the input, $k$ ($k\leq d$) is the number of hidden nodes, $\lambda$ is a conditioning property of the ground-truth parameter matrix between the input layer and the hidden layer, $\epsilon$ is the targeted precision and $n$ is the number of samples. To the best of our knowledge, this is the first work that provides recovery guarantees for 1NNs with both sample complexity and computational complexity linear in the input dimension and logarithmic in the precision.

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