Recovery Guarantees for One-hidden-layer Neural Networks

TL;DR

This paper establishes theoretical guarantees for recovering the parameters of one-hidden-layer neural networks using tensor initialization and gradient descent, with guarantees on sample and computational complexity for various activation functions.

Contribution

It provides the first recovery guarantees for 1-hidden-layer neural networks with sample complexity and computational complexity linear in input dimension and logarithmic in precision.

Findings

Tensor methods initialize parameters within the strong convexity region.

Gradient descent converges locally for smooth activations.

Sample complexity is proportional to input dimension and log of inverse precision.

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 $\mathit{local~strong~convexity}$ in the neighborhood of the ground-truth parameters for the 1NN squared-loss objective. 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 $\mathit{local~linear~convergence}$ 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 $\mathit{linear}$ in the input dimension and $\mathit{logarithmic}$ in the precision.