A Convergence Theory for Deep Learning via Over-Parameterization

Deep neural networks (DNNs) have demonstrated dominating performance in many fields, e.g., computer vision, natural language progressing, and robotics. Since AlexNet, the neural networks used in practice are going wider and deeper. On the theoretical side, a long line of works have been focusing on why we can train neural networks when there is only one hidden layer. The theory of multi-layer neural networks remains somewhat unsettled. We present a new theory to understand the convergence of training DNNs. We only make two assumptions: the inputs do not degenerate and the network is over-parameterized. The latter means the number of hidden neurons is sufficiently large: polynomial in $n$, the number of training samples and in $L$, the number of layers. We show on the training dataset, starting from randomly initialized weights, simple algorithms such as stochastic gradient descent attain 100% accuracy in classification tasks, or minimize $\ell_2$ regression loss in linear convergence rate, with a number of iterations that only scale polynomial in $n$ and $L$. Our theory applies to the widely-used but non-smooth ReLU activation, and to any smooth and possibly non-convex loss functions. In terms of network architectures, our theory at least applies to fully-connected neural networks, convolutional neural networks (CNN), and residual neural networks (ResNet).