Dynamics of learning in deep linear neural networks

Despite the widespread practical success of deep learning methods, our theoretical understanding of the dynamics of learning in deep neural networks remains quite sparse. We attempt to bridge the gap between the theory and practice of deep learning by systematically analyzing learning dynamics for the restricted case of deep linear neural networks. Despite the linearity of their input-output map, such networks have nonlinear gradient descent dynamics that change with the addition of each new hidden layer. We show that deep linear networks exhibit nonlinear learning phenomena similar to those seen in simulations of nonlinear networks, including long plateaus followed by rapid transitions to lower error solutions, and faster convergence from greedy unsupervised pretraining initial conditions than from random initial conditions. We provide an analytical description of these phenomena by finding new exact solutions to the nonlinear dynamics of deep learning. Our theoretical analysis also reveals the surprising finding that infinitely deep networks can be learned in finite time: for a special class of initial conditions on the weights, very deep networks incur only a finite delay in learning speed relative to shallow networks. We further show that, under certain conditions on the training data, unsupervised pre-training can find this special class of initial conditions, thereby providing analytical insight into the success of unsupervised pre-training in deep supervised learning tasks. Deep learning methods have realized impressive performance in a range of applications, from visual object classification [1, 2] to speech recognition [3] and natural language processing [4, 5]. These successes have been achieved despite the noted difficulty of training such deep architectures [6, 7, 8, 9]. Indeed, many explanations for the difficulty of deep learning have been advanced in the literature, including the presence of many local minima, low curvature regions due to saturating nonlinearities, and exponential growth or decay of back-propagated gradients [10, 11, 12, 13]. Furthermore, many neural network simulations have observed strikingly nonlinear learning dynamics, including long plateaus of little apparent improvement followed by almost stage-like transitions to better performance. However, a quantitative, analytical understanding of the rich dynamics of deep learning remains elusive. For example, what determines the time scales over which deep learning unfolds? How does training speed retard with depth? Under what conditions will greedy unsupervised pretraining speed up learning? And how do the final learned internal representations depend on the statistical regularities inherent in the training data?