Exact solutions to the nonlinear dynamics of learning in deep linear neural networks

Andrew Saxe, James L. McClelland, Surya Ganguli

Decision: submitted, no decision

Abstract: 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 on weights 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 as the depth of a network approaches infinity, learning speed remains finite: 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 pretraining can find this special class of initial conditions, thereby providing analytical insight into the success of unsupervised pretraining in deep supervised learning tasks.