Exact solutions to the nonlinear dynamics of learning in deep linear neural networks
Authors: Andrew Saxe, James L. McClelland, Surya Ganguli
ICLR 2014

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.


Peer review:
[Review]:
Thanks for the review, we have uploaded a new draft with expanded material on a new property, called dynamic isometry, that allows gradients to propagate from input to output in a deep linear net. We have added a comparison to and analysis of other random initialization schemes in both linear and nonlinear nets.

Also we'd like to point out that we do in fact study standard backpropagation. In the linear case, backpropagation gives exactly the equations used in the paper. Our form is simply a rearrangement of terms from what one usually sees (and the rearrangement relies on the linearity of the network). For example, the hidden unit activity h in standard back propagation can be written as h=W_1x in the linear case. The error is delta=(y-hat y), and in the linear case, hat y=W^{32}W^{21}x. Hence the change in W^{32} = delta h^T =(y-W^{32}W^{21}x)x^TW^{21}=(yx^T-W^{32}W^{21}xx^T)W^{21} which is our expression. A similar argument will yield our expression for the change in W^{21}. See our citation [16] for a similar derivation.