Information Geometry of Orthogonal Initializations and Training
This work addresses the optimization challenges in deep learning by providing insights into initialization and training mechanisms, though it is incremental as it builds on existing mean field theory and orthogonal initialization methods.
The paper investigates why orthogonal weight initializations speed up neural network training, showing a connection between gradient smoothness and the spectral radius of the Jacobian, and finds that maintaining orthogonality during training helps regardless of gradient smoothness, with experiments indicating that a low Fisher information matrix condition number does not predict faster learning.
Recently mean field theory has been successfully used to analyze properties of wide, random neural networks. It gave rise to a prescriptive theory for initializing feed-forward neural networks with orthogonal weights, which ensures that both the forward propagated activations and the backpropagated gradients are near $\ell_2$ isometries and as a consequence training is orders of magnitude faster. Despite strong empirical performance, the mechanisms by which critical initializations confer an advantage in the optimization of deep neural networks are poorly understood. Here we show a novel connection between the maximum curvature of the optimization landscape (gradient smoothness) as measured by the Fisher information matrix (FIM) and the spectral radius of the input-output Jacobian, which partially explains why more isometric networks can train much faster. Furthermore, given that orthogonal weights are necessary to ensure that gradient norms are approximately preserved at initialization, we experimentally investigate the benefits of maintaining orthogonality throughout training, from which we conclude that manifold optimization of weights performs well regardless of the smoothness of the gradients. Moreover, motivated by experimental results we show that a low condition number of the FIM is not predictive of faster learning.