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 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, and we conclude that manifold optimization of weights performs well regardless of the smoothness of the gradients. Moreover, we observe a surprising yet robust behavior of highly isometric initializationseven though such networks have a lower FIM condition number at initialization, and therefore by analogy to convex functions should be easier to optimize, experimentally they prove to be much harder to train with stochastic gradient descent. We conjecture that stability of model linearized around its initial parameters has implication beyond the lazy training regime-stability of the linearization impacts the degree to which the initial curvature is predictive of the local curvature during training.
This paper analyzes the connection between the spectrum of the layer-to-layer or input-output Jacobian matrices and the spectrum of the Fisher information matrix / Neural Tangent Kernel. By bounding the maximum eigenvalue of the Fisher in terms of the maximum squared singular value of the input-output Jacobian, this paper provides a partial explanation for the successful initialization procedures obeying "dynamical isometry". By additionally investigating optimization on the orthogonal weight manifold, this paper sheds light on the important of maintaining spectral uniformity throughout training.  These two analyses help fill in important gaps in the understanding of initialization, dynamical isometry, and the training of deep neural networks. For these reasons I recommend this paper for acceptance.
There are two aspects of the paper that could nevertheless use some clarification and improvement. First, unless I missed something, this paper does not provide any bounds on the condition number or the minimum eigenvalue of the Fisher or the NTK. It seems like the main arguments only depend on the maximum eigenvalues. Generally, I think the insights into the maximum eigenvalues are useful and important on their own, but perhaps some additional discussion up front clarifying which results were derived theoretically and which were observed empirically could be useful.
Second, it should be noted that the the networks trained in the experiments are likely in a regime that is well outside the NTK regime, in two important ways: the dataset is large compared to the width and the optimal learning rates may be large as well.
Overall, I think this is a good paper that adds important insights into the study of initialization, local geometry, and their effects on training speed.