Information Geometry of Orthogonal Initializations and Training
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 initializations— even 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 propose an explanation for this phenomenon by exploting connections between Fisher geometry and the recently introduced Neural Tangent Kernel.
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.
You are a member of the scientific community tasked with peer review. Review the following paper title and abstract.
### Title + Abstract
Information Geometry of Orthogonal Initializations and Training
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 initializations— even 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 propose an explanation for this phenomenon by exploting connections between Fisher geometry and the recently introduced Neural Tangent Kernel.
user
<think>Okay, to do a decent peer review, I should start with summarizing what the work is about. This includes novelty, contributions, clarity, and impact. I might include suggestions to issues that I find critical to the usefulness of the work. Starting with novelty, there are different types of novelty. The first aspect of novelty is scope and scale. Is the work covering a broader perspective than previous works? The second aspect is methodology. Is the proposed methodology novel compared to previous methods? Do they provide any significant advantage over previous methodologies? The third aspect is results. Are the results different compared previous work? Does this lead to any novel theories or an extension of current theories? Next is clarity, I will be reading the work in depth and analyze if any sentences can be potentially ambiguous or misleading. If any sentences are ambiguous, I will state them and give a suggestion on how to rephrase such sentence or how to make more clear to the audience. Lastly, impact is important. I will attempt to predict how such work will impact the scientific community. This might include the applications of such work and how the work might transform the way we study a certain topic. Finally, I will might only mention issues that are critical. This is to keep the review as concise and helpful as possible.</think>



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.
assistant
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.