Paper:
In this work, we describe a set of rules for the design and initialization of wellconditioned neural networks, guided by the goal of naturally balancing the diagonal blocks of the Hessian at the start of training. We show how our measure of conditioning of a block relates to another natural measure of conditioning, the ratio of weight gradients to the weights. We prove that for a ReLU-based deep multilayer perceptron, a simple initialization scheme using the geometric mean of the fan-in and fan-out satisfies our scaling rule. For more sophisticated architectures, we show how our scaling principle can be used to guide design choices to produce well-conditioned neural networks, reducing guess-work.

Review:
The authors propose a new initialization scheme for training neural networks. The initialization considers fan-in and fan-out, to regularize the range of singular values of the Hessian matrix, under several assumptions.
The proposed approach gives important insights for the problem of weight initialization in neural networks. Overall, the method makes sense. However, I have several concerns:
- The authors do not consider more recent neural network designs such as normalization layers, skip connections, etc. It would be great if the authors could discuss how these layers would change the derivation of the initialization method. Also, preliminary experimental results using these layers are needed. Additionally, to me, normalization layers [Huang et al. Decorrelated Batch Normalization. CVPR 2018] implicitly precondition the Hessian matrix as well. It would be great if the authors also compare their approach to [Huang et al. 2018].
- The authors compared to other initialization schemes such as [He et al., 2015] and [ Glorot and Bengio 2010]. But as the authors mentioned, there are approaches that scales backpropagation gradients also [Martens and Grosse, 2015; Grosse and Martens, 2016; Ba et al., 2017; George et al., 2018]. Since these methods are highly related to the proposed method, it would be great if the authors could show time complexities and performance differences of these methods as well.
- Experiments on the CIFAR-10 dataset with AlexNet seem not exciting: the proposed Preconditioned approach only outperforms the Fan-out approach marginally. I would say that training a [He et al. 2015]-initialized neural network for 500 more iterations than a preconditioned neural network, yields a similar or better loss.
Overall I think the work is very important and interesting. However, it lacks comprehensive comparison and consideration of more recent neural network layers.
Post Rebuttal Comments
I have read all reviewer comments and the author feedback. I appreciate that the authors addressed the skip connections in Appendix.
1. The authors agree that batch norm requires different initialization schemes that are not included in this paper.
2. I agree with the authors that their approach is complementary to the baseline optimization methods; and both approaches can be applied together. However, I still believe that it is informative to compare the two approaches because: (a). Both approaches address the same problem. Since the optimization based approach adds complexity and computational overhead to implementation, it would be great to show if using the proposed approach eliminates the need for the optimization based approach. (b). Is it necessary to use both approaches, or one of them is good enough?
3. I understand that strong experimental evidence is not always required. However, I believe that the new technical insights of the paper alone is not significant enough (part of the reasons in point 1). Thus I was expecting stronger experimental evidences.
Overall I agree with reviewer 1 that the topic is interesting, but in the paper’s current form, it is not ready. I keep my initial rating of weak reject.