Paper:
It has been noted in existing literature that over-parameterization in ReLU networks generally improves performance. While there could be several factors involved behind this, we prove some desirable theoretical properties at initialization which may be enjoyed by ReLU networks. Specifically, it is known that He initialization in deep ReLU networks asymptotically preserves variance of activations in the forward pass and variance of gradients in the backward pass for infinitely wide networks, thus preserving the flow of information in both directions. Our paper goes beyond these results and shows novel properties that hold under He initialization: i) the norm of hidden activation of each layer is equal to the norm of the input, and, ii) the norm of weight gradient of each layer is equal to the product of norm of the input vector and the error at output layer. These results are derived using the PAC analysis framework, and hold true for finitely sized datasets such that the width of the ReLU network only needs to be larger than a certain finite lower bound. As we show, this lower bound depends on the depth of the network and the number of samples, and by the virtue of being a lower bound, over-parameterized ReLU networks are endowed with these desirable properties. For the aforementioned hidden activation norm property under He initialization, we further extend our theory and show that this property holds for a finite width network even when the number of data samples is infinite. Thus we overcome several limitations of existing papers, and show new properties of deep ReLU networks at initialization.

Review:
This work considers random parameter initialization in neural networks (In particular the initialization presented in He et al.) and develops non-asymptotic bounds for the norms and gradients of neural networks during initialization. The authors show that the norms of the outputs and gradients (for gradients, under a different assumption on the dimension of the matrix) remain constant through the different layers. The results presented differ from previous work in that they give nice concentration bounds for such output and gradient norms. In addition the authors prove results in the case of infinite samples under the assumption that they arise from a finite dimensional space.
Overall the presentation is clean, but the results presented have the following issues.
1.Very similar in nature to previous results, for example the results presented in [1] Theorem 5.4 give very similar concentration results and use very similar mechanisms.
2. All the results for the infinite stream coming from a finite dimensional subspace do not address the fact that the training points usually do not lay in a linear subspace of dimension d << n. Further, there is a stringent requirement on d in such results and they fail to hold for even "fairly" small d.
3. In Theorem 2, the argument that the output through a layer of a rank-d linear subspace remains within a rank-d linear subspace does not seem correct, is it possible that it remains in the union of subsets each of which lies in a subspace of rank d?
4. The proofs for the gradients make assumptions that deviate from the initialization previously introduced. The work of Glorot et al. for example discusses the tradeoffs between the two assumptions and suggests the balancing between maintaining the input and output variance distributions.
Due to the following issues I choose to reject this work at this time.
Below are additional minor typos or issues in the paper.
On page 6, the sentence that starts with: "it is beneficial for the width": Suggesting to use neural networks of constant width seems a bit impractical.
The citation of Arpit et al. from 2017 seems possibly wrong since the paper cited discusses memorization and does not focus at the initialization of neural networks.
TYPOS:
page 1 last sentence before equation 1:  back backward? 
page 1 last sentence: repeated words: as as the the
page 1 last sentence: property --> properties?
page 2 section 2 paragraph 1: both these papers --> both papers, both of these papers?
page 4 sentence after the proof sketch paragraph: the sentence is difficult to read
[1] "Stochastic gradient descent optimizes over-parameterized deep relu networks", 2018, Difan Zou, Yuan Cao, Dongruo Zhou, Quanquan Gu