Venue: ICLR 2020 Conference

Reviews:

final_score: rating: 6: Weak Accept
review_content: Summary of the paper:
This is a theoretical paper that builds on top of Achille and Soatto (2018), Achille et al. (2019), McAllester (2013), and Berglund (2011),. The paper attempts to answer the relationship between the inductive bias of SGD, generalization of DNNs, and the invariance of learned representation from an information theoretical point of view. 
The paper mentioned many interesting links. In my opinion, the contributions are the following:
1. Invoking the theoretical result of Berglund (2011) to justify why Fisher information is relevant -- SGD tends to avoid local minima with high Fisher information. 
2. Relating the Fisher information and the stability of SGD to I(w; D).
3. Introducing the definition of effective information in the activation, and show that which is closely related to the Fisher information. 

About the rating:
This is basically a good paper but I have a few concerns: 
1. A large fraction of this paper are taken from Achille and Soatto (2018), Achille et al. (2019).  
2. In terms of impact, the paper is somehow incomplete -- it only demonstrates that the Fisher information is important, but the insights didn't lead to any substantial improvement over the current deep learning framework. 

Detailed comments:
1. In my opinion, defining "information in the weights for the task D" by KL(Q||P) is inaccurate. 
The weights themselves are information, which form a representation or a lossy compression of the data (which is also an information). 
According to the rate-distortion theory, what we care about is the amount of information the representation attains rather than "where" the information are. Therefore, we should talk about "rate" or "amount of information" or "mutual information" rather than "information" itself.
A missing reference regarding this point: 
Hu et al. "β-BNN: A Rate-Distortion Perspective on Bayesian Neural Networks." 2018,
which derives the information Lagrangian directly from rate-distortion theory. 
2. More discussions on Xu and Raginsky (2017) is expected, since it proposes to use I(w; D) as a generalization bound. It seems, in terms of generalization, minimizing I(w; D) is a sufficient condition while minimizing Fisher is a necessary condition.  
3. There are in fact 4 key aspects: sufficiency, minimality, invariance and generalization. It would be great to have a theorem to summarize the relationships between them.  
4. Could you elaborate on the footnote 3? 
review_title: Official Blind Review #2

final_score: rating: 8: Accept
review_content: The paper deals with where the information is in a deep network and how information is propagated when new data points are observed. The authors measure information in the weights of a DNN as the trade-off between network accuracy and weight complexity. They bring out the relationships between Shannon MI and Fisher Information and the connections to PAC-Bayes bound and invariance. The main result is that models of low information generalize better and are invariance-tolerant. 

The paper is very well written and concepts are theoretically-well documented.

In Definition 3.1 for the ‘Information in the Weights’, how does the complexity of the task vary with \beta? Is the Pareto curve provided in the paper? 
review_title: Official Blind Review #1

final_score: rating: 6: Weak Accept
review_content: This paper presents a theoretical account of information encoded within deep neural networks subject to information theoretic measures. In contrast to other efforts that examine information encoded in weights, this work emphasizes the effective information in the activations. This characterization is further related to information in the weights, and a theoretical justification is made for what this means with respect to properties of generalization and invariance in the network. 
The notion of attaching the weights that represent the training set to the activations that accord with the test set in a theoretical framework is interesting. In practice, I would have liked to see a bit more attachment of the theoretical formalisms to the empirical justification that follows the references. This, however, is a matter of personal bias as I don't typically produce papers that are principally theoretical contributions in my own work. Overall, the content of the paper seems sound and the theoretical and empirical justifications seem well founded but I also can't claim to be an expert in this area.
review_title: Official Blind Review #3

Meta-review: This paper is full of ideas. However, a logical argument is only as strong as its weakest link, and I believe the current paper has some weak links. For example, the attempt to tie the behavior of SGD to free energy minimization relies on unrealistic approximations. Second, the bounds based on limiting flat priors become trivial. The authors in-depth response to my own review was much appreciated, especially given its last minute appearance. Unfortunately, I was not convinced by the arguments. In part, the authors argue that the logical argument they are making is not sensitive to certain issues that I raised, but this only highlights for me that the argument being made is not very precise.  I can imagine a version of this work with sharper claims, built on clearly stated assumptions/conjectures about SGD's dynamics, RATHER THAN being framed as the consequences of clearly inaccurate approximations. The behavior of diffusions can be presented as evidence that the assumptions/conjectures (that cannot be proven at the moment, but which are needed to complete the logical argument) are reasonable. However, I am also not convinced that it is trivial to do this, and so the community must have a chance to review a major revision.
Decision: Reject