Paper:
The vulnerability to slight input perturbations is a worrying yet intriguing property of deep neural networks (DNNs). Though some efforts have been devoted to investigating the reason behind such adversarial behavior, the relation between standard accuracy and adversarial behavior of DNNs is still little understood. In this work, we reveal such relation by first introducing a metric characterizing the standard performance of DNNs. Then we theoretically show this metric can be disentangled into an information-theoretic non-robust component that is related to adversarial behavior, and a robust component. Then, we show by experiments that DNNs under standard training rely heavily on optimizing the non-robust component in achieving decent performance. We also demonstrate current state-ofthe-art adversarial training algorithms indeed try to robustify DNNs by preventing them from using the non-robust component to distinguish samples from different categories. Based on our findings, we take a step forward and point out the possible direction of simultaneously achieving decent standard generalization and adversarial robustness. It is hoped that our theory can further inspire the community to make more interesting discoveries about the relation between standard accuracy and adversarial robustness of DNNs.

Review:
The authors propose a new performance metric, CCKL, as an alternative to standard cross-entropy. At a high level the goal of this objective is to ensure that the classifier is confidently separating samples from different classes. They decompose this objective into its Taylor expansion terms and argue that, since adversarial vulnerability is a local property, it only depends on the first of these terms. They finally propose training models of higher capacity as a way of relying more on higher-order terms of the objective and hence improving adversarial robustness.
I found the core idea potentially interesting. It is an attempt at rigorously studying the tension between local stability (necessary for robustness) and the fact that different classes should be confidently separated. However, I have several concerns regarding the theoretical derivation and experimental evaluation.
First of all, I did not find the proposed CCKL objective well motivated. Theorem 3.1 only shows that standard training improves a _lower bound_ on the proposed objective. However, this only implies that this objective is being optimized as a byproduct of standard cross-entropy optimization. It is unclear for instance if this is a good performance indicator or whether training with this objective will even lead to a good classifier (for instance the classifier could learn a permutation of the labels instead). Therefore, I do not find the theoretical and experimental evidence in favor of studying this proposed objective convincing.
Moreover, the experimental evidence presented is quite weak. While the authors compare various quantities related to standard and robust models, they do not perform the ablations necessary to understand whether these properties are causally related to robustness of simply a byproduct of the specific training scheme.
Finally, the conclusion that larger capacity can improve both model accuracy and robustness has already been made in prior work (Kurakin et al. 2016, Madry et al. 2017). The intuition that added capacity helps by allowing the model to leverage higher order terms of the loss is interesting but a rather incremental contribution.
Overall, while the ideas presented in the paper could be of interest, they are not rigorously established and rather serve as high-level intuition at this moment. Moreover, the writing of the paper could be significantly improved. It took me quite some time to distill the above intuition despite the fact that it is a rather elementary consequence of the Taylor expansion (I am not sure what the discussion about information geometry adds to this work). I would thus encourage the authors to continue working on these ideas and their presentation. However, for now, I will have to recommend rejection.