Paper:
We investigate global adversarial robustness guarantees for machine learning models. Specifically, given a trained model we consider the problem of computing the probability that its prediction at any point sampled from the (unknown) input distribution is susceptible to adversarial attacks. Assuming continuity of the model, we prove measurability for a selection of local robustness properties used in the literature. We then show how concentration inequalities can be employed to compute global robustness with estimation error upper-bounded by , for any > 0 selected a priori. We utilise the methods to provide statistically sound analysis of the robustness/accuracy trade-off for a variety of neural networks architectures and training methods on MNIST, Fashion-MNIST and CIFAR. We empirically observe that robustness and accuracy tend to be negatively correlated for networks trained via stochastic gradient descent and with iterative pruning techniques, while a positive trend is observed between them in Bayesian settings.

Review:
This paper seeks to analyze the global robustness of neural networks, a concept defined in the paper. The authors show using concentration inequalities that the empirical local robustness approximates the global robustness. The authors investigate various other issues in the robustness literature, including the robustness/accuracy tradeoff, whether iterative pruning increases robustness, and the robustness of Bayesian networks.
I would vote for rejecting this paper for two key reasons. First, the notion of global robustness is not well-motivated (why do we want to compute this metric? What does it tell us that local robustness does not?). Second, the paper tries to do too many different things, and as a result does not give enough attention to any particular topic.
First, I believe it is up to the authors to motivate their study of global robustness further. While I acknowledge that a few prior works exists along these lines, I do not feel that this work provides much new insight into why global robustness is interesting to examine.
The authors go on to prove results showing that an empirical estimator of the local robustness will converge to the global robustness. The bounds require that the dataset size scales with eps^-2, where eps is the error. This is not terrible but also not great; for example, achieving 1% error requires a dataset size of 10^4 (realistically, even larger datasets would be required to achieve results with high probability).
Next, I would suggest that the authors avoid using the word “guarantees” if they are estimating empirical local robustness in an approximate (rather than exact) manner. Guarantees implies strict results, but the authors use a weak attack (FGSM) to approximate empirical local robustness. The results from FGSM could be far from optimal; the authors could use a stronger attack (e.g. PGD) in addition to changing the wording, or they could find provable guarantees using alternate methods.
Lastly, the authors try to tackle 3 extra questions beyond global robustness toward the end of the paper, and the last two questions are not properly fleshed out.
I like section 4.2, where the authors empirically show that networks that have better hyperparameters (for regular accuracy) tend to be less robust. This is a confirmation of a previously studied phenomena in the literature. Ideally, I would also appreciate it if the authors found the line of best fit to the dataset in addition to the plots provided. I would like a clarification on whether any of these networks were trained to be robust, although it appears that they were all trained normally. I would also like to see plot 2c (for the standard case of robustness of C(x_tilde) = C(x)), except for MNIST and CIFAR10 as well. I feel that the last-layer representation metric the authors analyze (f(x) is close to f(x_tilde)) could be misleading, as robustness on the last layer does not necessarily imply standard adversarial robustness.
Section 4.3 explores iterative pruning, but that seems fairly unrelated to the rest of the paper. Finally, Section 4.4 tries to show the opposite trend for Bayesian Neural Networks, but unfortunately the results for such networks do not yet scale beyond MNIST.
Additional Feedback:
- Why did you use R^emp and D^emp as opposed to just R^emp(g) and R^emp(g_bar)?
- In Figure 1, what is the dataset size |S|?
- In the last sentence of Section 4.3, I didn’t understand what you meant about the relationship between weight pruning and network regularization. Do you mean that weight regularization has no effect on robustness, just like iterative weight pruning?