Paper:
In adversarial attacks to machine-learning classifiers, small perturbations are added to input that is correctly classified. The perturbations yield adversarial examples, which are virtually indistinguishable from the unperturbed input, and yet are misclassified. In standard neural networks used for deep learning, attackers can craft adversarial examples from most input to cause a misclassification of their choice. We introduce a new type of network units, called RBFI units, whose non-linear structure makes them inherently resistant to adversarial attacks. On permutationinvariant MNIST, in absence of adversarial attacks, networks using RBFI units match the performance of networks using sigmoid units, and are slightly below the accuracy of networks with ReLU units. When subjected to adversarial attacks based on projected gradient descent or fast gradient-sign methods, networks with RBFI units retain accuracies above 75%, while ReLU or Sigmoid see their accuracies reduced to below 1%. Further, RBFI networks trained on regular input either exceed or closely match the accuracy of sigmoid and ReLU network trained with the help of adversarial examples. The non-linear structure of RBFI units makes them difficult to train using standard gradient descent. We show that RBFI networks of RBFI units can be efficiently trained to high accuracies using pseudogradients, computed using functions especially crafted to facilitate learning instead of their true derivatives.

Review:
This paper introduces a new neural network layer for the purposes of defending against "white-box" adversarial attacks (in which the adversary is provided access to the neural network parameters). The new network unit and its activation function are constructed in such a way that the local gradient is sparse and therefore is difficult to exploit to add adversarial shifts to the input. To train the networks in the presence of a sparse gradient signal, the authors introduce a "pseudogradient", and optimize this proxy-gradient to optimize the parameters. This training procedure shows competitive performance (after training) on the permutation-invariant MNIST dataset versus other more standard network architectures, but is more robust to both adversarial attacks and random noise.
High-level comments:
- Using only a single dataset, and one on which the classification problem is rather easy, is cause for concern. I would need to see performance on another dataset, like CIFAR 10, to be more convinced that this is a general pipeline. In Sec 4, the authors mention that, using the pseudogradient, "one may be concerned that ... we may converge ... and yet, we are not at a minimum of the loss function". They claim that "in practice it does not seem to be a problem" on their experiments. This claim is a bit weak considering only a single, simple dataset was used for training. It is not obvious to me that this would succeed for more complex datasets.
- I would also like to see an additional set of adversarial attacks that are "RBFI-aware". A motivated attacker who is aware of this technique might replace the gradient in the adversarial attack with the pseudogradient instead; I expect such an attack would be effective. While problematic in general, I do not think this is necessarily an overall weakness of the paper (since we, the community, should be investigating methods like these to obfuscate the process of exploiting neural network models), but I would still like to see results showing the impact/performance of adversarial training over the pseudo-gradient. (I do not expect this will be very much effort.)
- What is the purpose of showing robustness of your network models to random noise? It is nice/interesting to see that your results are more robust to random noise, but what is the intuition for why your network performs better?
Wording and minor comments:
- The abstract is rather lengthy, but should probably contain somewhere a spelling-out of RBFI, since it informs the reader that the radial basis function (with infinity-norm) is the structure of the new network unit.
- Sec 4: "...indicate that pseudogradients work much better than regular gradients" :: Please be more clear that this is context specific "...than regular gradients for training RBFI networks".
- Sec. 4 :: Try to be consistent to how you specify "z" in this section, you alternate between the 'infinity-norm' definition and the 'max' definition from Eq. (2). Try to homogenize these.
- In general, the paper was well-proofed and well-written and was easy to read (high clarity).
- To my knowledge, this work is a rather unique foray into solving this problem (original).
Overall, I think this work is an interesting idea to address a rather important concern in the Deep Learning community. While the idea has merit, the small set of experiments in this paper is not sufficiently compelling for me to immediately recommend publication. With a bit more work put into exploring the performance of this method on other datasets, this paper could be made more complete. (Also, since I am aware that space is limited, some of the details on the adversarial attacks from other publications can probably be moved to an appendix.)