Paper:
Published as a conference paper at ICLR 2018 CERTIFIED DEFENSES AGAINST ADVERSARIAL EX- AMPLES
While neural networks have achieved high accuracy on standard image classification benchmarks, their accuracy drops to nearly zero in the presence of small adversarial perturbations to test inputs. Defenses based on regularization and adversarial training have been proposed, but often followed by new, stronger attacks that defeat these defenses. Can we somehow end this arms race? In this work, we study this problem for neural networks with one hidden layer. We first propose a method based on a semidefinite relaxation that outputs a certificate that for a given network and test input, no attack can force the error to exceed a certain value. Second, as this certificate is differentiable, we jointly optimize it with the network parameters, providing an adaptive regularizer that encourages robustness against all attacks. On MNIST, our approach produces a network and a certificate that no attack that perturbs each pixel by at most = 0.1 can cause more than 35% test error.

Review:
This paper develops a new differentiable upper bound on the performance of classifier when the adversarial input in l_infinity is assumed to be applied.
While the attack model is quite general, the current bound is only valid for linear and NN with one hidden layer model, so the result is quite restrictive.
However the new bound is an "upper" bound of the worst-case performance which is very different from the conventional sampling based "lower" bounds. Therefore minimizing this upper bound together with a classification loss makes perfect sense and provides a theoretically sound approach to train a robust classifier.
This paper provides a gradient of this new upper bound with respect to model parameters so we can apply the usual first order optimization scheme to this joint optimization (loss + upper bound).
In conclusion, I recommend this paper to be accepted, since it presents a new and feasible direction of a principled approach to train a robust classifier, and the paper is clearly written and easy to follow.
 
There are possible future directions to be developed.
1. Apply the sum-of-squares (SOS) method.
The paper's SDP relaxation is the straightforward relaxation of Quadratic Program (QP), and in terms of SOS relaxation hierarchy, it is the first hierarchy. One can increase the complexity going beyond the first hierarchy, and this should provides a computationally more challenging but tighter upper bound.
The paper already mentions about this direction and it would be interesting to see the experimental results.
2. Develop a similar relaxation for deep neural networks.
The author already mentioned that they are pursuing this direction. While developing the result to the general deep neural networks might be hard, residual networks maybe fine thanks to its structure.