Several recent results provide theoretical insights into the phenomena of adversarial examples. Existing results, however, are often limited due to a gap between the simplicity of the models studied and the complexity of those deployed in practice. In this work, we strike a better balance by considering a model that involves learning a representation while at the same time giving a precise generalization bound and a robustness certificate. We focus on the hypothesis class obtained by combining a sparsity-promoting encoder coupled with a linear classifier, and show an interesting interplay between the expressivity and stability of the (supervised) representation map and a notion of margin in the feature space. We bound the robust risk (to $\ell_2$-bounded perturbations) of hypotheses parameterized by dictionaries that achieve a mild encoder gap on training data. Furthermore, we provide a robustness certificate for end-to-end classification. We demonstrate the applicability of our analysis by computing certified accuracy on real data, and compare with other alternatives for certified robustness.

Additional Feedback: Overall, the work achieves new and interesting theoretical results for the model being studied. My main worry is the lack of experimental results on the encoder gap for datasets beyond MNIST, especially given that the size/existence of the encoder gap is crucial to the theoretical results and is an assumption made in the theoretical claims. Thus, I would highly recommend at least evaluating the encoder gap for other (more complex than MNIST) datasets. Many techniques that work well on MNIST may not work on other datasets due to MNIST's relative simplicity. For example, a network that binarizes pixel values (converts everything below 0.5 to 0, everything above to 1) and then classifies the result is quite adversarially robust, but the same technique will not work for more complex datasets.

Several recent results provide theoretical insights into the phenomena of adversarial examples. Existing results, however, are often limited due to a gap between the simplicity of the models studied and the complexity of those deployed in practice. In this work, we strike a better balance by considering a model that involves learning a representation while at the same time giving a precise generalization bound and a robustness certificate. We focus on the hypothesis class obtained by combining a sparsity-promoting encoder coupled with a linear classifier, and show an interesting interplay between the expressivity and stability of the (supervised) representation map and a notion of margin in the feature space. We bound the robust risk (to \ell_2 -bounded perturbations) of hypotheses parameterized by dictionaries that achieve a mild encoder gap on training data.