Shafahi et al. discuss fundamental limits of adversarial robustness, showing that adversarial examples are – to some extent – inevitable. Specifically, for the unit sphere, the unit cube as well as for different attacks (e.g., sparse attacks and dense attacks), the authors show that adversarial examples likely exist. The provided theoretical arguments also provide some insights on which problems are more (or less) robust. For example, more concentrated class distributions seem to be more robust by construction. Overall, these insights lead the authors to several interesting conclusions: First, the results are likely to extent to datasets which actually live on low-dimensional manifolds of the unit sphere/cube. Second, it needs to be differentiated between the existence adversarial examples and our ability to compute them efficiently. Making it harder to compute adversarial examples might, thus, be a valid defense mechanism. And third, the results suggest that lower-dimensional data might be less susceptible to adversarial examples.
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