Limitations of adversarial robustness: strong No Free Lunch Theorem
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This manuscript presents some new results on adversarial robustness in machine learning, a very important yet largely open problem. We show that if conditioned on a class label the data distribution satisfies the generalized Talagrand transportation-cost inequality (for example, this condition is satisfied if the conditional distribution has density which is log-concave), any classifier can be adversarially fooled with high probability once the perturbations are slightly greater than the natural noise level in the problem. We call this result The Strong "No Free Lunch" Theorem as some recent results (Tsipras et al. 2018, Fawzi et al. 2018, etc.) on the subject can be immediately recovered as very particular cases. Our theoretical bounds are demonstrated on both simulated and real data (MNIST). These bounds readily extend to distributional ro- bustness (with 0/1 loss). We conclude the manuscript with some speculation on possible future research directions.
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