  This manuscript presents some new impossibility 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
$W_2$ Talagrand transportation-cost inequality (for example, this condition is
satisfied if the conditional distribution has density which is log-concave; is
the uniform measure on a compact Riemannian manifold with positive Ricci
curvature, 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). We conclude the
manuscript with some speculation on possible future research directions.
