  Recently, Montasser et al. [2019] showed that finite VC dimension is not
sufficient for proper adversarially robust PAC learning. In light of this
hardness, there is a growing effort to study what type of relaxations to the
adversarially robust PAC learning setup can enable proper learnability. In this
work, we initiate the study of proper learning under relaxations of the
worst-case robust loss. We give a family of robust loss relaxations under which
VC classes are properly PAC learnable with sample complexity close to what one
would require in the standard PAC learning setup. On the other hand, we show
that for an existing and natural relaxation of the worst-case robust loss,
finite VC dimension is not sufficient for proper learning. Lastly, we give new
generalization guarantees for the adversarially robust empirical risk
minimizer.
