  We initiate the study of computability requirements for adversarially robust
learning. Adversarially robust PAC-type learnability is by now an established
field of research. However, the effects of computability requirements in
PAC-type frameworks are only just starting to emerge. We introduce the problem
of robust computable PAC (robust CPAC) learning and provide some simple
sufficient conditions for this. We then show that learnability in this setup is
not implied by the combination of its components: classes that are both CPAC
and robustly PAC learnable are not necessarily robustly CPAC learnable.
Furthermore, we show that the novel framework exhibits some surprising effects:
for robust CPAC learnability it is not required that the robust loss is
computably evaluable! Towards understanding characterizing properties, we
introduce a novel dimension, the computable robust shattering dimension. We
prove that its finiteness is necessary, but not sufficient for robust CPAC
learnability. This might yield novel insights for the corresponding phenomenon
in the context of robust PAC learnability, where insufficiency of the robust
shattering dimension for learnability has been conjectured, but so far a
resolution has remained elusive.
