We study the problem of maximizing a continuous DR-submodular function that
is not necessarily smooth. We prove that the continuous greedy algorithm
achieves an [(1-1/e)\OPT-\epsilon] guarantee when the function is monotone
and H\"older-smooth, meaning that it admits a H\"older-continuous gradient. For
functions that are non-differentiable or non-smooth, we propose a variant of
the mirror-prox algorithm that attains an [(1/2)\OPT-\epsilon] guarantee. We
apply our algorithmic frameworks to robust submodular maximization and
distributionally robust submodular maximization under Wasserstein ambiguity. In
particular, the mirror-prox method applies to robust submodular maximization to
obtain a single feasible solution whose value is at least (1/2)\OPT-\epsilon.
For distributionally robust maximization under Wasserstein ambiguity, we deduce
and work over a submodular-convex maximin reformulation whose objective
function is H\"older-smooth, for which we may apply both the continuous greedy
and the mirror-prox algorithms