Non-smooth and Hölder-smooth Submodular Maximization
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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-ϵ] guarantee when the function is monotone and Hölder-smooth, meaning that it admits a Hölder-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-ϵ] 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-ϵ. For distributionally robust maximization under Wasserstein ambiguity, we deduce and work over a submodular-convex maximin reformulation whose objective function is Hölder-smooth, for which we may apply both the continuous greedy method and the mirror-prox method.
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