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.

翻译：我们研究最大化不一定光滑的连续DR-子模函数问题。我们证明，当函数为单调且赫尔德光滑（即其梯度具有赫尔德连续性）时，连续贪婪算法可实现 $[(1-1/e)\OPT-\epsilon]$ 的保证。针对不可微或非光滑函数，我们提出了一种镜像近似算法变体，该算法可达到 $[(1/2)\OPT-\epsilon]$ 的保证。我们将算法框架应用于瓦瑟斯坦模糊集下的鲁棒子模最大化和分布鲁棒子模最大化。特别地，镜像近似方法可应用于鲁棒子模最大化，获得一个值至少为 $(1/2)\OPT-\epsilon$ 的可行解。对于瓦瑟斯坦模糊集下的分布鲁棒最大化问题，我们推导并研究了一个子模-凸极大极小重构形式，其目标函数为赫尔德光滑，可同时适用连续贪婪算法和镜像近似算法。