Non-Smooth, H\"older-Smooth, and Robust Submodular Maximization

TL;DR

This paper develops algorithms for maximizing non-smooth, H"older-smooth, and robust submodular functions, providing approximation guarantees and applying them to robust and distributionally robust scenarios.

Contribution

It introduces a continuous greedy algorithm for H"older-smooth functions and a mirror-prox variant for non-smooth cases, extending submodular maximization to robust settings.

Findings

Continuous greedy achieves (1-1/e) approximation for H"older-smooth functions.

Mirror-prox attains 1/2 approximation for non-smooth functions.

Algorithms apply to robust and distributionally robust submodular maximization problems.

Abstract

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.