Stronger Approximation Guarantees for Non-Monotone γ-Weakly DR-Submodular Maximization
This addresses a fundamental optimization problem in machine learning with incremental improvements in approximation guarantees for non-monotone weakly submodular functions.
The paper tackles the problem of maximizing non-monotone γ-weakly DR-submodular functions over down-closed convex bodies, achieving an approximation algorithm with a guarantee that smoothly depends on γ, recovering a 0.401 factor for γ=1 and improving upon previous bounds for γ<1.
Maximizing submodular objectives under constraints is a fundamental problem in machine learning and optimization. We study the maximization of a nonnegative, non-monotone $γ$-weakly DR-submodular function over a down-closed convex body. Our main result is an approximation algorithm whose guarantee depends smoothly on $γ$; in particular, when $γ=1$ (the DR-submodular case) our bound recovers the $0.401$ approximation factor, while for $γ<1$ the guarantee degrades gracefully and, it improves upon previously reported bounds for $γ$-weakly DR-submodular maximization under the same constraints. Our approach combines a Frank-Wolfe-guided continuous-greedy framework with a $γ$-aware double-greedy step, yielding a simple yet effective procedure for handling non-monotonicity. This results in state-of-the-art guarantees for non-monotone $γ$-weakly DR-submodular maximization over down-closed convex bodies.