Stronger Approximation Guarantees for Non-Monotone γ-Weakly DR-Submodular Maximization

TL;DR

The paper addresses maximizing a nonnegative, non-monotone $\gamma$-weakly DR-submodular function over a down-closed convex body, proposing a projection-free algorithm that yields a $\Phi_\gamma$-dependent approximation. It fuses a $\gamma$-weighted Frank–Wolfe guided continuous greedy step with a $\gamma$-aware double-greedy procedure, and optimizes a convex mixture of their certificates to produce the guarantee $\Phi_\gamma$. The main results show that the algorithm achieves $F(ALG) \ge \Phi_\gamma \cdot \max_{\mathbf{y}\in P} F(\mathbf{y}) - O(\delta L D^2)$, with $\Phi_\gamma$ strictly exceeding the baseline $\kappa(\gamma)=\gamma e^{-\gamma}$ for all $\gamma\in(0,1)$ and matching the DR bound $0.401$ at $\gamma=1$. The framework is fully first-order and scalable, relying on grid-based optimization over a small parameter set, and it provides a smooth interpolation between the weakly DR and DR regimes. This advances non-monotone continuous submodular maximization with practical, large-scale applicability in fields like online allocation and probabilistic modeling.

Abstract

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.

Figures (1)

• Figure 1: Approximation guarantee versus weakly-DR parameter. The horizontal axis is the weakly-DR parameter $\gamma\in(0,1]$ and the vertical axis is the approximation factor. We plot our optimized guarantee $\Phi_\gamma$ (blue curve) alongside the non-monotone weakly-DR baseline $\kappa(\gamma)=\gamma e^{-\gamma}$ (orange curve). Across the entire regime $\gamma\in(0,1)$, $\Phi_\gamma$ strictly exceeds $\kappa(\gamma)$, and at $\gamma=1$ (full DR) our curve reaches $0.401$, matching the current best bound. Selected parameter choices $(\alpha,r,t_s)$ used to construct $\Phi_\gamma$ are reported in Table \ref{['tab:phi-params']}.

Theorems & Definitions (62)

• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 3.1
• Theorem 3.2
• Theorem 3.3
• Corollary 3.4: Box maximization
• proof
• Lemma 4.1
• Theorem 4.2