Upper-Linearizability of Online Non-Monotone DR-Submodular Maximization over Down-Closed Convex Sets
arXiv:2602.20578v1 Announce Type: new Abstract: We study online maximization of non-monotone Diminishing-Return(DR)-submodular functions over down-closed convex sets, a regime where existing projection-free online methods suffer from suboptimal regret and limited feedback guarantees. Our main contribution is a new structural result showing that this class is $1/e$-linearizable under carefully designed exponential reparametrization, scaling parameter, and surrogate potential, enabling a reduction to online linear optimization. As a result, we obtain $O(T^{1/2})$ static regret with a single gradient query per round and unlock adaptive and dynamic regret guarantees, together with improved rates under semi-bandit, bandit, and zeroth-order feedback. Across all feedback models, our bounds strictly improve the state of the art.
arXiv:2602.20578v1 Announce Type: new Abstract: We study online maximization of non-monotone Diminishing-Return(DR)-submodular functions over down-closed convex sets, a regime where existing projection-free online methods suffer from suboptimal regret and limited feedback guarantees. Our main contribution is a new structural result showing that this class is $1/e$-linearizable under carefully designed exponential reparametrization, scaling parameter, and surrogate potential, enabling a reduction to online linear optimization. As a result, we obtain $O(T^{1/2})$ static regret with a single gradient query per round and unlock adaptive and dynamic regret guarantees, together with improved rates under semi-bandit, bandit, and zeroth-order feedback. Across all feedback models, our bounds strictly improve the state of the art.
Executive Summary
This article introduces a novel approach to online maximization of non-monotone DR-submodular functions over down-closed convex sets, achieving $1/e$-linearizability and $O(T^{1/2})$ static regret with improved rates under various feedback models. The authors' contributions have significant implications for online optimization, enabling more efficient and adaptive algorithms. The study's results surpass existing state-of-the-art bounds, providing a substantial advancement in the field. The methodology's potential applications are vast, ranging from machine learning to operations research. Overall, the article presents a groundbreaking framework for tackling complex optimization problems, offering a promising direction for future research.
Key Points
- ▸ Upper-linearizability of non-monotone DR-submodular functions
- ▸ Exponential reparametrization and surrogate potential for $1/e$-linearizability
- ▸ Achieving $O(T^{1/2})$ static regret with a single gradient query per round
Merits
Novel Structural Result
The authors' discovery of $1/e$-linearizability enables a reduction to online linear optimization, significantly improving regret bounds and feedback guarantees.
Demerits
Complexity of Reparametrization
The carefully designed exponential reparametrization and surrogate potential may be challenging to implement in practice, potentially limiting the approach's applicability.
Expert Commentary
The article's contribution to the field of online optimization is substantial, as it addresses a significant gap in the existing literature. The authors' innovative approach to achieving $1/e$-linearizability and reducing the problem to online linear optimization is a testament to the power of careful mathematical analysis. The implications of this study are far-reaching, with potential applications in various domains. However, further research is necessary to explore the practical implementation of the proposed methodology and to investigate its limitations in more complex scenarios.
Recommendations
- ✓ Further investigation into the practical implementation of the proposed reparametrization and surrogate potential
- ✓ Exploration of the approach's applicability to other optimization problems and domains
