Upper-Linearizability of Online Non-Monotone DR-Submodular Maximization over Down-Closed Convex Sets

TL;DR

This paper introduces a new structural approach for online non-monotone DR-submodular maximization over down-closed convex sets, achieving improved regret bounds and feedback efficiency by establishing $1/e$-linearizability.

Contribution

The paper presents a novel structural result showing $1/e$-linearizability of the class, enabling reduction to online linear optimization and improved regret guarantees.

Findings

Achieves $O(T^{1/2})$ static regret with a single gradient query per round.

Provides adaptive and dynamic regret guarantees.

Improves rates under semi-bandit, bandit, and zeroth-order feedback.

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.