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

2026-02-24

AI Summary

This work addresses the problem of maximizing online non-monotone DR-submodular functions over downward-closed convex sets, where existing projection-free methods suffer from suboptimal regret bounds and restrictive feedback assumptions. By introducing exponential reparameterization, scaling parameters, and a surrogate potential function, the authors present the first reduction of this problem to online linear optimization, establishing a $1/e$-linearization framework. Their algorithm achieves $O(T^{1/2})$ static regret with only a single gradient query per round. The approach unifies support for multiple feedback models—including semi-bandit, full-information, and zeroth-order settings—and provides both adaptive and dynamic regret guarantees. The resulting performance strictly improves upon the best-known results in the literature.

Technology Category

• Search and Optimization: Non-convex Optimization
• Machine Learning: Online Learning & Bandits
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

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.

Problem

• online optimization
• DR-submodular maximization
• non-monotone
• down-closed convex sets
• regret bounds

Innovation

• DR-submodular maximization
• online linearization
• adaptive regret
• dynamic regret
• projection-free optimization