Online Non-Monotone DR-Submodular Maximization Matching the Offline $0.401$ Factor

2026-09-02

AI Summary

本文解决了在线非单调DR-子模函数最大化问题，通过设计一种新的在线学习算法，在全信息反馈模型下达到0.401的近似比。

Abstract

We study online maximization of nonnegative, non-monotone DR-submodular functions over compact convex down-closed subsets of the $d$-dimensional unit cube. The best known constructive offline approximation factor is $0.401$ under the corresponding meta-solvability assumptions, whereas comparable adversarial online guarantees had remained at $1/e$. We show that this factor is also achievable online. In the post-decision full-information value-oracle model, our algorithm attains factor $0.401$ with sublinear approximate regret when oracle feedback is conditionally unbiased and bounded. The online algorithm does not run the offline construction on a changing objective. Instead, it replaces the offline objective-dependent box step by a weighted online learner that controls the required residual terms cumulatively. An exact asymmetric balance theorem preserves the offline coefficients despite adversarial variation. The direct implementation has $O(T^{3/4})$ regret and uses $O(dT^{1/4})$ oracle calls per round. More generally, for every $δ\in[0,1/4]$, batching gives $O(T^δ)$ calls per round and $O(T^{4/5-δ/5})$ regret, including a one-call $O(T^{4/5})$ endpoint. Under a positive-anchor condition, randomized blocking retains factor $0.401$ with $O(T^{5/6})$ one-point bandit regret.

Problem

• non-monotone DR-submodular
• online maximization
• approximation factor
• compact convex down-closed subsets

Innovation

• online non-monotone DR-submodular maximization
• 0.401 approximation factor
• weighted online learner
• sublinear approximate regret
• batching