在线非单调DR-子模最大化：匹配离线0.401因子

中文摘要 AI 辅助

我们研究在d维单位立方体的紧凸下闭子集上，非负、非单调DR-子模函数的在线最大化问题。在相应的元可解性假设下，已知的最优构造性离线近似因子为0.401，而此前可比拟的对抗性在线保证仅停留在1/e。我们证明该因子也可在在线场景中实现。在决策后全信息值神谕模型中，当神谕反馈为条件无偏且有界时，我们的算法可达到0.401因子，并具有次线性近似遗憾。该在线算法不会在变化的目标上运行离线构造，而是用加权在线学习器替代依赖离线目标的框步，以累积方式控制所需的残差项。精确的非对称平衡定理可在对抗性变化下保留离线系数。直接实现的遗憾为O(T^{3/4})，每轮使用O(dT^{1/4})次神谕调用。更一般地，对每个δ∈[0,1/4]，批处理可使每轮神谕调用次数为O(T^δ)，遗憾为O(T^{4/5-δ/5})，包括单调用的O(T^{4/5})端点。在正锚条件下，随机阻塞可在O(T^{5/6})单臂老虎机遗憾下保留0.401因子。

英文摘要

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.

发表机构

• Purdue University（普渡大学）

机构由 AI 辅助整理，请以论文原文为准。