The combinatorial multi-armed bandit (CMAB) is a cornerstone of sequential decision-making framework, dominated by two algorithmic families: UCB-based and adversarial methods such as follow the regularized leader (FTRL) and online mirror descent (OMD). However, prominent UCB-based approaches like CUCB suffer from additional regret factor $\log T$ that is detrimental over long horizons, while adversarial methods such as EXP3.M and HYBRID impose significant computational overhead. To resolve this trade-off, we introduce the Combinatorial Minimax Optimal Strategy in the Stochastic setting (CMOSS). CMOSS is a computationally efficient algorithm that achieves an instance-independent regret of $O\big( (\log k)\sqrt{kmT}\big )$ when $k\leq \frac{m}{2}$ and $O\big((m-k)\sqrt{\log k\log(m-k)T}\big )$ when $k>\frac{m}{2}$ under semi-bandit feedback, where $m$ is the number of arms and $k$ is the maximum cardinality of a feasible action. Crucially, this result eliminates the dependency on $\log T$ and matches the established lower bounds of $Ω\big(\sqrt{kmT}\big)$ when $k\leq \frac{m}{2}$ and $Ω\big((m-k)\sqrt{\log (\frac{m}{m-k}) T}\big)$ when $k>\frac{m}{2}$ up to logarithmic terms of $k$ and $m$. We then extend our analysis to show that CMOSS is also applicable to cascading feedback. Experiments on synthetic and real-world datasets validate that CMOSS consistently outperforms benchmark algorithms in both regret and runtime efficiency.

翻译：组合多臂赌博机（CMAB）是序列决策框架的基石，主要由两类算法族主导：基于UCB的方法以及对抗性方法，如跟随正则化领导者（FTRL）和在线镜像下降（OMD）。然而，诸如CUCB等著名的基于UCB的方法会引入额外的遗憾因子$\log T$，这在长时域中是有害的；而对抗性方法如EXP3.M和HYBRID则带来显著的计算开销。为解决这一权衡，我们提出了随机设定下的组合极小极大最优策略（CMOSS）。CMOSS是一种计算高效的算法，在半赌博机反馈下，当$k\leq \frac{m}{2}$时实现实例无关的遗憾$O\big( (\log k)\sqrt{kmT}\big )$，当$k>\frac{m}{2}$时实现$O\big((m-k)\sqrt{\log k\log(m-k)T}\big )$，其中$m$为臂的数量，$k$为可行动作的最大基数。关键的是，这一结果消除了对$\log T$的依赖，并且当$k\leq \frac{m}{2}$时匹配了已知下界$Ω\big(\sqrt{kmT}\big)$，当$k>\frac{m}{2}$时匹配了$Ω\big((m-k)\sqrt{\log (\frac{m}{m-k}) T}\big)$，仅相差$k$和$m$的对数项。我们进一步扩展分析，表明CMOSS也适用于级联反馈。在合成和真实世界数据集上的实验验证了CMOSS在遗憾和运行时间效率上均持续优于基准算法。