The combinatorial pure exploration (CPE) in the stochastic multi-armed bandit setting (MAB) is a well-studied online decision-making problem: A player wants to find the optimal \emph{action} $\boldsymbol{\pi}^*$ from \emph{action class} $\mathcal{A}$, which is a collection of subsets of arms with certain combinatorial structures. Though CPE can represent many combinatorial structures such as paths, matching, and spanning trees, most existing works focus only on binary action class $\mathcal{A}\subseteq\{0, 1\}^d$ for some positive integer $d$. This binary formulation excludes important problems such as the optimal transport, knapsack, and production planning problems. To overcome this limitation, we extend the binary formulation to real, $\mathcal{A}\subseteq\mathbb{R}^d$, and propose a new algorithm. The only assumption we make is that the number of actions in $\mathcal{A}$ is polynomial in $d$. We show an upper bound of the sample complexity for our algorithm and the action class-dependent lower bound for R-CPE-MAB, by introducing a quantity that characterizes the problem's difficulty, which is a generalization of the notion \emph{width} introduced in Chen et al.[2014].

翻译：组合纯探索（CPE）在随机多臂赌博机（MAB）设定中是一个被广泛研究的在线决策问题：玩家希望从具有特定组合结构的臂子集集合——即动作类$\mathcal{A}$中，找到最优动作$\boldsymbol{\pi}^*$。尽管CPE能够涵盖路径、匹配和生成树等多种组合结构，但现有研究大多仅关注于二元动作类$\mathcal{A}\subseteq\{0, 1\}^d$（其中$d$为正整数）。这种二元表述排除了诸如最优传输、背包及生产规划等重要问题。为克服这一局限，我们将二元表述扩展至实数域$\mathcal{A}\subseteq\mathbb{R}^d$，并提出一种新算法。我们所做的唯一假设是$\mathcal{A}$中动作的数量关于$d$呈多项式级增长。通过引入一个刻画问题难度的量（该量是对Chen等人[2014]提出的"宽度"概念的推广），我们给出了算法样本复杂度的上界，以及R-CPE-MAB问题中与动作类相关的下界。