We study the Combinatorial Thompson Sampling policy (CTS) for combinatorial multi-armed bandit problems (CMAB), within an approximation regret setting. Although CTS has attracted a lot of interest, it has a drawback that other usual CMAB policies do not have when considering non-exact oracles: for some oracles, CTS has a poor approximation regret (scaling linearly with the time horizon $T$) [Wang and Chen, 2018]. A study is then necessary to discriminate the oracles on which CTS could learn. This study was started by Kong et al. [2021]: they gave the first approximation regret analysis of CTS for the greedy oracle, obtaining an upper bound of order $\mathcal{O}(\log(T)/\Delta^2)$, where $\Delta$ is some minimal reward gap. In this paper, our objective is to push this study further than the simple case of the greedy oracle. We provide the first $\mathcal{O}(\log(T)/\Delta)$ approximation regret upper bound for CTS, obtained under a specific condition on the approximation oracle, allowing a reduction to the exact oracle analysis. We thus term this condition REDUCE2EXACT, and observe that it is satisfied in many concrete examples. Moreover, it can be extended to the probabilistically triggered arms setting, thus capturing even more problems, such as online influence maximization.

翻译：我们研究组合汤普森采样策略（CTS）在组合多臂老虎机问题（CMAB）中的近似遗憾设定。尽管CTS引起了广泛关注，但它存在一个其他常见CMAB策略在考虑非精确预言机时不具有的缺陷：对于某些预言机，CTS的近似遗憾表现较差（随时间范围$T$线性增长）[Wang and Chen, 2018]。因此有必要区分CTS能够学习的预言机类型。Kong等人[2021]开启了这项研究：他们首次对贪心预言机下的CTS进行了近似遗憾分析，获得了阶为$\mathcal{O}(\log(T)/\Delta^2)$的上界，其中$\Delta$为最小奖励差距。本文旨在将这一研究推进到超越简单贪心预言机的情形。我们首次给出了CTS的$\mathcal{O}(\log(T)/\Delta)$近似遗憾上界，该上界在近似预言机满足特定条件时成立，从而可简化为精确预言机分析。我们将此条件称为REDUCE2EXACT，并观察到它在许多具体实例中成立。此外，该条件可扩展至概率触发臂设定，从而涵盖更多问题，例如在线影响力最大化。