We consider the problem of latent bandits with cluster structure where there are multiple users, each with an associated multi-armed bandit problem. These users are grouped into \emph{latent} clusters such that the mean reward vectors of users within the same cluster are identical. At each round, a user, selected uniformly at random, pulls an arm and observes a corresponding noisy reward. The goal of the users is to maximize their cumulative rewards. This problem is central to practical recommendation systems and has received wide attention of late \cite{gentile2014online, maillard2014latent}. Now, if each user acts independently, then they would have to explore each arm independently and a regret of $\Omega(\sqrt{\mathsf{MNT}})$ is unavoidable, where $\mathsf{M}, \mathsf{N}$ are the number of arms and users, respectively. Instead, we propose LATTICE (Latent bAndiTs via maTrIx ComplEtion) which allows exploitation of the latent cluster structure to provide the minimax optimal regret of $\widetilde{O}(\sqrt{(\mathsf{M}+\mathsf{N})\mathsf{T}})$, when the number of clusters is $\widetilde{O}(1)$. This is the first algorithm to guarantee such strong regret bound. LATTICE is based on a careful exploitation of arm information within a cluster while simultaneously clustering users. Furthermore, it is computationally efficient and requires only $O(\log{\mathsf{T}})$ calls to an offline matrix completion oracle across all $\mathsf{T}$ rounds.

翻译：我们研究具有簇结构的潜在赌博机问题，其中存在多个用户，每个用户关联一个多臂赌博机问题。这些用户被划分为潜在簇，使得同一簇内用户的平均奖励向量相同。在每一轮中，均匀随机选取一个用户，该用户拉动一个臂并观测到相应的含噪奖励。用户的目标是最大化累积奖励。该问题对于实际推荐系统至关重要，近期受到广泛关注。如果每个用户独立行动，则他们必须独立探索每个臂，此时$\Omega(\sqrt{\mathsf{MNT}})$的遗憾不可避免，其中$\mathsf{M}$和$\mathsf{N}$分别为臂和用户的数量。为此，我们提出LATTICE（通过矩阵补全的潜在赌博机），该方法能够利用潜在簇结构，在簇数量为$\widetilde{O}(1)$时实现$\widetilde{O}(\sqrt{(\mathsf{M}+\mathsf{N})\mathsf{T}})$的极小化最优遗憾。这是首个保证如此强遗憾界的算法。LATTICE在聚类用户的同时，巧妙地利用簇内臂信息。此外，该算法计算高效，在整个$\mathsf{T}$轮中仅需调用$O(\log{\mathsf{T}})$次离线矩阵补全预言机。