The Competing Bandits framework is a recently emerging area that integrates multi-armed bandits in online learning with stable matching in game theory. While conventional models assume that all players and arms are constantly available, in real-world problems, their availability can vary arbitrarily over time. In this paper, we formulate this setting as Sleeping Competing Bandits. To analyze this problem, we naturally extend the regret definition used in existing competing bandits and derive regret bounds for the proposed model. We propose an algorithm that simultaneously achieves an asymptotic regret bound of $\mathrm{O}\left(NK\log T_{i}/Δ^2\right)$ under reasonable assumptions, where $N$ is the number of players, $K$ is the number of arms, $T_{i}$ is the number of rounds of each player $p_i$, and $Δ$ is the minimum reward gap. We also provide a regret lower bound of $\mathrmΩ\left( N(K-N+1)\log T_{i}/Δ^2 \right)$ under the same assumptions. This implies that our algorithm is asymptotically optimal in the regime where the number of arms $K$ is relatively larger than the number of players $N$.

翻译：竞争赌博机框架是近年来兴起的一个领域，它将在线学习中的多臂赌博机与博弈论中的稳定匹配相结合。传统模型假设所有参与者和臂始终可用，但在现实问题中，它们的可用性可能随时间任意变化。本文将此设定形式化为沉睡竞争赌博机。为分析该问题，我们自然扩展了现有竞争赌博机中使用的遗憾定义，并推导出所提出模型的遗憾界。我们提出一种算法，在合理假设下同时实现了渐近遗憾界 $\mathrm{O}\left(NK\log T_{i}/Δ^2\right)$，其中 $N$ 为参与者数量，$K$ 为臂数，$T_{i}$ 为每个参与者 $p_i$ 的轮次，$Δ$ 为最小奖励差距。我们还提供了相同假设下的遗憾下界 $\mathrm{Ω}\left( N(K-N+1)\log T_{i}/Δ^2 \right)$。这表明，在臂数 $K$ 相对大于参与者数 $N$ 的 regimes 中，我们的算法是渐近最优的。