Regret Analysis of Sleeping Competing Bandits

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 $\mathrm{O}(NK\log T_i\mathrm{/}{\mathrm{\Delta }}^2)$ 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 $\mathrm{\Delta }$ is the minimum reward gap. We also provide a regret lower bound of $\mathrm{\Omega }(N(K-N+1)\log T_i\mathrm{/}{\mathrm{\Delta }}^2)$ 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 $N$.