Restless multi-armed bandits (RMAB) have been widely used to model sequential decision making problems with constraints. The decision maker (DM) aims to maximize the expected total reward over an infinite horizon under an "instantaneous activation constraint" that at most B arms can be activated at any decision epoch, where the state of each arm evolves stochastically according to a Markov decision process (MDP). However, this basic model fails to provide any fairness guarantee among arms. In this paper, we introduce RMAB-F, a new RMAB model with "long-term fairness constraints", where the objective now is to maximize the long term reward while a minimum long-term activation fraction for each arm must be satisfied. For the online RMAB-F setting (i.e., the underlying MDPs associated with each arm are unknown to the DM), we develop a novel reinforcement learning (RL) algorithm named Fair-UCRL. We prove that Fair-UCRL ensures probabilistic sublinear bounds on both the reward regret and the fairness violation regret. Compared with off-the-shelf RL methods, our Fair-UCRL is much more computationally efficient since it contains a novel exploitation that leverages a low-complexity index policy for making decisions. Experimental results further demonstrate the effectiveness of our Fair-UCRL.

我们介绍了一种名为RMAB-F的新的多臂老虎机模型，它具有长期公平约束；并开发了一种名为Fair-UCRL的强化学习算法，证明了其能够在奖励遗憾和公平违反遗憾方面保持概率亚线性边界。

具有长期公平约束的在线不安定多臂赌博机