Modified Index Policies for Multi-Armed Bandits with Network-like Markovian Dependencies

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Sequential decision-making in dynamic and interconnected environments is a cornerstone of numerous applications, ranging from communication networks and finance to distributed blockchain systems and IoT frameworks. The multi-armed bandit (MAB) problem is a fundamental model in this domain that traditionally assumes independent and identically distributed (iid) rewards, which limits its effectiveness in capturing the inherent dependencies and state dynamics present in some real-world scenarios. In this paper, we lay a theoretical framework for a modified MAB model in which each arm’s reward is generated by a hidden Markov process. In our model, each arm undergoes Markov state transitions independent of play in a way that results in varying reward distributions and heightened uncertainty in reward observations. The number of states for each arm can be up to three states. A key challenge arises from the fact that the underlying states governing each arm’s rewards remain hidden at the time of selection. To address this, we adapt traditional index-based policies and develop a modified index approach tailored to accommodate Markovian transitions and enhance selection efficiency for our model. Our proposed proposed Markovian Upper Confidence Bound (MC-UCB) policy achieves logarithmic regret. Comparative analysis with the classical UCB algorithm reveals that MC-UCB consistently achieves approximately a 15% reduction in cumulative regret. This work provides significant theoretical insights and lays a robust foundation for future research aimed at optimizing decision-making processes in complex, networked systems with hidden state dependencies.