Switching Latent Bandits

Alessio Russo, Alberto Maria Metelli, Marcello Restelli

Abstract: We consider a Latent Bandit problem where the latent state keeps changing in time according to an underlying Markov Chain and every state is represented by a specific Bandit instance. At each step, the agent chooses an arm and observes a random reward but is unaware of which MAB he is currently pulling. As typical in Latent Bandits, we assume to know the reward distribution of the arms of all the Bandit instances. Within this setting, our goal is to learn the transition matrix determined by the Markov process, so as to minimize the cumulative regret. We propose a technique to solve this estimation problem that exploits the properties of Markov Chains and results in solving a system of linear equations. We present an offline method that chooses the best subset of possible arms that can be used for matrix estimation, and we ultimately introduce the SL-EC learning algorithm based on an Explore Then Commit strategy that builds a belief representation of the current state and optimizes the instantaneous regret at each step. This algorithm achieves a regret of the order $\widetilde{\mathcal{O}}(T^{2/3})$ with $T$ being the interaction horizon. Finally, we illustrate the effectiveness of the approach and compare it with state-of-the-art algorithms for non-stationary bandits.