Asymptotic Randomised Control with applications to bandits

We consider a general multi-armed bandit problem with correlated (and simple
contextual and restless) elements, as a relaxed control problem. By introducing
an entropy regularisation, we obtain a smooth asymptotic approximation to the
value function. This yields a novel semi-index approximation of the optimal
decision process. This semi-index can be interpreted as explicitly balancing an
exploration-exploitation trade-off as in the optimistic (UCB) principle where
the learning premium explicitly describes asymmetry of information available in
the environment and non-linearity in the reward function. Performance of the
resulting Asymptotic Randomised Control (ARC) algorithm compares favourably
well with other approaches to correlated multi-armed bandits.