Gaussian Process Bandits with Aggregated Feedback

We consider the continuum-armed bandits problem, under a novel setting of recommending the best arms within a fixed budget under aggregated feedback. This is motivated by applications where the precise rewards are impossible or expensive to obtain, while an aggregated reward or feedback, such as the average over a subset, is available. We constrain the set of reward functions by assuming that they are from a Gaussian Process and propose the Gaussian Process Optimistic Optimisation (GPOO) algorithm. We adaptively construct a tree with nodes as subsets of the arm space, where the feedback is the aggregated reward of representatives of a node. We propose a new simple regret notion with respect to aggregated feedback on the recommended arms. We provide theoretical analysis for the proposed algorithm, and recover single point feedback as a special case. We illustrate GPOO and compare it with related algorithms on simulated data.