BelMan: Bayesian Bandits on the Belief–Reward Manifold

We propose a generic, Bayesian, information geometric approach to the exploration–exploitation trade-off in multi-armed bandit problems. Our approach, BelMan, uniformly supports pure exploration, exploration–exploitation, and two-phase bandit problems. The knowledge on bandit arms and their reward distributions is summarised by the barycentre of the joint distributions of beliefs and rewards of the arms, the pseudobelief-reward, within the beliefs-rewards manifold. BelMan alternates information projection and reverse information projection, i.e., projection of the pseudobelief-reward onto beliefs-rewards to choose the arm to play, and projection of the resulting beliefs-rewards onto the pseudobelief-reward. It introduces a mechanism that infuses an exploitative bias by means of a focal distribution, i.e., a reward distribution that gradually concentrates on higher rewards. Comparative performance evaluation with state-of-the-art algorithms shows that BelMan is not only competitive but can also outperform other approaches in specific setups, for instance involving many arms and continuous rewards.