Upper Confidence Interval Strategies for Multi-Armed Bandits with Entropy Rewards

We introduce a multi-armed bandit problem with information-based rewards. At each round, a player chooses an arm, observes a symbol, and receives an unobserved reward in the form of the symbol's self-information. The player aims to maximize the expected total reward associated with the entropy values of the arms played. We propose two algorithms based on upper confidence bounds (UCB) for this model. The first algorithm optimistically corrects the bias term in the entropy estimation. The second algorithm relies on data-dependent UCBs that adapt to sources with small entropy values. We provide performance guarantees by upper bounding the expected regret of each of the algorithms, and compare their asymptotic behavior to the Lai-Robbins lower bound. Finally, we provide numerical results illustrating the regret of the algorithms presented.