Optimism in the face of uncertainty supported by a statistically-designed multi-armed bandit algorithm

The principle of optimism in the face of uncertainty is known as a heuristic in sequential decision-making problems. Overtaking method based on this principle is an effective algorithm to solve multi-armed bandit problems. It was defined by a set of some heuristic patterns of the formulation in the previous study. The objective of the present paper is to redefine the value functions of Overtaking method and to unify the formulation of them. The unified Overtaking method is associated with upper bounds of confidence intervals of expected rewards on statistics. The unification of the formulation enhances the universality of Overtaking method. Consequently we newly obtain Overtaking method for the exponentially distributed rewards, numerically analyze it, and show that it outperforms UCB algorithm on average. The present study suggests that the principle of optimism in the face of uncertainty should be regarded as the statistics-based consequence of the law of large numbers for the sample mean of rewards and estimation of upper bounds of expected rewards, rather than as a heuristic, in the context of multi-armed bandit problems.