The checklist follows the references. Please read the checklist guidelines carefully for information on how to answer these questions. You are strongly encouraged to include a justification to your answer, either by referencing the appropriate section of your paper or providing a brief inline description. Please do not modify the questions and only use the provided macros for your answers. Note that the Checklist section does not count towards the page limit. In your paper, please delete this instructions block and only keep the Checklist section heading above along with the questions/answers below.

Starting with the Thomspon sampling algorithm, recent years have seen a resurgence of interest in Bayesian algorithms for the Multi-armed Bandit (MAB) problem. These algorithms seek to exploit prior information on arm biases and while several have been shown to be regret optimal, their design has not emerged from a principled approach. In contrast, if one cared about Bayesian regret discounted over an infinite horizon at a fixed, pre-specified rate, the celebrated Gittins index theorem offers an optimal algorithm. Unfortunately, the Gittins analysis does not appear to carry over to minimizing Bayesian regret over all sufficiently large horizons and computing a Gittins index is onerous relative to essentially any incumbent index scheme for the Bayesian MAB problem. The present paper proposes a sequence of'optimistic' approximations to the Gittins index. We show that the use of these approximations in concert with the use of an increasing discount factor appears to offer a compelling alternative to state-of-the-art index schemes proposed for the Bayesian MAB problem in recent years by offering substantially improved performance with little to no additional computational overhead. In addition, we prove that the simplest of these approximations yields frequentist regret that matches the Lai-Robbins lower bound, including achieving matching constants.

We introduce the Blackwell discount factor for Markov Decision Processes (MDPs). Classical objectives for MDPs include discounted, average, and Blackwell opti-mality.

We introduce the Blackwell discount factor for Markov Decision Processes (MDPs). Classical objectives for MDPs include discounted, average, and Blackwell opti-mality.

Weproveconvergence for this method under standard assumptions and demonstrate empirically that it indeed enables lowerdiscount factors forapproximate reinforcement-learning methods.