ACloser Look at the Worst-case Behavior of Multi-armed Bandit Algorithms

One of the key drivers of complexity in the classical (stochastic) multi-armed bandit (MAB) problem is the difference between mean rewards in the top two arms, also known as the instance gap. The celebrated Upper Confidence Bound (UCB) policy is among the simplest optimism-based MAB algorithms that naturally adapts to this gap: for a horizon of play n, it achieves optimal O(logn) regret in instances with "large" gaps, and a near-optimal O p nlogn minimax regret when the gap can be arbitrarily "small." This paper provides new results on the arm-sampling behavior of UCB, leading to several important insights. Among these, it is shown that arm-sampling rates under UCB are asymptotically deterministic, regardless of the problem complexity.