  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(log n) regret in instances with "large" gaps, and a near-optimal O(\sqrt{n
log n}) 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. This
discovery facilitates new sharp asymptotics and a novel alternative proof for
the O(\sqrt{n log n}) minimax regret of UCB. Furthermore, the paper also
provides the first complete process-level characterization of the MAB problem
under UCB in the conventional diffusion scaling. Among other things, the
"small" gap worst-case lens adopted in this paper also reveals profound
distinctions between the behavior of UCB and Thompson Sampling, such as an
"incomplete learning" phenomenon characteristic of the latter.
