  This work is motivated by the growing demand for reproducible machine
learning. We study the stochastic multi-armed bandit problem. In particular, we
consider a replicable algorithm that ensures, with high probability, that the
algorithm's sequence of actions is not affected by the randomness inherent in
the dataset. We observe that existing algorithms require $O(1/\rho^2)$ times
more regret than nonreplicable algorithms, where $\rho$ is the level of
nonreplication. However, we demonstrate that this additional cost is
unnecessary when the time horizon $T$ is sufficiently large for a given $\rho$,
provided that the magnitude of the confidence bounds is chosen carefully. We
introduce an explore-then-commit algorithm that draws arms uniformly before
committing to a single arm. Additionally, we examine a successive elimination
algorithm that eliminates suboptimal arms at the end of each phase. To ensure
the replicability of these algorithms, we incorporate randomness into their
decision-making processes. We extend the use of successive elimination to the
linear bandit problem as well. For the analysis of these algorithms, we propose
a principled approach to limiting the probability of nonreplication. This
approach elucidates the steps that existing research has implicitly followed.
Furthermore, we derive the first lower bound for the two-armed replicable
bandit problem, which implies the optimality of the proposed algorithms up to a
$\log\log T$ factor for the two-armed case.
