  The multi-armed bandit (MAB) problem is a widely studied model in the field
of reinforcement learning. This paper considers two cases of the classical MAB
model -- the light-tailed reward distributions and the heavy-tailed,
respectively. For the light-tailed (i.e. sub-Gaussian) case, we propose the
UCB1-LT policy, achieving the optimal $O(\log T)$ of the order of regret
growth. For the heavy-tailed case, we introduce the extended robust UCB policy,
which is an extension of the UCB policies proposed by Bubeck et al. (2013) and
Lattimore (2017). The previous UCB policies require the knowledge of an upper
bound on specific moments of reward distributions, which can be hard to acquire
in some practical situations. Our extended robust UCB eliminates this
requirement while still achieving the optimal regret growth order $O(\log T)$,
thus providing a broadened application area of the UCB policies for the
heavy-tailed reward distributions.
