  Heavy-tailed distributions naturally arise in several settings, from finance
to telecommunications. While regret minimization under subgaussian or bounded
rewards has been widely studied, learning with heavy-tailed distributions only
gained popularity over the last decade. In this paper, we consider the setting
in which the reward distributions have finite absolute raw moments of maximum
order $1+\epsilon$, uniformly bounded by a constant $u<+\infty$, for some
$\epsilon \in (0,1]$. In this setting, we study the regret minimization problem
when $\epsilon$ and $u$ are unknown to the learner and it has to adapt. First,
we show that adaptation comes at a cost and derive two negative results proving
that the same regret guarantees of the non-adaptive case cannot be achieved
with no further assumptions. Then, we devise and analyze a fully data-driven
trimmed mean estimator and propose a novel adaptive regret minimization
algorithm, AdaR-UCB, that leverages such an estimator. Finally, we show that
AdaR-UCB is the first algorithm that, under a known distributional assumption,
enjoys regret guarantees nearly matching those of the non-adaptive heavy-tailed
case.
