  In linear stochastic bandits, it is commonly assumed that payoffs are with
sub-Gaussian noises. In this paper, under a weaker assumption on noises, we
study the problem of \underline{lin}ear stochastic {\underline b}andits with
h{\underline e}avy-{\underline t}ailed payoffs (LinBET), where the
distributions have finite moments of order $1+\epsilon$, for some $\epsilon\in
(0,1]$. We rigorously analyze the regret lower bound of LinBET as
$\Omega(T^{\frac{1}{1+\epsilon}})$, implying that finite moments of order 2
(i.e., finite variances) yield the bound of $\Omega(\sqrt{T})$, with $T$ being
the total number of rounds to play bandits. The provided lower bound also
indicates that the state-of-the-art algorithms for LinBET are far from optimal.
By adopting median of means with a well-designed allocation of decisions and
truncation based on historical information, we develop two novel bandit
algorithms, where the regret upper bounds match the lower bound up to
polylogarithmic factors. To the best of our knowledge, we are the first to
solve LinBET optimally in the sense of the polynomial order on $T$. Our
proposed algorithms are evaluated based on synthetic datasets, and outperform
the state-of-the-art results.
