  In this paper, we study the problem of stochastic linear bandits with finite
action sets. Most of existing work assume the payoffs are bounded or
sub-Gaussian, which may be violated in some scenarios such as financial
markets. To settle this issue, we analyze the linear bandits with heavy-tailed
payoffs, where the payoffs admit finite $1+\epsilon$ moments for some
$\epsilon\in(0,1]$. Through median of means and dynamic truncation, we propose
two novel algorithms which enjoy a sublinear regret bound of
$\widetilde{O}(d^{\frac{1}{2}}T^{\frac{1}{1+\epsilon}})$, where $d$ is the
dimension of contextual information and $T$ is the time horizon. Meanwhile, we
provide an $\Omega(d^{\frac{\epsilon}{1+\epsilon}}T^{\frac{1}{1+\epsilon}})$
lower bound, which implies our upper bound matches the lower bound up to
polylogarithmic factors in the order of $d$ and $T$ when $\epsilon=1$. Finally,
we conduct numerical experiments to demonstrate the effectiveness of our
algorithms and the empirical results strongly support our theoretical
guarantees.
