  This paper investigates the problem of generalized linear bandits with
heavy-tailed rewards, whose $(1+\epsilon)$-th moment is bounded for some
$\epsilon\in (0,1]$. Although there exist methods for generalized linear
bandits, most of them focus on bounded or sub-Gaussian rewards and are not
well-suited for many real-world scenarios, such as financial markets and
web-advertising. To address this issue, we propose two novel algorithms based
on truncation and mean of medians. These algorithms achieve an almost optimal
regret bound of $\widetilde{O}(dT^{\frac{1}{1+\epsilon}})$, where $d$ is the
dimension of contextual information and $T$ is the time horizon. Our
truncation-based algorithm supports online learning, distinguishing it from
existing truncation-based approaches. Additionally, our mean-of-medians-based
algorithm requires only $O(\log T)$ rewards and one estimator per epoch, making
it more practical. Moreover, our algorithms improve the regret bounds by a
logarithmic factor compared to existing algorithms when $\epsilon=1$. Numerical
experimental results confirm the merits of our algorithms.
