  In stochastic low-rank matrix bandit, the expected reward of an arm is equal
to the inner product between its feature matrix and some unknown $d_1$ by $d_2$
low-rank parameter matrix $\Theta^*$ with rank $r \ll d_1\wedge d_2$. While all
prior studies assume the payoffs are mixed with sub-Gaussian noises, in this
work we loosen this strict assumption and consider the new problem of
\underline{low}-rank matrix bandit with \underline{h}eavy-\underline{t}ailed
\underline{r}ewards (LowHTR), where the rewards only have finite $(1+\delta)$
moment for some $\delta \in (0,1]$. By utilizing the truncation on observed
payoffs and the dynamic exploration, we propose a novel algorithm called LOTUS
attaining the regret bound of order $\tilde
O(d^\frac{3}{2}r^\frac{1}{2}T^\frac{1}{1+\delta}/\tilde{D}_{rr})$ without
knowing $T$, which matches the state-of-the-art regret bound under sub-Gaussian
noises~\citep{lu2021low,kang2022efficient} with $\delta = 1$. Moreover, we
establish a lower bound of the order $\Omega(d^\frac{\delta}{1+\delta}
r^\frac{\delta}{1+\delta} T^\frac{1}{1+\delta}) = \Omega(T^\frac{1}{1+\delta})$
for LowHTR, which indicates our LOTUS is nearly optimal in the order of $T$. In
addition, we improve LOTUS so that it does not require knowledge of the rank
$r$ with $\tilde O(dr^\frac{3}{2}T^\frac{1+\delta}{1+2\delta})$ regret bound,
and it is efficient under the high-dimensional scenario. We also conduct
simulations to demonstrate the practical superiority of our algorithm.
