  This paper investigates the problem of non-stationary linear bandits, where
the unknown regression parameter is evolving over time. Existing studies
develop various algorithms and show that they enjoy an
$\widetilde{\mathcal{O}}(T^{2/3}P_T^{1/3})$ dynamic regret, where $T$ is the
time horizon and $P_T$ is the path-length that measures the fluctuation of the
evolving unknown parameter. In this paper, we discover that a serious technical
flaw makes their results ungrounded, and then present a fix, which gives an
$\widetilde{\mathcal{O}}(T^{3/4}P_T^{1/4})$ dynamic regret without modifying
original algorithms. Furthermore, we demonstrate that instead of using
sophisticated mechanisms, such as sliding window or weighted penalty, a simple
restarted strategy is sufficient to attain the same regret guarantee.
Specifically, we design an UCB-type algorithm to balance exploitation and
exploration, and restart it periodically to handle the drift of unknown
parameters. Our approach enjoys an $\widetilde{\mathcal{O}}(T^{3/4}P_T^{1/4})$
dynamic regret. Note that to achieve this bound, the algorithm requires an
oracle knowledge of the path-length $P_T$. Combining the bandits-over-bandits
mechanism by treating our algorithm as the base learner, we can further achieve
the same regret bound in a parameter-free way. Empirical studies also validate
the effectiveness of our approach.
