  As a metric to measure the performance of an online method, dynamic regret
with switching cost has drawn much attention for online decision making
problems. Although the sublinear regret has been provided in many previous
researches, we still have little knowledge about the relation between the
dynamic regret and the switching cost. In the paper, we investigate the
relation for two classic online settings: Online Algorithms (OA) and Online
Convex Optimization (OCO). We provide a new theoretical analysis framework,
which shows an interesting observation, that is, the relation between the
switching cost and the dynamic regret is different for settings of OA and OCO.
Specifically, the switching cost has significant impact on the dynamic regret
in the setting of OA. But, it does not have an impact on the dynamic regret in
the setting of OCO. Furthermore, we provide a lower bound of regret for the
setting of OCO, which is same with the lower bound in the case of no switching
cost. It shows that the switching cost does not change the difficulty of online
decision making problems in the setting of OCO.
