  In this paper, we investigate an online prediction strategy named as
Discounted-Normal-Predictor (Kapralov and Panigrahy, 2010) for smoothed online
convex optimization (SOCO), in which the learner needs to minimize not only the
hitting cost but also the switching cost. In the setting of learning with
expert advice, Daniely and Mansour (2019) demonstrate that
Discounted-Normal-Predictor can be utilized to yield nearly optimal regret
bounds over any interval, even in the presence of switching costs. Inspired by
their results, we develop a simple algorithm for SOCO: Combining online
gradient descent (OGD) with different step sizes sequentially by
Discounted-Normal-Predictor. Despite its simplicity, we prove that it is able
to minimize the adaptive regret with switching cost, i.e., attaining nearly
optimal regret with switching cost on every interval. By exploiting the
theoretical guarantee of OGD for dynamic regret, we further show that the
proposed algorithm can minimize the dynamic regret with switching cost in every
interval.
