  In this paper, we present an improved analysis for dynamic regret of strongly
convex and smooth functions. Specifically, we investigate the Online Multiple
Gradient Descent (OMGD) algorithm proposed by Zhang et al. (2017). The original
analysis shows that the dynamic regret of OMGD is at most
$\mathcal{O}(\min\{\mathcal{P}_T,\mathcal{S}_T\})$, where $\mathcal{P}_T$ and
$\mathcal{S}_T$ are path-length and squared path-length that measures the
cumulative movement of minimizers of the online functions. We demonstrate that
by an improved analysis, the dynamic regret of OMGD can be improved to
$\mathcal{O}(\min\{\mathcal{P}_T,\mathcal{S}_T,\mathcal{V}_T\})$, where
$\mathcal{V}_T$ is the function variation of the online functions. Note that
the quantities of $\mathcal{P}_T, \mathcal{S}_T, \mathcal{V}_T$ essentially
reflect different aspects of environmental non-stationarity -- they are not
comparable in general and are favored in different scenarios. Therefore, the
dynamic regret presented in this paper actually achieves a
\emph{best-of-three-worlds} guarantee and is strictly tighter than previous
results.
