Improved Analysis for Dynamic Regret of Strongly Convex and Smooth Functions

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.