We study the problem of dynamic regret minimization in online convex optimization, in which the objective is to minimize the difference between the cumulative loss of an algorithm and that of an arbitrary sequence of comparators. While the literature on this topic is very rich, a unifying framework for the analysis and design of these algorithms is still missing. In this paper, \emph{we show that dynamic regret minimization is equivalent to static regret minimization in an extended decision space}. Using this simple observation, we show that there is a frontier of lower bounds trading off penalties due to the variance of the losses and penalties due to variability of the comparator sequence, and provide a framework for achieving any of the guarantees along this frontier. As a result, we prove for the first time that adapting to the squared path-length of an arbitrary sequence of comparators to achieve regret $R_{T}(u_{1},\dots,u_{T})\le O(\sqrt{T\sum_{t} \|u_{t}-u_{t+1}\|^{2}})$ is impossible. However, we prove that it is possible to adapt to a new notion of variability based on the locally-smoothed squared path-length of the comparator sequence, and provide an algorithm guaranteeing dynamic regret of the form $R_{T}(u_{1},\dots,u_{T})\le \tilde O(\sqrt{T\sum_{i}\|\bar u_{i}-\bar u_{i+1}\|^{2}})$. Up to polylogarithmic terms, the new notion of variability is never worse than the classic one involving the path-length.

翻译：我们研究在线凸优化中的动态遗憾最小化问题，其目标是最小化算法累积损失与任意比较器序列累积损失之间的差异。尽管该主题的文献非常丰富，但用于分析和设计此类算法的统一框架仍然缺失。本文证明，动态遗憾最小化等价于扩展决策空间中的静态遗憾最小化。基于这一简单观察，我们揭示了在损失函数方差与比较器序列变异度之间权衡的下界前沿，并提供了实现该前沿上任一保证的理论框架。由此，我们首次证明：通过适应任意比较器序列的平方路径长度以实现遗憾$R_{T}(u_{1},\dots,u_{T})\le O(\sqrt{T\sum_{t} \|u_{t}-u_{t+1}\|^{2}})$是不可能的。然而，我们证明可以适应基于比较器序列局部平滑平方路径长度的新型变异度概念，并提出了保证形如$R_{T}(u_{1},\dots,u_{T})\le \tilde O(\sqrt{T\sum_{i}\|\bar u_{i}-\bar u_{i+1}\|^{2}})$的动态遗憾的算法。在多项式对数项范围内，该新型变异度度量始终不劣于经典的路径长度度量。