We consider the problem of universal dynamic regret minimization under exp-concave and smooth losses. We show that appropriately designed Strongly Adaptive algorithms achieve a dynamic regret of $\tilde O(d^2 n^{1/5} C_n^{2/5} \vee d^2)$, where $n$ is the time horizon and $C_n$ a path variational based on second order differences of the comparator sequence. Such a path variational naturally encodes comparator sequences that are piecewise linear -- a powerful family that tracks a variety of non-stationarity patterns in practice (Kim et al, 2009). The aforementioned dynamic regret rate is shown to be optimal modulo dimension dependencies and poly-logarithmic factors of $n$. Our proof techniques rely on analysing the KKT conditions of the offline oracle and requires several non-trivial generalizations of the ideas in Baby and Wang, 2021, where the latter work only leads to a slower dynamic regret rate of $\tilde O(d^{2.5}n^{1/3}C_n^{2/3} \vee d^{2.5})$ for the current problem.

翻译：我们研究在指数凹和光滑损失函数下的通用动态遗憾最小化问题。我们证明了适当设计的强自适应算法能够达到$\tilde O(d^2 n^{1/5} C_n^{2/5} \vee d^2)$的动态遗憾界，其中$n$为时间范围，$C_n$为基于比较序列二阶差分的路径变分量。这种路径变分量自然地编码了分段线性比较序列——该序列族在实际应用中能够跟踪多种非平稳模式（Kim等人，2009）。该动态遗憾率在模维度依赖性和n的多对数因子意义下被证明是最优的。我们的证明技术依赖于对离线最优解的KKT条件的分析，并需要对Baby and Wang，2021中的思想进行若干非平凡推广——该工作针对当前问题仅能给出$\tilde O(d^{2.5}n^{1/3}C_n^{2/3} \vee d^{2.5})$的较慢动态遗憾率。