We consider the problem of the Zinkevich (2003)-style dynamic regret minimization in online learning with exp-concave losses. We show that whenever improper learning is allowed, a Strongly Adaptive online learner achieves the dynamic regret of $\tilde O^*(n^{1/3}C_n^{2/3} \vee 1)$ where $C_n$ is the total variation (a.k.a. path length) of the an arbitrary sequence of comparators that may not be known to the learner ahead of time. Achieving this rate was highly nontrivial even for squared losses in 1D where the best known upper bound was $O(\sqrt{nC_n} \vee \log n)$ (Yuan and Lamperski, 2019). Our new proof techniques make elegant use of the intricate structures of the primal and dual variables imposed by the KKT conditions and could be of independent interest. Finally, we apply our results to the classical statistical problem of locally adaptive non-parametric regression (Mammen, 1991; Donoho and Johnstone, 1998) and obtain a stronger and more flexible algorithm that do not require any statistical assumptions or any hyperparameter tuning.

翻译：我们考虑了Zinkevich (2003年) 的动态在网上学习中以解剖损失的方式最大限度地减低遗憾的问题。我们表明,只要允许不适当的学习,强烈适应性在线学习者就会获得1美元(元1/3}C_n ⁇ 2/3}\vee 1美元)的动态遗憾,其中C_n美元是参照方任意顺序的总变数(a.k.a.路径长度),而学习者可能不会事先知道这一点。实现这一比率对于1D的平方损失来说是高度非三重性的,即使在1D中最知名的上限为$O(sqrt{nC_n}\vee\log n) (Yuan和Lamperski, 2019年)。我们的新证明技术优雅地利用了KKT条件所强加的原始和双重变数的复杂结构(a.k.a.a.路径长度),可能具有独立的兴趣。最后,我们将我们的结果应用于典型的当地适应性非参数回归的统计问题(Mammen,1991年;Donho和Johnststone,1998年) 并获得任何更强大和弹性的统计分析。