We develop algorithms for online linear regression which achieve optimal static and dynamic regret guarantees \emph{even in the complete absence of prior knowledge}. We present a novel analysis showing that a discounted variant of the Vovk-Azoury-Warmuth forecaster achieves dynamic regret of the form $R_{T}(\vec{u})\le O\left(d\log(T)\vee \sqrt{dP_{T}^{\gamma}(\vec{u})T}\right)$, where $P_{T}^{\gamma}(\vec{u})$ is a measure of variability of the comparator sequence, and show that the discount factor achieving this result can be learned on-the-fly. We show that this result is optimal by providing a matching lower bound. We also extend our results to \emph{strongly-adaptive} guarantees which hold over every sub-interval $[a,b]\subseteq[1,T]$ simultaneously.

翻译：本文提出了一种在线线性回归算法，该算法在完全缺乏先验知识的情况下仍能达到最优的静态与动态遗憾界。我们提出了一种新颖的分析方法，证明Vovk-Azoury-Warmuth预测器的折扣变体能够实现形式为$R_{T}(\vec{u})\le O\left(d\log(T)\vee \sqrt{dP_{T}^{\gamma}(\vec{u})T}\right)$的动态遗憾，其中$P_{T}^{\gamma}(\vec{u})$是比较器序列变异性的度量，并且实现该结果的折扣因子可以在线学习获得。我们通过提供一个匹配的下界证明该结果是最优的。此外，我们将结果扩展到具有强适应性的保证，该保证在每个子区间$[a,b]\subseteq[1,T]$上同时成立。