Motivated by time series forecasting, we study Online Linear Optimization (OLO) under the coupling of two problem structures: the domain is unbounded, and the performance of an algorithm is measured by its dynamic regret. Handling either of them requires the regret bound to depend on certain complexity measure of the comparator sequence -- specifically, the comparator norm in unconstrained OLO, and the path length in dynamic regret. In contrast to a recent work (Jacobsen & Cutkosky, 2022) that adapts to the combination of these two complexity measures, we propose an alternative complexity measure by recasting the problem into sparse coding. Adaptivity can be achieved by a simple modular framework, which naturally exploits more intricate prior knowledge of the environment. Along the way, we also present a new gradient adaptive algorithm for static unconstrained OLO, designed using novel continuous time machinery. This could be of independent interest.

翻译：受时间序列预测的启发，我们研究了在线线性优化（OLO）问题中两种问题结构的耦合：定义域无界，且算法性能通过动态遗憾衡量。处理这两种结构均需使遗憾界依赖于比较器序列的特定复杂度度量——具体而言，无约束OLO中的比较器范数，以及动态遗憾中的路径长度。与近期工作（Jacobsen & Cutkosky, 2022）适应这两种复杂度度量的组合不同，我们通过将问题重新表述为稀疏编码，提出了一种替代的复杂度度量。适应性可通过一个简单的模块化框架实现，该框架自然利用了环境中更精细的先验知识。在此过程中，我们还提出了一种适用于静态无约束OLO的新型梯度自适应算法，该算法采用新颖的连续时间机制设计，可能具有独立的研究意义。