Motivated by the challenge of nonstationarity in sequential decision making, we study Online Convex Optimization (OCO) under the coupling of two problem structures: the domain is unbounded, and the comparator sequence $u_1,\ldots,u_T$ is arbitrarily time-varying. As no algorithm can guarantee low regret simultaneously against all comparator sequences, handling this setting requires moving from minimax optimality to comparator adaptivity. That is, sensible regret bounds should depend on certain complexity measures of the comparator relative to one's prior knowledge. This paper achieves a new type of these adaptive regret bounds via a sparse coding framework. The complexity of the comparator is measured by its energy and its sparsity on a user-specified dictionary, which offers considerable versatility. Equipped with a wavelet dictionary for example, our framework improves the state-of-the-art bound (Jacobsen & Cutkosky, 2022) by adapting to both ($i$) the magnitude of the comparator average $||\bar u||=||\sum_{t=1}^Tu_t/T||$, rather than the maximum $\max_t||u_t||$; and ($ii$) the comparator variability $\sum_{t=1}^T||u_t-\bar u||$, rather than the uncentered sum $\sum_{t=1}^T||u_t||$. Furthermore, our analysis is simpler due to decoupling function approximation from regret minimization.

翻译：受序列决策中非平稳性挑战的驱动，本研究探讨了两种问题结构耦合下的在线凸优化：定义域无界，且比较器序列$u_1,\ldots,u_T$可任意时变。由于不存在能够对所有比较器序列同时保证低遗憾的算法，处理该场景需要从极小极大最优性转向比较器自适应性——即合理的遗憾界应取决于比较器相对于先验知识的特定复杂度度量。本文通过稀疏编码框架实现了一种新型自适应遗憾界。比较器的复杂度通过其能量及在用户指定字典上的稀疏度来度量，这赋予了框架显著的多功能性。例如，配备小波字典后，本框架改进了现有最优界（Jacobsen & Cutkosky, 2022），可同时自适应：（i）比较器均值$||\bar u||=||\sum_{t=1}^Tu_t/T||$的幅值（而非最大值$\max_t||u_t||$）；（ii）比较器变异性$\sum_{t=1}^T||u_t-\bar u||$（而非非中心化求和$\sum_{t=1}^T||u_t||$）。此外，由于将函数逼近与遗憾最小化解耦，我们的分析更加简洁。