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，我们研究了在两种问题结构耦合下的在线凸优化（OCO）：域无界，且比较序列$u_1,\ldots,u_T$任意时变。由于没有任何算法能同时对所有比较序列保证低遗憾，处理这一设置需要从极小化最优性转向比较器自适应性。换言之，合理的遗憾界应依赖于比较器相对于先验知识的特定复杂性度量。本文通过稀疏编码框架实现了一种新型的自适应遗憾界。比较器的复杂性通过其在用户指定字典上的能量和稀疏性来度量，这提供了显著的灵活性。例如，配备小波字典时，我们的框架通过同时适应（$i$）比较器平均值的幅度$||\bar u||=||\sum_{t=1}^Tu_t/T||$（而非最大值$\max_t||u_t||$）以及（$ii$）比较器变异性$\sum_{t=1}^T||u_t-\bar u||$（而非非中心化总和$\sum_{t=1}^T||u_t||$），改进了当前最优界（Jacobsen & Cutkosky, 2022）。此外，由于将函数逼近与遗憾最小化解耦，我们的分析更为简洁。