通过稀疏编码实现无约束动态遗憾
Unconstrained Dynamic Regret via Sparse Coding
- Harvard University（哈佛大学）
- Boston University（波士顿大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文针对无界域上任意时变比较序列的在线凸优化问题，提出基于稀疏编码框架的自适应遗憾界，通过比较器在用户指定字典上的能量和稀疏性度量复杂度，并利用小波字典改进了现有最优界。
AI中文摘要：
受序贯决策中非平稳性挑战的驱动，本文研究在两种问题结构耦合下的在线凸优化（OCO）：定义域无界，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，处理该设置需要从极小化极大最优性转向比较器自适应性。也就是说，合理的遗憾界应依赖于比较器相对于先验知识的某些复杂度度量。本文通过稀疏编码框架实现了一类新型的自适应遗憾界。比较器的复杂度通过其在用户指定字典上的能量和稀疏性来度量，这提供了相当大的通用性。例如，配备小波字典后，我们的框架改进了当前最优界（Jacobsen & Cutkosky, 2022），具体体现在适应于：（$i$）比较器均值的幅度 $||\bar u||=||\sum_{t=1}^Tu_t/T||$，而非最大值 $\max_t||u_t||$；以及（$ii$）比较器的变异性 $\sum_{t=1}^T||u_t-\bar u||$，而非未中心化的和 $\sum_{t=1}^T||u_t||$。此外，由于将函数逼近与遗憾最小化解耦，我们的分析更为简洁。
英文摘要：
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.
