发表机构
Nanjing University（南京大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究区间遗憾与动态遗憾的关系，证明最优区间遗憾不蕴含最优动态遗憾，并提出通过扩大可行域与域转换损失实现最优动态遗憾，首次为指数凹函数给出高效算法。
AI 中文摘要
近年来，非平稳在线学习受到了广泛关注，因为静态遗憾不足以指导变化环境中的算法设计。为解决这一局限性，区间遗憾和动态遗憾被引入，作为两种代表性的性能指标，从互补的方向强化静态遗憾。区间遗憾要求在线算法在每个局部时间区间内实现具有竞争力的静态遗憾，而动态遗憾则针对任意时变比较器序列评估性能。尽管这些指标很重要，但它们之间的关系长期以来一直不明确。先前的工作通常基于局部保证自然应能导出全局保证的直觉，将区间遗憾视为更强的概念。因此，人们普遍猜想，具有最优区间遗憾的算法应自动达到最优动态遗憾。在本文中，我们首先建立了一个否定性结果，反驳了这种指标级蕴含的直觉。具体来说，对于凸函数和曲函数（包括指数凹函数和强凸函数），我们证明了存在这样的实例：具有最优区间遗憾的算法仍无法达到最优动态遗憾。然后，我们展示了如何利用局部自适应性来获得最优动态遗憾。特别是，可以通过在包含原始凸可行域的扩大欧几里得球上调用区间遗憾最小化过程，并使用合适的域转换替代损失，来实现最优动态遗憾。这种归约适用于凸函数和曲函数。作为副产品，我们为指数凹函数获得了第一个具有最优动态遗憾的恰当且高效的算法，改进了先前的结果，同时显著简化了分析。
英文摘要
Non-stationary online learning has attracted much attention in recent years, as static regret is insufficient to guide algorithm design in changing environments. To address this limitation, interval regret and dynamic regret have been introduced as two representative performance metrics that strengthen static regret in complementary directions. Interval regret requires an online algorithm to achieve competitive static regret over every local time interval, whereas dynamic regret evaluates performance against an arbitrary sequence of time-varying comparators. Despite their importance, the relation between these metrics has long remained unclear. Prior work has often regarded interval regret as the stronger notion, based on the intuition that local guarantees should naturally induce global guarantees. Consequently, it is widely conjectured that an algorithm with optimal interval regret should automatically attain optimal dynamic regret. In this paper, we first establish a negative result that refutes this intuition of a metric-level implication. Specifically, for both convex and curved functions (including exp-concave and strongly convex functions), we show that there exist instances in which an algorithm with optimal interval regret nevertheless fails to achieve optimal dynamic regret. We then show how to leverage local adaptivity to obtain optimal dynamic regret. In particular, optimal dynamic regret can be attained by invoking an interval regret minimization process over an enlarged Euclidean ball containing the original convex feasible domain and using a suitable domain-converted surrogate loss. This reduction applies to both convex and curved functions. As a byproduct, we obtain the first proper and efficient algorithm with optimal dynamic regret for exp-concave functions, improving prior results while significantly simplifying the analysis.
