发表机构
University of Washington; University of California, San Diego; State Key Laboratory for Novel Software Technology, Nanjing University; Nanjing University（华盛顿大学; 加利福尼亚大学圣迭戈分校; 南京大学计算机软件新技术全国重点实验室; 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
本文提出通用动态投资组合，将通用投资组合推广至动态比较器序列，引入结构感知度量以克服路径长度粗糙性，实现更优动态遗憾率。
AI 中文摘要
Cover 的通用投资组合（Cover, 1991）在事后能够匹配最佳恒定再平衡投资组合的表现。我们将这一框架推广到与任意比较器序列 $\mathbf{u}_1,\ldots,\mathbf{u}_T$ 竞争，从而引出了对数损失下的动态遗憾最小化问题，而现有方法由于梯度可能无界而失效。对数损失是 exp-凹的，这种曲率性质在经典上能为静态遗憾带来快速率，然而我们表明这一优势在动态设置中通常会消失。特别地，线性损失型的 $\sqrt{TP_T}$ 依赖是不可避免的，其中 $P_T=\sum_{t=2}^T\lVert\mathbf{u}_t-\mathbf{u}_{t-1}\rVert_1$ 是标准路径长度。这一限制源于 $P_T$ 的粗糙性，它掩盖了比较器序列更精细的空间和时间结构。因此，我们引入了两种结构感知度量——用于空间结构的 Jensen-Shannon 距离和用于时间结构的 JS$^q$ 路径长度——在比较器序列具有有利结构时，可以实现更快的速率。为了同时针对两种度量获得尖锐的界，我们开发了通用动态投资组合，这是一种无参数方法，结合了新的 Dirichlet Hedge 算法和固定份额更新，同时在最坏情况下保留了接近最优的 $P_T$ 保证。最后，在额外的有界梯度假设下，我们表明 OPS 在所有比较器序列上实现了更快的 $T^{1/3}P_T^{2/3}$ 动态遗憾率。我们通过一个可处理的适当算法达到了这一速率，该算法更广泛地适用于任意紧致凸域上的通用在线 exp-凹优化。
英文摘要
Cover's Universal Portfolio (Cover, 1991) matches the performance of the best constant rebalanced portfolio in hindsight. We generalize this framework to compete with an arbitrary comparator sequence $\mathbf{u}_1,\ldots,\mathbf{u}_T$, leading to a dynamic regret minimization problem for the log loss where existing methods break down due to potentially unbounded gradients. The log loss is exp-concave, a curvature property that classically yields fast rates for static regret, yet we show that this advantage generally disappears in the dynamic setting. In particular, a linear-loss-type $\sqrt{TP_T}$ dependence is unavoidable, where $P_T=\sum_{t=2}^T\lVert\mathbf{u}_t-\mathbf{u}_{t-1}\rVert_1$ is the standard path length. This limitation stems from the coarse nature of $P_T$, which obscures finer spatial and temporal structure of the comparator sequence. We therefore introduce two structure-aware measures---the Jensen-Shannon distance for spatial structure and the JS$^q$-path length for temporal structure---under which faster rates are attainable when the comparator sequence has favorable structure. To achieve sharp bounds for both measures simultaneously, we develop Universal Dynamic Portfolio, a parameter-free method that combines a new Dirichlet Hedge algorithm with a fixed-share update, while retaining a near-optimal $P_T$ guarantee in the worst case. Finally, under an additional bounded-gradient assumption, we show that OPS admits the faster $T^{1/3}P_T^{2/3}$ dynamic regret rate over all comparator sequences. We attain this rate with a tractable proper algorithm that applies more broadly to general online exp-concave optimization over arbitrary compact convex domains.
