Universal Dynamic Portfolios

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.