Universal Dynamic Portfolios

arXiv:2609.34643v1 Announce Type: new Abstract: 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}^TlVertmathbf{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.

Source: arXiv cs.LG | 2026-09-29