Universal Dynamic Portfolios

2026-09-28

AI Summary

This study addresses the challenge of minimizing logarithmic loss in dynamic portfolio selection, where unbounded gradients render regret control difficult and traditional static curvature advantages become ineffective. To overcome these limitations, this work proposes a structure-aware metric alongside a parameter-free algorithm. Specifically, it introduces Jensen-Shannon divergence and path-length metrics to characterize environmental dynamics. By integrating Dirichlet Hedge, fixed-share updates, and online convex optimization techniques, the authors construct an adaptive learning framework that operates without requiring predefined parameters. These contributions transcend existing theoretical limitations by establishing tighter dynamic regret bounds and achieving accelerated convergence rates under structured comparator sequences, thereby significantly advancing both the theoretical foundations and practical performance of dynamic portfolio selection.

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 $P_T=\sum_{t=2}^T\lVert\mathbf{u}_t-\mathbf{u}_{t-1}\rVert_1$ 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.

Problem

• dynamic regret minimization
• universal portfolio
• log loss
• exp-concave optimization
• path length

Innovation

• Dynamic Regret Minimization
• Universal Portfolio
• Exp-Concave Optimization
• Dirichlet Hedge
• Structure-Aware Measures