Cover meets Robbins while Betting on Bounded Data: ln n Regret and Almost Sure ln ln n Regret
TL;DR
A simple hedging principle shows how splitting initial wealth between two strategies achieves a best-of-both-worlds guarantee by retaining the smaller of their regrets and the larger of their growth rates.
Abstract
Consider betting against a sequence of $[0,1]$-valued observations using bets that are fair if the data have a conditional mean $m_0 \in (0,1)$. Cover's universal portfolio algorithm achieves a worst-case regret of $O(\ln n)$ compared to the best constant bet in hindsight, and this logarithmic order is unimprovable against adversarial data. We construct an explicit full-support Robbins mixture that combines insights from Robbins and Cover. It retains $O(\ln n)$ regret on every path while also eventually satisfying $R_n\le(1+o(1))\ln\ln V_n$ on every path with bounded mixture wealth and $V_n\to\infty$, where $V_n$ is an intrinsic variance process. It also attains the optimal asymptotic wealth growth rate under iid alternatives. Under the conditional-mean claim and almost-sure divergence of $V_n$, the sharper regret bound holds almost surely, and its leading constant $1$ is unimprovable. We also give explicit finite-time, path-wise regret bounds. A simple hedging principle shows how splitting initial wealth between two strategies achieves a best-of-both-worlds guarantee by retaining the smaller of their regrets and the larger of their growth rates. We contrast our results to those for unbounded data in Agrawal and Ramdas (2026). Their worst-case regret is unbounded. However, hedging yields almost-sure iterated-logarithmic regret together with optimal growth. Finally, our wealth process witnesses a sharp game-theoretic upper law of iterated logarithm, analogous to Shafer and Vovk (2005).
