Cover meets Robbins while Betting on Bounded Data: $\ln n$ Regret and Almost Sure $\ln\ln n$ Regret
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 wea...
| Authors | Shubhada Agrawal, Aaditya Ramdas |
|---|---|
| Primary category | cs.LG |
| Published | 22 April 2026 |
| Last updated at source | 13 October 2026 |
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