Consider betting against a sequence of data in $[0,1]$, where one is allowed to make any bet that is fair if the data have a conditional mean $m_0 \in (0,1)$. Cover's universal portfolio algorithm delivers a worst-case regret of $O(\ln n)$ compared to the best constant bet in hindsight, and this bound is unimprovable against adversarially generated data. In this work, we present a novel mixture betting strategy that combines insights from Robbins and Cover, and exhibits a different behavior: it eventually produces a regret of $O(\ln \ln n)$ on \emph{almost} all paths (a measure-one set of paths if each conditional mean equals $m_0$ and intrinsic variance increases to $\infty$), but has an $O(\log n)$ regret on the complement (a measure zero set of paths). Our paper appears to be the first to point out the value in hedging two very different strategies to achieve a best-of-both-worlds adaptivity to stochastic data and protection against adversarial data. We contrast our results to those in~\cite{agrawal2025regret} for a sub-Gaussian mixture on unbounded data: their worst-case regret has to be unbounded, but a similar hedging delivers both an optimal betting growth-rate and an almost sure $\ln\ln n$ regret on stochastic data. Finally, our strategy witnesses a sharp game-theoretic upper law of the iterated logarithm, analogous to~\cite{shafer2005probability}.

翻译：考虑对$[0,1]$中的数据序列进行下注，允许在数据条件均值$m_0 \in (0,1)$下进行任何公平下注。Cover通用投资组合算法相较于事后最优常数下注，能实现最坏情况下的$O(\ln n)$遗憾，且这一界对对抗性生成数据而言不可改进。本文提出一种融合Robbins与Cover思想的新型混合下注策略，展现出不同行为：在\emph{几乎}所有路径上（即每个条件均值等于$m_0$且内在方差趋于无穷时的测度为一的路径集），最终产生$O(\ln \ln n)$遗憾，而在其补集（测度为零的路径集）上则保持$O(\log n)$遗憾。本文首次指出，对冲两种截然不同的策略能实现“两全其美”的适应性——对随机数据的最优表现与对对抗性数据的保护。我们将结果与~\cite{agrawal2025regret}中对无界数据的次高斯混合策略进行对比：后者最坏情况遗憾必然无界，但相似的对冲策略既能实现最优下注增长率，又能在随机数据上达到几乎必然的$\ln\ln n$遗憾。最后，我们的策略验证了尖锐的博弈论迭代对数上律，与~\cite{shafer2005probability}中的结论类似。