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}.

Bandit and reinforcement learning (RL) algorithms demonstrated a wide range of successful real-life applications [Silver et al., 2016, 2017, Mnih et al., 2013, Berner et al., 2019, Vinyals et al., 2019, Mnih et al., 2015, Degrave et al., 2022]. Past works have theoretically studied the regret or sample complexity of various interactive decision making problems, such as contextual bandits, reinforcement learning (RL), partially observable Markov decision process (see Azar et al. [2017], Jin et al. [2018], Dong et al. [2021], Li et al. [2019], Agarwal et al. [2014], Foster and Rakhlin [2020], Jin et al. [2020], and references therein). Recently, Foster et al. [2021] present a unified algorithmic principle for achieving the minimax regret--the optimal regret for the worst-case problem instances. However, minimax regret bounds do not necessarily always present a full picture of the statistical complexity of the problem. They characterize the complexity of the most difficult instances, but potentially many other instances are much easier. An ideal algorithm should adapt to the complexity of a particular instance and incur smaller regrets on easy instances than the worst-case instances. Thus, an ideal regret bound should be instance-dependent, that is, depending on some properties of each instance. Prior works designed algorithms with instance-dependent regret bounds that are stronger than minimax regret bounds, but they are often not directly comparable because they depend on different properties of the instances, such as the gap conditions and the variance of the value function [Zanette and Brunskill, 2019, Xu et al., 2021, Foster et al., 2020, Tirinzoni et al., 2021]. A more ambitious and challenging goal is to design instance-optimal algorithms that outperform, on every instance, all consistent algorithms (those achieving non-trivial regrets on all instances).

The clusters of a distribution are often defined by the connected components of a density level set. However, this definition depends on the user-specified level. We address this issue by proposing a simple, generic algorithm, which uses an almost arbitrary level set estimator to estimate the smallest level at which there are more than one connected components. In the case where this algorithm is fed with histogram-based level set estimates, we provide a finite sample analysis, which is then used to show that the algorithm consistently estimates both the smallest level and the corresponding connected components. We further establish rates of convergence for the two estimation problems, and last but not least, we present a simple, yet adaptive strategy for determining the width-parameter of the involved density estimator in a data-depending way.