Cover meets Robbins while Betting on Bounded Data: $\ln n$ Regret and Almost Sure $\ln\ln n$ Regret

2026-04-22

AI Summary

This study addresses the online betting problem for sequences of outcomes in the interval [0,1], aiming to achieve optimal regret guarantees simultaneously in both adversarial and stochastic environments. To this end, the authors propose an adaptive hybrid strategy that integrates Cover’s universal portfolio framework with Robbins’ stochastic approximation ideas, dynamically balancing between the two approaches via a novel first-passage mechanism. The method guarantees a worst-case logarithmic regret of O(log n) against any adversarial sequence, while almost surely attaining a doubly logarithmic regret bound of O(ln ln n) on almost all stochastic paths whose conditional means are constant and conditional variances diverge. This result achieves a theoretically optimal trade-off and reveals a profound connection to the law of the iterated logarithm in game-theoretic probability.

Technology Category

• Machine Learning: Online Learning & Bandits
• Reasoning under Uncertainty: Stochastic Optimization
• Game Theory and Economic Paradigms: Adversarial Learning

Application Category

• Economics, Online Markets and Human Computation: Uses of LLMs and GenAI for marketplace design, bidding, and strategic interactions
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs

Abstract

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}.

Problem

• regret minimization
• universal betting
• adversarial data
• stochastic data
• almost sure convergence

Innovation

• universal betting
• log-log regret
• almost sure convergence
• best-of-both-worlds
• iterated logarithm law