Prediction with Expert Advice: Anytime Regret with Many Experts Matches the Fixed-Time Constant

Abstract:Prediction with expert advice is a fundamental problem in online learning. When the time horizon $T$ is known in advance, the minimax cumulative regret over $n$ experts is asymptotically $\sqrt{\frac{T \ln n}{2}}$. This is achieved by the Multiplicative Weights Update algorithm with a learning rate tuned to $T$, and is known to be tight. If instead the regret bound is required to hold simultaneously at every time $t$, the best known guarantee has been $\sqrt{t \ln n}$---a factor of $\sqrt{2}$ worse---and it has remained unknown whether this factor of $\sqrt{2}$ is necessary. We show that it is not. We give an algorithm, requiring no knowledge of the horizon, whose cumulative regret satisfies $R_t \le \bigl(1 + O(\sqrt{\ln \ln n / \ln n})\bigr)\sqrt{t \ln n / 2}$ simultaneously for every $t \ge 1$.

Machine Learning (stat.ML); Machine Learning (cs.LG)

arXiv:2609.27206 [stat.ML]

(or arXiv:2609.27206v1 [stat.ML] for this version)

https://doi.org/10.48550/arXiv.2609.27206 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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