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

2026-09-22

AI Summary

本文解决了在线学习中专家建议预测问题，提出了一种无需预先知道时间范围的算法，其累积遗憾与已知时间范围的最佳结果相匹配。

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 $t \ge 1$.

Problem

• Prediction with Expert Advice
• Anytime Regret
• Online Learning

Innovation

• Prediction with Expert Advice
• Anytime Regret
• Minimax Cumulative Regret
• Multiplicative Weights Update