关于组合半老虎机问题中扰动跟随领导者策略的注记
Note on Follow-the-Perturbed-Leader in Combinatorial Semi-Bandit Problems
- Kyoto University（京都大学）
- RIKEN AIP（理化学研究所（Advanced Institute for Scientific Technology））
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文研究规模不变组合半老虎机问题中FTPL策略的最优性与复杂度，证明其在Fréchet和Pareto分布下的遗憾界，并提出条件几何重采样以降低计算复杂度。
AI中文摘要：
本文研究扰动跟随领导者（FTPL）策略在规模不变组合半老虎机问题中的最优性与复杂度。近期，Honda等人（2023）和Lee等人（2024）证明了在具有Fréchet型分布的标准多臂老虎机问题中，FTPL能够实现两全其美（BOBW）最优性。然而，FTPL在组合半老虎机问题中的最优性仍不清楚。本文考虑在规模不变半老虎机设定下带有几何重采样（GR）的FTPL的遗憾界，证明FTPL在Fréchet分布下分别取得$O\left(\sqrt{m^2 d^\frac{1}{\alpha}T}+\sqrt{mdT}\right)$的遗憾，并在对抗性设定下以Pareto分布取得$O\left(\sqrt{mdT}\right)$的最佳可能遗憾界。此外，我们将条件几何重采样（CGR）扩展到规模不变半老虎机设定，将计算复杂度从原始GR的$O(d^2)$降低到$O\left(md\left(\log(d/m)+1\right)\right)$，且不牺牲FTPL的遗憾性能。
英文摘要：
This paper studies the optimality and complexity of Follow-the-Perturbed-Leader (FTPL) policy in size-invariant combinatorial semi-bandit problems. Recently, Honda et al. (2023) and Lee et al. (2024) showed that FTPL achieves Best-of-Both-Worlds (BOBW) optimality in standard multi-armed bandit problems with Fréchet-type distributions. However, the optimality of FTPL in combinatorial semi-bandit problems remains unclear. In this paper, we consider the regret bound of FTPL with geometric resampling (GR) in size-invariant semi-bandit setting, showing that FTPL respectively achieves $O\left(\sqrt{m^2 d^\frac{1}αT}+\sqrt{mdT}\right)$ regret with Fréchet distributions, and the best possible regret bound of $O\left(\sqrt{mdT}\right)$ with Pareto distributions in adversarial setting. Furthermore, we extend the conditional geometric resampling (CGR) to size-invariant semi-bandit setting, which reduces the computational complexity from $O(d^2)$ of original GR to $O\left(md\left(\log(d/m)+1\right)\right)$ without sacrificing the regret performance of FTPL.
