This paper studies the optimality and complexity of Follow-the-Perturbed-Leader (FTPL) policy in $m$-set semi-bandit problems. FTPL has been studied extensively as a promising candidate of an efficient algorithm with favorable regret for adversarial combinatorial semi-bandits. Nevertheless, the optimality of FTPL has still been unknown unlike Follow-the-Regularized-Leader (FTRL) whose optimality has been proved for various tasks of online learning. In this paper, we extend the analysis of FTPL with geometric resampling (GR) to $m$-set semi-bandits, which is a special case of combinatorial semi-bandits, showing that FTPL with Fréchet and Pareto distributions with certain parameters achieves the best possible regret of $O(\sqrt{mdT})$ in adversarial setting. We also show that FTPL with Fréchet and Pareto distributions with a certain parameter achieves a logarithmic regret for stochastic setting, meaning the Best-of-Both-Worlds optimality of FTPL for $m$-set semi-bandit problems. Furthermore, we extend the conditional geometric resampling to $m$-set semi-bandits for efficient loss estimation in FTPL, reducing the computational complexity from $O(d^2)$ of the original geometric resampling to $O(md(\log(d/m)+1))$ without sacrificing the regret performance.

翻译：本文研究了$m$集半赌博问题中跟随扰动领导者（FTPL）策略的最优性与计算复杂度。FTPL作为一种在对抗性组合半赌博问题中具有优越遗憾界的高效算法，已得到广泛研究。然而，与已在多种在线学习任务中被证明最优性的跟随正则化领导者（FTRL）不同，FTPL的最优性始终未被证实。本文通过将几何重采样（GR）的FTPL分析扩展至$m$集半赌博问题（组合半赌博问题的特例），证明采用特定参数的Fréchet分布与Pareto分布的FTPL在对抗性环境下可实现$O(\sqrt{mdT})$的最优遗憾界。同时，我们证明采用特定参数的Fréchet分布与Pareto分布的FTPL在随机环境下可实现对数遗憾界，这意味着FTPL在$m$集半赌博问题上达到了最佳双世界最优性。此外，我们将条件几何重采样技术扩展至$m$集半赌博问题，以提升FTPL中损失估计的效率，在保持遗憾性能不变的前提下，将计算复杂度从原始几何重采样的$O(d^2)$降低至$O(md(\log(d/m)+1))$。