Note on Follow-the-Perturbed-Leader in Combinatorial Semi-Bandit Problems

TL;DR

This paper analyzes the optimality and computational complexity of Follow-the-Perturbed-Leader (FTPL) in combinatorial semi-bandit problems, demonstrating regret bounds and efficiency improvements with geometric resampling techniques.

Contribution

It establishes regret bounds for FTPL with geometric resampling in semi-bandit settings and introduces an extension to reduce computational complexity.

Findings

FTPL achieves near-optimal regret bounds in semi-bandit problems.

Conditional geometric resampling reduces computational complexity from O(d^2) to O(md(log(d/m)+1)).

FTPL with geometric resampling attains optimal regret bounds in adversarial settings.

Abstract

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 Fr\'{e}chet-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 $O\left(\sqrt{m^2 d^\frac{1}{\alpha}T}+\sqrt{mdT}\right)$ regret with Fr\'{e}chet 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.