Note on Follow-the-Perturbed-Leader in Combinatorial Semi-Bandit Problems
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915403894095872 |
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| author | Chen, Botao Honda, Junya |
| author_facet | Chen, Botao Honda, Junya |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_12490 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Note on Follow-the-Perturbed-Leader in Combinatorial Semi-Bandit Problems Chen, Botao Honda, Junya Machine Learning 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. |
| title | Note on Follow-the-Perturbed-Leader in Combinatorial Semi-Bandit Problems |
| topic | Machine Learning |
| url | https://arxiv.org/abs/2506.12490 |