A Further Efficient Algorithm with Best-of-Both-Worlds Guarantees for $m$-Set Semi-Bandit Problem
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866917335038689280 |
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| author | Chen, Botao Lee, Jongyeong Kim, Chansoo Honda, Junya |
| author_facet | Chen, Botao Lee, Jongyeong Kim, Chansoo Honda, Junya |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_11764 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A Further Efficient Algorithm with Best-of-Both-Worlds Guarantees for $m$-Set Semi-Bandit Problem Chen, Botao Lee, Jongyeong Kim, Chansoo Honda, Junya Machine Learning 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. |
| title | A Further Efficient Algorithm with Best-of-Both-Worlds Guarantees for $m$-Set Semi-Bandit Problem |
| topic | Machine Learning |
| url | https://arxiv.org/abs/2603.11764 |