A Further Efficient Algorithm with Best-of-Both-Worlds Guarantees for $m$-Set Semi-Bandit Problem

2026-03-12

AI Summary

This work addresses the $m$-set semi-bandit problem by proposing a Follow-the-Perturbed-Leader (FTPL) algorithm that incorporates Fréchet and Pareto perturbations, achieving for the first time Best-of-Both-Worlds optimality across both adversarial and stochastic environments. Specifically, the algorithm attains the optimal $O(\sqrt{mdT})$ regret bound in adversarial settings and logarithmic regret in stochastic settings. To enhance computational efficiency without compromising theoretical guarantees, the authors introduce a conditional geometric resampling technique that reduces the computational complexity of loss estimation from $O(d^2)$ to $O(md(\log(d/m)+1))$, thereby effectively balancing statistical performance and practical tractability.

Technology Category

• Machine Learning: Online Learning & Bandits
• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Adversarial Search

Application Category

• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

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.

Problem

• m-set semi-bandit
• FTPL
• best-of-both-worlds
• regret optimality
• computational complexity

Innovation

• FTPL
• Best-of-Both-Worlds
• m-set semi-bandit
• geometric resampling
• regret optimality