Improved Best-of-Both-Worlds Guarantees for Multi-Armed Bandits: FTRL with General Regularizers and Multiple Optimal Arms

• URL: http://arxiv.org/abs/2302.13534v2
• Date: Thu, 26 Oct 2023 17:21:36 GMT
• Title: Improved Best-of-Both-Worlds Guarantees for Multi-Armed Bandits: FTRL with General Regularizers and Multiple Optimal Arms
• Authors: Tiancheng Jin, Junyan Liu, Haipeng Luo
• Abstract summary: We study the problem of designing adaptive multi-armed bandit algorithms that optimally perform in both the setting and the adversarial setting simultaneously. We show that uniqueness is unnecessary for FTRL with a broad family of regularizers and a new learning rate schedule.
• Score: 41.06668954462585
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We study the problem of designing adaptive multi-armed bandit algorithms that perform optimally in both the stochastic setting and the adversarial setting simultaneously (often known as a best-of-both-world guarantee). A line of recent works shows that when configured and analyzed properly, the Follow-the-Regularized-Leader (FTRL) algorithm, originally designed for the adversarial setting, can in fact optimally adapt to the stochastic setting as well. Such results, however, critically rely on an assumption that there exists one unique optimal arm. Recently, Ito (2021) took the first step to remove such an undesirable uniqueness assumption for one particular FTRL algorithm with the $\frac{1}{2}$-Tsallis entropy regularizer. In this work, we significantly improve and generalize this result, showing that uniqueness is unnecessary for FTRL with a broad family of regularizers and a new learning rate schedule. For some regularizers, our regret bounds also improve upon prior results even when uniqueness holds. We further provide an application of our results to the decoupled exploration and exploitation problem, demonstrating that our techniques are broadly applicable.

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