← 研踪 PaperTrailarXiv:2610.39457 · v1
Achieving Minimax-Optimal First-Order and Best-of-Both-Worlds Regret Simultaneously in Multi-Armed Bandits
首次提交 2026-10-29 14:28
版本更新 2026-10-29 14:28
雷达观测 2026-10-30 11:25
摘要
We study first-order and best-of-both-worlds regret guarantees in multi-armed bandits. Pogodin and Lattimore (2019) proposed an algorithm that combines minimax-optimal worst-case regret with a first-order guarantee whose loss-dependent term is $\mathcal{O}(\sqrt{K L^\star}\log T)$, where $T$ denotes the horizon, $L^\star$ denotes the smallest expected cumulative loss among fixed arms, and $K$ denotes the number of arms. However, a logarithmic gap remains between this loss-dependent term and the optimal first-order scale. In this paper, we remove this logarithmic factor by proposing an efficient and parameter-free algorithm that achieves the minimax optimal first-order regret guarantee $\mathcal{O}(\sqrt{K L^\star})$. Technically, our algorithm is built upon FTRL with a Tsallis entropy regularizer (Zimmert and Seldin,2021), with a learning rate adapted to the learner's cumulative loss and a sampling distribution determined by a scalar fixed point. This construction controls the combined estimation and stability cost in terms of the learner's loss. Moreover, we show that the same algorithm achieves a best-of-both-worlds guarantee, with $\mathcal{O}(\sum_{i\ne i^\star}\frac{\log T}{Δ_i})$ regret for i.i.d. losses with a unique best arm $i^\star$, where $Δ_i$ denotes the suboptimality gap of arm $i$.
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v1 · 2026-10-29 14:28 · 最新观测版本
Achieving Minimax-Optimal First-Order and Best-of-Both-Worlds Regret Simultaneously in Multi-Armed Bandits
We study first-order and best-of-both-worlds regret guarantees in multi-armed bandits. Pogodin and Lattimore (2019) proposed an algorithm that combines minimax-optimal worst-case regret with a first-order guarantee whose loss-dependent term is $\mathcal{O}(\sqrt{K L^\star}\log T)$, where $T$ denotes the horizon, $L^\star$ denotes the smallest expected cumulative loss among fixed arms, and $K$ denotes the number of arms. However, a logarithmic gap remains between this loss-dependent term and the optimal first-order scale. In this paper, we remove this logarithmic factor by proposing an efficient and parameter-free algorithm that achieves the minimax optimal first-order regret guarantee $\mathcal{O}(\sqrt{K L^\star})$. Technically, our algorithm is built upon FTRL with a Tsallis entropy regularizer (Zimmert and Seldin,2021), with a learning rate adapted to the learner's cumulative loss and a sampling distribution determined by a scalar fixed point. This construction controls the combined estimation and stability cost in terms of the learner's loss. Moreover, we show that the same algorithm achieves a best-of-both-worlds guarantee, with $\mathcal{O}(\sum_{i\ne i^\star}\frac{\log T}{Δ_i})$ regret for i.i.d. losses with a unique best arm $i^\star$, where $Δ_i$ denotes the suboptimality gap of arm $i$.
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