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# Scale-free Adversarial Reinforcement Learning

*Proceedings of Thirty Seventh Conference on Learning Theory*, PMLR 247:1068-1101, 2024.

#### Abstract

This paper initiates the study of scale-free learning in Markov Decision Processes (MDPs), where the scale of rewards/losses is unknown to the learner. We design a generic algorithmic framework, \underline{S}cale \underline{C}lipping \underline{B}ound (\texttt{SCB}), and instantiate this framework in both the adversarial Multi-armed Bandit (MAB) setting and the adversarial MDP setting. Through this framework, we achieve the first minimax optimal expected regret bound and the first high-probability regret bound in scale-free adversarial MABs, resolving an open problem raised in \cite{hadiji2020adaptation}. On adversarial MDPs, our framework also give birth to the first scale-free RL algorithm with a $\tilde{\mathcal{O}}(\sqrt{T})$ high-probability regret guarantee.