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# Multi-User Reinforcement Learning with Low Rank Rewards

*Proceedings of the 40th International Conference on Machine Learning*, PMLR 202:25627-25659, 2023.

#### Abstract

We consider collaborative multi-user reinforcement learning, where multiple users have the same state-action space and transition probabilities but different rewards. Under the assumption that the reward matrix of the $N$ users has a low-rank structure – a standard and practically successful assumption in the collaborative filtering setting – we design algorithms with significantly lower sample complexity compared to the ones that learn the MDP individually for each user. Our main contribution is an algorithm which explores rewards collaboratively with $N$ user-specific MDPs and can learn rewards efficiently in two key settings: tabular MDPs and linear MDPs. When $N$ is large and the rank is constant, the sample complexity per MDP depends logarithmically over the size of the state-space, which represents an exponential reduction (in the state-space size) when compared to the standard “non-collaborative” algorithms. Our main technical contribution is a method to construct policies which obtain data such that low rank matrix completion is possible (without a generative model). This goes beyond the regular RL framework and is closely related to mean field limits of multi-agent RL.