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# Tesseract: Tensorised Actors for Multi-Agent Reinforcement Learning

*Proceedings of the 38th International Conference on Machine Learning*, PMLR 139:7301-7312, 2021.

#### Abstract

Reinforcement Learning in large action spaces is a challenging problem. This is especially true for cooperative multi-agent reinforcement learning (MARL), which often requires tractable learning while respecting various constraints like communication budget and information about other agents. In this work, we focus on the fundamental hurdle affecting both value-based and policy-gradient approaches: an exponential blowup of the action space with the number of agents. For value-based methods, it poses challenges in accurately representing the optimal value function for value-based methods, thus inducing suboptimality. For policy gradient methods, it renders the critic ineffective and exacerbates the problem of the lagging critic. We show that from a learning theory perspective, both problems can be addressed by accurately representing the associated action-value function with a low-complexity hypothesis class. This requires accurately modelling the agent interactions in a sample efficient way. To this end, we propose a novel tensorised formulation of the Bellman equation. This gives rise to our method Tesseract, which utilises the view of Q-function seen as a tensor where the modes correspond to action spaces of different agents. Algorithms derived from Tesseract decompose the Q-tensor across the agents and utilise low-rank tensor approximations to model the agent interactions relevant to the task. We provide PAC analysis for Tesseract based algorithms and highlight their relevance to the class of rich observation MDPs. Empirical results in different domains confirm the gains in sample efficiency using Tesseract as supported by the theory.