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# Coarse-Grained Smoothness for Reinforcement Learning in Metric Spaces

*Proceedings of The 26th International Conference on Artificial Intelligence and Statistics*, PMLR 206:1390-1410, 2023.

#### Abstract

Principled decision-making in continuous state–action spaces is impossible without some assumptions. A common approach is to assume Lipschitz continuity of the Q-function. We show that, unfortunately, this property fails to hold in many typical domains. We propose a new coarse-grained smoothness definition that generalizes the notion of Lipschitz continuity, is more widely applicable, and allows us to compute significantly tighter bounds on Q-functions, leading to improved learning. We provide a theoretical analysis of our new smoothness definition, and discuss its implications and impact on control and exploration in continuous domains.