Kinetic theory of decentralized learning for smart active matter

Smart active matter has the ability to control its motion guided by individual policies to achieve collective goals. We introduce a theoretical framework to study a decentralized learning process in which agents can locally exchange policies to adapt their behavior and maximize a predefined reward function. We use our formalism to derive explicit hydrodynamic equations for the policy dynamics. We apply the theory to two different microscopic models where policies correspond either to fixed parameters similar to evolutionary dynamics, or to state-dependent controllers known from the field of robotics. We find good agreement between theoretical predictions and agent-based simulations. By deriving fundamental control parameters and uncertainty relations, our work lays the foundations for a statistical physics analysis of decentralized learning.