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In this paper, we consider the policy evaluation problem in multi-agentreinforcement learning (MARL) and derive exact closed-form formulas for thefinite-time mean-squared estimation errors of decentralized temporal difference(TD) learning with linear function approximation. Our analysis hinges upon thefact that the decentralized TD learning method can be viewed as a Markov jumplinear system (MJLS). Then standard MJLS theory can be applied to quantify themean and covariance matrix of the estimation error of the decentralized TDmethod at every time step. Various implications of our exact formulas on thealgorithm performance are also discussed. An interesting finding is that undera necessary and sufficient stability condition, the mean-squared TD estimationerror will converge to an exact limit at a specific exponential rate.

Exact Formulas for Finite-Time Estimation Errors of Decentralized Temporal Difference Learning with Linear Function Approximation