Characterizing the Exact Behaviors of Temporal Difference Learning Algorithms Using Markov Jump Linear System Theory
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In this paper, we provide a unified analysis of temporal difference learning algorithms with linear function approximators by exploiting their connections to Markov jump linear systems (MJLS). We tailor the MJLS theory developed in the control community to characterize the exact behaviors of the first and second order moments of a large family of temporal difference learning algorithms. For both the IID and Markov noise cases, we show that the evolution of some augmented versions of the mean and covariance matrix of TD learning exactly follows the trajectory of a deterministic linear time-invariant (LTI) dynamical system. Applying the well-known LTI system theory, we obtain closed-form expressions for the mean and covariance matrix of TD learning at any time step. We provide a tight matrix spectral radius condition to guarantee the convergence of the covariance matrix of TD learning, and perform a perturbation analysis to characterize the dependence of the TD behaviors on learning rate. For the IID case, we provide an exact formula characterizing how the mean and covariance matrix of TD learning converge to the steady state values at a linear rate. For the Markov case, we use our formulas to explain how the behaviors of TD learning algorithms are affected by learning rate and various properties of the underlying Markov chain.
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