We consider the discrete-time infinite-horizon optimal control problem formalized by Markov Decision Processes. We revisit the work of Bertsekas and Ioffe, that introduced $lambda$ Policy Iteration, a family of algorithms parameterized by $lambda$ that generalizes the standard algorithms Value Iteration and Policy Iteration, and has some deep connections with the Temporal Differences algorithm TD($lambda$) described by Sutton and Barto. We deepen the original theory developped by the authors by providing convergence rate bounds which generalize standard bounds for Value Iteration described for instance by Puterman. Then, the main contribution of this paper is to develop the theory of this algorithm when it is used in an approximate form and show that this is sound. Doing so, we extend and unify the separate analyses developped by Munos for Approximate Value Iteration and Approximate Policy Iteration. Eventually, we revisit the use of this algorithm in the training of a Tetris playing controller as originally done by Bertsekas and Ioffe. We provide an original performance bound that can be applied to such an undiscounted control problem. Our empirical results are different from those of Bertsekas and Ioffe (which were originally qualified as paradoxical and intriguing), and much more conform to what one would expect from a learning experiment. We discuss the possible reason for such a difference.
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