This paper describes a novel machine learning framework for solving sequential decision problems called Markov decision processes (MDPs) by iteratively computing low-dimensional representations and approximately optimal policies. A unified mathematical framework for learning representation and optimal control in MDPs is presented based on a class of singular operators called Laplacians, whose matrix representations have nonpositive off-diagonal elements and zero row sums. Exact solutions of discounted and average-reward MDPs are expressed in terms of a generalized spectral inverse of the Laplacian called the Drazin inverse. A generic algorithm called representation policy iteration (RPI) is presented which interleaves computing low-dimensional representations and approximately optimal policies. Two approaches for dimensionality reduction of MDPs are described based on geometric and reward-sensitive regularization, whereby low-dimensional representations are formed by diagonalization or dilation of Laplacian operators. Model-based and model-free variants of the RPI algorithm are presented; they are also compared experimentally on discrete and continuous MDPs. Some directions for future work are finally outlined.
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