Low-rank MDP Approximation via Moment Coupling
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We propose a novel method—based on local moment matching—to approximate the value function of a Markov Decision Process. The method is grounded in recent work by Braverman et al (2020) that relates the solution of the Bellman equation to that of a PDE where, in the spirit of the central limit theorem, the transition matrix is reduced to its local first and second moments. Solving the PDE is not required by our method. Instead we construct a "sister" Markov chain whose two local transition moments are (approximately) identical with those of the focal chain. Because they share these moments, the original chain and its "sister" are coupled through the PDE, a coupling that facilitates optimality guarantees. We show how this view can be embedded into the existing aggregation framework of ADP, providing a disciplined mechanism to tune the aggregation and disaggregation probabilities. This embedding into aggregation also reveals how the approximation's accuracy depends on a certain local linearity of the value function. The computational gains arise from the reduction of the effective state space from N to N^1/2+ϵ is as one might intuitively expect from approximations grounded in the central limit theorem.
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