Low-discrepancy sampling for approximate dynamic programming with local approximators

Approximate dynamic programming (ADP) relies, in the continuous-state case, on both a flexible class of models for the approximation of the value functions and a smart sampling of the state space for the numerical solution of the recursive Bellman equations. In this paper, low-discrepancy sequences, commonly employed for number-theoretic methods, are investigated as a sampling scheme in the ADP context when local models, such as the Nadaraya-Watson (NW) ones, are employed for the approximation of the value function. The analysis is carried out both from a theoretical and a practical point of view. In particular, it is shown that the combined use of low-discrepancy sequences and NW models enables the convergence of the ADP procedure. Then, the regular structure of the low-discrepancy sampling is exploited to derive a method for automatic selection of the bandwidth of NW models, which yields a significant saving in the computational effort with respect to the standard cross validation approach. Simulation results concerning an inventory management problem are presented to show the effectiveness of the proposed techniques.