Approximation method for discrete Markov decision models with a large state space

TL;DR

This paper introduces the SLSTD approximation method for large discrete Markov decision models, effectively reducing computational costs and overcoming the curse of dimensionality.

Contribution

The paper presents a novel SLSTD method that approximates the value function across large state spaces efficiently, with proven consistency and asymptotic normality.

Findings

Reduces computation time by over 99% in some cases

Successfully approximates value functions in high-dimensional models

Proven statistical properties of the estimates

Abstract

To solve discrete Markov decision models with a large number of dimensions is always difficult (and at times, impossible), because size of state space and computation cost increases exponentially with the number of dimensions. This phenomenon is called "The Curse of Dimensionality," and it prevents us from using models with many state variables. To overcome this problem, we propose a new approximation method, named statistical least square temporal difference (SLSTD) method, that can solve discrete Markov decision models with large state space. SLSTD method approximate the value function on the whole state space at once, and obtain optimal approximation weight by minimizing temporal residuals of the Bellman equation. Furthermore, a stochastic approximation method enables us to optimize the problem with low computational cost. SLSTD method can solve DMD models with large state space more easily than other existing methods, and in some cases, reduces the computation time by over 99 percent. We also show the parameter estimated by SLSTD is consistent and asymptotically normally distributed.