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Mean field for Markov Decision Processes: from Discrete to Continuous Optimization
Nicolas Gast1, 2, Bruno Gaujal1, Jean-Yves Le Boudec2

We study the convergence of Markov Decision Processes made of a large number of objects to optimization problems on ordinary differential equations (ODE). We show that the optimal reward of such a Markov Decision Process, satisfying a Bellman equation, converges to the solution of a continuous Hamilton-Jacobi-Bellman (HJB) equation based on the mean field approximation of the Markov Decision Process. We give bounds on the difference of the rewards, and a constructive algorithm for deriving an approximating solution to the Markov Decision Process from a solution of the HJB equations. We illustrate the method on three examples pertaining respectively to investment strategies, population dynamics control and scheduling in queues are developed. They are used to illustrate and justify the construction of the controlled ODE and to show the gain obtained by solving a continuous HJB equation rather than a large discrete Bellman equation.
1:  INRIA Grenoble Rhône-Alpes / LIG laboratoire d'Informatique de Grenoble - MESCAL
2:  EPFL - Ecole Polytechnique Fédérale de Lausanne
Mean Field – Hamilton-Jacobi-Bellman – Optimal Control – Markov Decision Processes