An efficient Kullback-Leibler optimization algorithm for probabilistic control design

This paper addresses the problem of iterative optimization of the Kullback-Leibler (KL) divergence on discrete (finite) probability spaces. Traditionally, the problem is formulated in the constrained optimization framework and is tackled by gradient like methods. Here, it is shown that performing the KL optimization in a Riemannian space equipped with the Fisher metric provides three major advantages over the standard methods: 1. The Fisher metric turns the original constrained optimization into an unconstrained optimization problem; 2. The optimization using a Fisher metric behaves asymptotically as a Newton method and shows very fast convergence near the optimum; 3. The Fisher metric is an intrinsic property of the space of probability distributions and allows a formally correct interpretation of a (natural) gradient as the steepest-descent method. Simulation results are presented.