We present two nonparametric approaches to Kullback-Leibler (KL) control, or
linearly-solvable Markov decision problem (LMDP) based on Gaussian processes
(GP) and Nystr\"{o}m approximation. Compared to recently developed parametric
methods, the proposed data-driven frameworks feature accurate function
approximation and efficient on-line operations. Theoretically, we derive the
mathematical connection of KL control based on dynamic programming with earlier
work in control theory which relies on information theoretic dualities for the
infinite time horizon case. Algorithmically, we give explicit optimal control
policies in nonparametric forms, and propose on-line update schemes with
budgeted computational costs. Numerical results demonstrate the effectiveness
and usefulness of the proposed frameworks