  A learning based method for obtaining feedback laws for nonlinear optimal
control problems is proposed. The learning problem is posed such that the open
loop value function is its optimal solution.
  This infinite dimensional, function space, problem, is approximated by a
polynomial ansatz and its convergence is analyzed. An $\ell_1$ penalty term is
employed, which combined with the proximal point method, allows to find sparse
solutions for the learning problem. The approach requires multiple evaluations
of the elements of the polynomial basis and of their derivatives. In order to
do this efficiently a graph-theoretic algorithm is devised. Several examples
underline that the proposed methodology provides a promising approach for
mitigating the curse of dimensionality which would be involved in case the
optimal feedback law was obtained by solving the Hamilton Jacobi Bellman
equation.
