Neural approximations in discounted infinite-horizon stochastic optimal control problems

Neural approximations of the optimal stationary closed-loop control strategies for discounted infinite-horizon stochastic optimal control problems are investigated. It is shown that for a family of such problems, the minimal number of network parameters needed to achieve a desired accuracy of the approximate solution does not grow exponentially with the number of state variables. In such a way, neural-network approximation mitigates the so-called “curse of dimensionality”. The obtained theoretical results point out the potentialities of neural-network based approximation in the framework of sequential decision problems with continuous state, control, and disturbance spaces.