Before revision:  We then propose a modified Reinforcement Learning algorithm to solve for the policy rule that achieves the best value in the pre-specified class. The algorithm is easily implementable and computationally efficient, with speedups achieved through multiple reinforcement learning agents simultaneously learning the problem in parallel processes. By exploiting the properties of the PDEs , we show that the average social welfare attained by the estimated policy rule converges at a n^{-1/2} rate to the maximum attainable within the specified class of policy functions ;
After revision:  The estimated policy rule is the one with the maximal estimated value function. Using the theory of viscosity solutions to PDEs we show that the average social welfare attained by the estimated policy rule converges at a n^{-1/2} rate to the maximum attainable within the specified class of policy functions ;