Before revision:  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:  By exploiting the properties of the PDEs , we show that the policy regret decays at a n^{-1/2} rate to the maximum attainable within the specified class of policy functions ;