Asymptotically efficient adaptive allocation schemes for controlled i.i.d. processes: finite parameter space

The authors consider a controlled i.i.d. (independently identically distributed) process whose distribution is parametrized by an unknown parameter theta belonging to some known parameter space Theta , and a one-step reward associated with each pair of control and the following state of the process. The objective is to maximize the expected value of the sum of one-step rewards over an infinite horizon. By introducing the loss associated with a control scheme, it is shown that the problem is equivalent to minimizing this loss. Uniformly good adaptive control schemes are defined and emphasized. A lower bound on the loss associated with any uniformly good control scheme is developed. Finally, an adaptive control scheme is constructed whose loss equals the lower bound, and is therefore asymptotically efficient.<>