Theoretical Analysis of Value-Iteration-Based Q-Learning with Approximation Errors

In this paper, the value-iteration-based Q-Iearning algorithm with approximation errors is analyzed theoretically. First, based on an upper bound of the approximation errors caused by the Q-function approximator, we get the lower and upper bound functions of the iterative Q-function, which proves that the limit of the approximate Q-function sequence is bounded. Then, we develop a stability condition for the termination of the iterative algorithm, for ensuring that the current control policy derived from the resulting approximate Q-function is stabilizing. Also, we establish an upper bound function of the approximation errors, which is caused by the policy function approximator, to guarantee that the approximate control policy is stabilizing. Finally, the numerical results verifies the theoretical results with a simulation example.