Arbitrary high precision output tracking is one of the most desirable control objectives found in industrial applications regardless of measurement errors. The main purpose of this paper is to supply to the iterative learning control (ILC) designer guidelines to select the corresponding learning gain in order to achieve this control objective. For example, if certain conditions are met, then it is necessary for the learning gain to converge to zero in the learning iterative domain. In particular, this paper presents necessary and sufficient conditions for boundedness of trajectories and uniform tracking in presence of measurement noise and a class of random reinitialization errors for a simple ILC algorithm. The system under consideration is a class of discrete-time affine non-linear systems with arbitrary relative degree and arbitrary number of system inputs and outputs. The state function does not need to satisfy a Lipschitz condition. This work also provides a recursive algorithm that generates the appropriate learning gain functions that meet the arbitrary high precision output tracking objective. The resulting tracking output error is shown to converge to zero at a rate inversely proportional to square root of the number of learning iterations in presence of measurement noise and a class of reinitialization errors. Two illustrative numerical examples are presented.