Probabilistic Bounds in Tracking a Discrete-Time Varying Process

This paper investigates the minimization problem, under a practical setting where the scalar-valued loss function Lk is intrinsically time-varying, and therefore the corresponding minimizer sequence is also time-varying. We focus on the tracking capability of the recursive estimates generated by the constant-gain stochastic gradient algorithm. It is shown that the trajectory of a limiting nonautonomous ordinary differential equation can be associated with the iterates generated by the stochastic gradient descent algorithm with constant gain. The main tool in establishing the connection is the formula for variation of parameters. The established probabilistic bound is applicable in tracking a discrete-time varying process, e.g., a jump process. A synthetic example illustrates the accountability of the trajectory characterization depends on the magnitude of the noise, the drift magnitude, and the constant gain.