Convergence and Error Bounds for Passive Stochastic Algorithms Using Vanishing Step Size

Abstract

This work is concerned with passive stochastic approximation (PSA) algorithms having vanishing or decreasing step size and window width. Unlike the traditional stochastic approximation methods, the passive stochastic approximation algorithms utilize passive strategies. Under the framework of PSA, not only the measurement noise is unobservable, but also the “state” {xn} is a randomly generated sequence. In our formulation, both the observation noise and the randomly generated {xn} are correlated random processes. Under rather general conditions, w.p.1. convergence of the algorithms is established. Then upper bounds on estimation errors are obtained. It is shown that the bounds depend on the smoothness of the function under consideration in an essential way, which reveals another distinct feature of the passive algorithms.