The mathematics of noise-free SPSA

We consider discrete-time fixed gain stochastic approximation processes that are defined in terms of a random field that is identically zero at some point θ*. The boundedness of the estimator process is enforced by a resetting mechanism. Under appropriate technical conditions the estimator sequence is shown to converge to θ* with geometric rate almost surely. This result is in striking contrast to classical stochastic approximation theory where the typical convergence rate is n^-1/2. For the proof a discrete-time version of the ODE-method is developed and used, and the techniques of Gerencser (1996) are extended. The paper is motivated by the study of simultaneous perturbation stochastic approximation (SPSA) methods applied to noise-free problems and to direct adaptive control