Mokkadem, Abdelkader et Pelletier, Mariane, Convergence rate and averaging of nonlinear two-time-scale stochastic approximation algorithms, Annals of applied probability , 16(3), 2006, pp. 1671-1702
The first aim of this paper is to establish the weak convergence rate of nonlinear two-time-scale stochastic approximation algorithms. Its second aim is to introduce the averaging principle in the context of two-time-scale stochastic approximation algorithms. We first define the notion of asymptotic efficiency in this framework, then introduce the averaged two-time-scale stochastic approximation algorithm, and finally establish its weak convergence rate. We show, in particular, that both components of the averaged two-time-scale stochastic approximation algorithm simultaneously converge at the optimal rate .n.