Convergence and applications of stochastic approximation with state-dependent noise

The purpose of stochastic approximation (SA) is to find the roots of f(·) when the unknown function f(·) can be observed with noise. In this paper the pathwise convergence of SA algorithms is considered for the case where the observation noise may depend on state, by which we mean the estimate for the sought-for root of f(·). The conditions imposed on the observation noise are the weakest in comparison with the existing ones. The superiority of the noise condition given in the paper over those used in Chen (1994) and Chen and Zhu (1996) consists in that the present condition is directly verifiable, needless to pay attention to the behavior of the algorithm. The conditions imposed on f(·) are general: f(·) is only required to be measurable and locally bounded. Applications to optimization and signal processing demonstrate the strong points of the convergence theorems given in the paper