Convergence of stochastic approximation under general noise and stability conditions

The almost sure convergence of stochastic approximation algorithms under general noise and stability conditions is considered in this paper. First, the stochastic approximation algorithm with additive, state-dependent noise is analyzed and sufficient conditions for its convergence are derived. Then, the algorithm with nonadditive noise is examined and sufficient conditions for its convergence are obtained using the results obtained for the additive noise case. The convergence of the algorithm with nonadditive noise is considered for the case where the noise is correlated and satisfies the strong or uniform mixing property. Finally, the results derived for the nonadditive noise case are applied to the analysis of the gradient based learning algorithm for feedforward neural networks and sufficient conditions for its convergence are derived.