Convergence and stability of distributed stochastic iterative processes

A mathematical framework is proposed for convergence analysis of stochastic iterative processes arising in applications of pseudogradient optimization algorithms. The framework is based upon the concepts of stochastic difference inequalities and vector Lyapunov functions, which are ideally suited for reducing the dimensionality problem arising in testing convergence of distributed parallel schemes. By applying the M-matrix conditions to a test matrix having a dimension equal to the number of the processor in the scheme, one can use the framework to select suitable scaling factors for each individual processor, producing a satisfactory convergence rate of the overall iterative process. The results provide new convergence tests for distributed iterative processes arising in decentralized extremal regulation, adaptation, and parameter estimation schemes.>