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. >

[1]  J. Blum.  Multidimensional Stochastic Approximation Methods , 1954 .

[2]  J. Gillis,et al.  Matrix Iterative Analysis , 1961 .

[3]  S. Sugiyama.  Difference inequalities and their applications to stability problems , 1971 .

[4]  D. Siljak,et al.  On stability of discrete composite systems , 1973 .

[5]  Dragoslav D. Šiljak,et al.  Exponential stability of large-scale discrete systems , 1974 .

[6]  Dragoslav D. Šiljak,et al.  Large-Scale Dynamic Systems: Stability and Structure , 1978 .

[7]  Gérard M. Baudet,et al.  Asynchronous Iterative Methods for Multiprocessors , 1978, JACM.

[8]  D. Siljak,et al.  Lotka-Volterra equations: Decomposition, stability, and structure , 1980 .

[9]  Dimitri Bertsekas,et al.  Distributed dynamic programming , 1981, 1981 20th IEEE Conference on Decision and Control including the Symposium on Adaptive Processes.

[10]  Yehuda Wallach,et al.  Alternating Sequential/Parallel Processing , 1982, Lecture Notes in Computer Science.

[11]  David J. Evans,et al.  Parallel Processing Systems , 1982 .

[12]  Dragoslav D. Šiljak,et al.  An inclusion principle for hereditary systems , 1984 .

[13]  Georges Bitsoris,et al.  Stability analysis of discrete-time dynamical systems via positive upper aggregated systems , 1984 .

[14]  Random Difference Inequalities , 1985 .

[15]  Dragoslav D. Siljak,et al.  Nested ε-decompositions and clustering of complex systems , 1986, Autom..

[16]  John N. Tsitsiklis,et al.  Parallel and distributed iterative algorithms : a selective survey , 1988 .

[17]  D. Siljak,et al.  Nested epsilon decompositions of linear systems: weakly coupled and overlapping blocks , 1991 .