Abstract
This work presents a comprehensive survey of the development of pseudogradient stochastic approximation algorithms with randomized input disturbance, considers the problems of their applicability in optimization problems with linear constraints, and discusses new possibilities to use them for multiagent control for load balancing of nodes in computational networks. Justifications of the algorithms’ correctness and their optimal convergence rate are based on the foundational works of B.T. Polyak.
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Polyak, B.T., Vvedenie v optimizatsiyu, Moscow: Nauka, 1983. Translated into English under the title Introduction to Optimization, New York: Optimization Software, 1987.
Schweppe, F.C., Uncertain Dynamic Systems, New York: Prentice Hall, 1973.
Fel’dbaum, A.A., On Dual Control Problems, in Metody optimizatsii avtomaticheskikh sistem (Optimization Methods for Automated Systems), Moscow: Nauka, 1972, pp. 89–108.
Ljung, L., System Identification: Theory for the User, Englewood Cliffs: Prentice Hall, 1987. Translated under the title Identifikatsiya sistem. Teoriya dlya pol’zovatelya, Moscow: Nauka, 1991.
Tsypkin, Ya.Z., Informatsionnaya teoriya identifikatsii (Information Theory of Identification), Moscow: Nauka, 1995.
Bai, E.W., Nagpal, K.M., and Tempo, R., Bounded-Error Parameter Estimation: Noise Models and Recursive Algorithms, Automatica, 1996, vol. 32. pp. 985–999.
Polyak, B.T. and Shcherbakov, P.S., Robastnaya ustoichivost’ i upravlenie (Robust Stability and Control), Moscow: Nauka, 2002.
Polyak, B.T., Khlebnikov, M.V., and Shcherbakov, P.S., Upravlenie lineinymi sistemami pri vneshnikh vozmushcheniyakh. Tekhnika lineinykh matrichnykh neravenstv (Control over Linear Systems under External Interference. Linear Matrix Inequalities), Moscow: LENAND, 2014.
Sokolov, V.F., Estimating Performance of the Robust Control System under Unknown Upper Disturbance Boundaries and Measurement Noise, Autom. Remote Control, 2010, vol. 71, no. 9, pp. 1741–1756.
Calafiore, G. and Polyak, B.T., Stochastic Algorithms for Exact and Approximate Feasibility of Robust LMIs, IEEE Trans. Automat. Control, 2001, vol. 46, pp. 1755–1759.
Granichin, O.N. and Polyak, B.T., Randomizirovannye algoritmy otsenivaniya i optimizatsii pri pochti proizvol’nykh pomekhakh (Randomized Estimation and Optimization Algorithms for Almost Arbitrary Noise), Moscow: Nauka, 2003.
Tempo, R., Calafiore, G., and Dabbene, F., Randomized Algorithms for Analysis and Control of Uncertain Systems: With Applications, New York: Springer-Verlag, 2013.
Granichin, O., Volkovich, V., and Toledano-Kitai, D., Randomized Algorithms in Automatic Control and Data Mining, Springer, 2014.
Robbins, H. and Monro, S., A Stochastic Approximation Method, Ann. Math. Statist., 1951, vol. 22, pp. 400–407.
Kiefer, J. and Wolfowitz, J., Statistical Estimation on the Maximum of a Regression Function, Ann. Math. Statist., 1952, vol. 23, pp. 462–466.
Blum, J.R., Multidimensional Stochastic Appoximation, Ann. Math. Statist., 1954, vol. 9, pp. 737–744.
Granichin, O.N., A Stochastic Recursive Procedure with Correlated Noises in the Observation that Employs Trial Perturbations at the Input, Vestn. Leningr. Univ., Mat., 1989, vol. 22, no. 1, pp. 27–31.
Granichin, O.N., Procedure of Stochastic Approximation with Disturbances at the Input, Autom. Remote Control, 1992, vol. 53, no. 2, part 1, pp. 232–237.
Polyak, B.T. and Tsybakov, A.B., Optimal Orders of Accuracy for Search Algorithms of Stochastic Optimization, Probl. Peredachi Inf., 1990, vol. 26, no. 2, pp. 126–133.
Polyak, V.T. and Tsybakov, A.V., On Stochastic Approximation with Arbitrary Noise (the KW Case), in Topics in Nonparametric Estimation, Khasminskii, R.Z., Ed., Adv. Sov. Math., Providence: Am. Math. Soc., 1992, no. 12, pp. 107–113.
Rastrigin, L.A., Statisticheskie metody poiska (Statistical Search Methods), Moscow: Nauka, 1968.
Spall, J.C., Multivariate Stochastic Approximation Using a Simultaneous Perturbation Gradient Approximation, IEEE Trans. Automat. Control, 1992, vol. 37, no. 3, pp. 332–341.
Spall, J.C., A One—Measurement Form of Simultaneous Perturbation Stochastic Approximation, Automatica, 1997, vol. 33, pp. 109–112.
Polyak, B.T., Convergence and Rate of Convergence of Recursive Stochastic Algorithms. I, Autom. Remote Control, 1976, vol. 37, no. 12, part 1, pp. 1858–1868.
Polyak, B.T., Convergence and Convergence Rate of Iterative Stochastic Algorithms. II. The Linear Case, Autom. Remote Control, 1977, vol. 38, no. 4, part 2, pp. 537–542.
Tsypkin, Ya.Z., Adaptatsiya i obuchenie v avtomaticheskikh sistemakh (Adaptation and Learning in Automated Systems), Moscow: Nauka, 1968.
Polyak, B.T. and Tsypkin, Ya.Z., Pseudogradient Algorithms of Adaptation and Learning, Autom. Remote Control, 1973, vol. 34, no. 3, part 1, pp. 377–398.
Polyak, B.T. and Tsypkin, Ya.Z., Adaptive Estimation Algorithms (Convergence, Optimality, Stability), Autom. Remote Control, 1979, vol. 40, no. 3, part 1, pp. 378–389.
Polyak, B.T. and Tsypkin, Ya.Z., Optimal Pseudogradient Adaptation Procedure, Autom. Remote Control, 1980, vol. 41, no. 8, part 1, pp. 1101–1110.
Polyak, B.T. and Tsypkin, Ya.Z., Robust Pseudogradient Adaptation Procedure, Autom. Remote Control, 1980, vol. 41, no. 10, part 2, pp. 1404–1409.
Tsypkin, Ya.Z. and Poznyak, A.S., Optimal Stochastic Optimization Search Algorithms, Dokl. Akad. Nauk USSR, 1981, vol. 260, no. 3, pp. 550–553.
Polyak, B.T. and Tsypkin, Ya.Z., Gradient Methods of Stochastic Optimization, Izmer., Kontrol’, Avtomatiz., 1989, no. 3, pp. 50–54.
Nazin, A.V., Polyak, B.T., and Tsybakov, A.B., Passive Stochastic Approximation, Autom. Remote Control, 1989, vol. 50, no. 11, part 2, pp. 1563–1569.
Polyak, B.T., New Method of Stochastic Approximation Type, Autom. Remote Control, 1990, vol. 51, no. 7, part 2, pp. 937–946.
Polyak, V.T. and Yuditskij, A.V., Acceleration of Stochastic Approximation Procedures by Averaging, SIAM J. Control Optim., 1992, vol. 30, no. 4, pp. 838–855.
Kushner, H.J. and Yin, G.G., Stochastic Approximation Algorithms and Applications, New York: Springer-Verlag, 2002.
Borkar, V.S., Stochastic Approximation. A Dynamical Systems Viewpoint, Cambridge: Cambridge Univ. Press, 2008.
Granichin, O., Gurevich, L., and Vakhitov, A., Discrete-Time Minimum Tracking Based on Stochastic Approximation Algorithm with Randomized Differences, 48th Conf. Decision Control, Shanghai, China, 2009, pp. 5763–5767.
Granichin, O.N., and Amelina, N.O., Simultaneous Perturbation Stochastic Approximation for Tracking under Unknown but Bounded Interference, IEEE Trans. Automat. Control, 2015, vol. 60, no. 5.
Tsitsiklis, J., Bertsekas, D., and Athans, M., Distributed Asynchronous Deterministic and Stochastic Gradient Optimization Algorithms, IEEE Trans. Automat. Control, 1986, vol. 31, no. 9, pp. 803–812.
Amelina, N.O. and Fradkov, A.L., Approximate Consensus in the Dynamic Stochastic Network with Incomplete Information and Measurement Delays, Autom. Remote Control, 2012, vol. 73, no. 11, pp. 1765–1783.
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Original Russian Text © O.N. Granichin, 2015, published in Avtomatika i Telemekhanika, 2015, No. 5, pp. 43–59.
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Granichin, O.N. Stochastic approximation search algorithms with randomization at the input. Autom Remote Control 76, 762–775 (2015). https://doi.org/10.1134/S0005117915050033
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DOI: https://doi.org/10.1134/S0005117915050033