Abstract
Multidimensional stochastic optimization plays an important role in analysis and control of many technical systems. To solve the challenging multidimensional problems of optimization, it was suggested to use the randomized algorithms of stochastic approximation with perturbed input which are not just simple, but also provide consistent estimates of the unknown parameters for observations in “almost arbitrary” noise. Optimal methods of choosing the parameters of algorithms were motivated.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
Kiefer, J. and Wolfowitz, J., Statistical Estimation on the Maximum of a Regression Function, Ann. Math. Statist., 1952, vol. 23, pp. 462-466.
Spall, J.C., Multivariate Stochastic Approximation Using a Simultaneous Perturbation Gradient Approximation, IEEE Trans. Autom. Control, 1992, vol. 37, pp. 332-341.
Granichin, O.N., Stochastic Approximation with Input Perturbation in Dependent Observation Noise, Vestn. LGU, 1989, vol. 1, no.4, pp. 27-31.
Wazan, M.T., Stochastic Approximation, Cambridge: Cambridge Univ. Press, 1969. Translated under the title Stokhasticheskaya approksimatsiya, Moscow: Mir, 1972.
Nevel'son, M.B. and Khas'minskii, R.Z., Stokhasticheskaya approksimatsiya i rekurrentnoe otsenivanie (Stochastic Approximation and Recurrent Estimation), Moscow: Nauka, 1972.
Ermol'ev, Yu.M., Metody stokhasticheskogo programmirovaniya (Methods of Stochastic Programming), Moscow: Nauka, 1976.
Katkovnik, V.Ya., Lineinye otsenki i stokhasticheskie zadachi optimizatsii (Linear Estimates and Stochastic Problems of Optimization), Moscow: Nauka, 1976.
Polyak, B.T., Vvedenie v optimizatsiyu, Moscow: Nauka, 1983. Translated under the title Introduction to Optimization, New York: Optimization Software, 1987.
Fomin, V.N., Rekurrentnoe otsenivanie i adaptivnaya fil'tratsiya (Recurrent Estimation and Adaptive Filtration), Moscow: Nauka, 1984.
Mikhalevich, V.S., Gupal, A.M., and Norkin, V.I., Metody nevypukloi optimizatsii (Methods of Nonconvex Optimization), Moscow: Nauka, 1987.
Kushner, H.J. and Yin, G.G., Stochastic Approximation Algorithms and Applications, New York: Springer, 1997.
Fabian, V., Stochastic Approximation of Minima with Improved Asymptotic Speed, Ann. Math. Statist., 1967, vol. 38, pp. 191-200.
Polyak, B.T. and Tsybakov, A.B., Optimal Orders of Accuracy of the Searching Algorithms of Stochastic Approximation, Probl. Peredachi Inf., 1990, no. 2, pp. 45-53.
Chen, H.F., Lower Rate of Convergence for Locating a Maximum of a Function, Ann. Statist., 1988, vol. 16, pp. 1330-1334.
Granichin, O.N., Randomized Algorithms of Stochastic Approximation in Arbitrary Noise, Avtom. Telemekh., 2002, no. 2, pp. 44-55.
Wang, I.-J. and Chong, E., A Deterministic Analysis of Stochastic Approximation with Randomized Directions, IEEE Trans. Autom. Control, 1998, vol. 43, pp. 1745-1749.
Chen, H.F., Duncan, T.E., and Pasik-Duncan, B., A Kiefer-Wolfowitz Algorithm with Randomized Differences, IEEE Trans. Autom. Control, 1999, vol. 44, no.3, pp. 442-453.
Polyak, B.T. and Tsybakov, A.B., On Stochastic Approximation with Arbitrary Noise (the KW Case), in Topics in Nonparametric Estimation, Adv. Sov. Math., Am. Math. Soc., Khasminskii, R.Z., Ed., 1992, no. 12, pp. 107-113.
Gerencser, L., Convergence Rate of Moments in Stochastic Approximation with Simultaneous Perturbation Gradient Approximation and Resetting, IEEE Trans. Autom. Control, 1999, vol. 44, pp. 894-905.
Granichin, O.N., Estimating the Parameters of a Linear Regression in Arbitrary Noise, Avtom. Telemekh., 2002, no. 1, pp. 30-41.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Granichin, O.N. Optimal Convergence Rate of the Randomized Algorithms of Stochastic Approximation in Arbitrary Noise. Automation and Remote Control 64, 252–262 (2003). https://doi.org/10.1023/A:1022263014535
Issue Date:
DOI: https://doi.org/10.1023/A:1022263014535