Abstract
The constant stepsize analog of Gelfand–Mitter type discrete-time stochastic recursive algorithms is shown to track an associated stochastic differential equation in the strong sense, i.e., with respect to an appropriate divergence measure.
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Gelfand, S. B., and Mitter, S. K., Recursive Stochastic Algorithms for Global Optimization in R N, SIAM Journal on Control and Optimization, Vol. 29, pp. 999–1018, 1991.
Gelfand, S. B., and Mitter, S. K., Metropolis-Type Annealing Algorithm for Global Optimization in R d, SIAM Journal on Control and Optimization, Vol. 31, pp. 111–131, 1993.
Borkar, V. S., Probability Theory: An Advanced Course, Springer Verlag, New York, New York, 1996.
Hirsch, M. W., Convergent Activation Dynamics in Continuous-Time Networks, Neural Networks, Vol. 2, pp. 331–349, 1989.
Liptser, R. S., and Shiryayev, A. N., Statistics of Random Processes, I: General Theory, Springer Verlag, New York, New York, 1977.
Bhattacharya, R. N., Asymptotic Behavior of Several Dimensional Diffusions, Stochastic Nonlinear Systems, Edited by L. Arnold and R. Lefever, Springer Verlag, Berlin, Germany, pp. 86–99, 1981.
Borkar, V. S., and Ghosh, M. K., Ergodic Control of Multidimensional Diffusions, II: Adaptive Control, Applied Mathematics and Optimization, Vol. 21, pp. 191–220, 1990.
Kushner, H. J., Approximation and Weak Convergence Methods for Random Processes, MIT Press, Cambridge, Massachusetts, 1984.
Meyn, S. P., and Tweedie, R. L., Markov Chains and Stochastic Stability, Springer Verlag, New York, New York, 1994.
Aronson, D. G., Bounds for the Fundamental Solution of a Parabolic Equation, Bulletin of the American Mathematical Society, Vol. 73, pp. 890–896, 1967.
Borkar, V. S., A Remark on the Attainable Distributions of Controlled Diffusions, Stochastics, Vol. 18, pp. 17–23, 1986.
GŸongi, I., Mimicking the One-Dimensional Marginal Distribution of Processes Having an Itô Differential, Probability Theory and Related Fields, Vol. 71, pp. 501–516, 1986.
Ladyzenskaja, O. A., Solonnikov, V. A., and Ural'ceva, N. N., Linear and Quasilinear Equations of Parabolic Type, Translations of Mathematical Monographs, American Mathematical Society, Providence, Rhode Island, Vol. 23, 1968.
Cover, T. M., and Thomas, J. A., Elements of Information Theory, John Wiley, New York, New York, 1991.
Deuschel, J. D., and Stroock, D., Large Deviations, Academic Press, New York, New York, 1989.
Pflug, G., Searching for the Best: Stochastic Approximation, Simulated Annealing, and Related Procedures, Identification, Adaptation, Learning, Edited by S. Bittanti and G. Picci, Springer Verlag, Berlin, Germany, pp. 514–549, 1996.
Gelfand, S. B., Personal Communication, 1996.
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Borkar, V.S., Mitter, S.K. A Strong Approximation Theorem for Stochastic Recursive Algorithms. Journal of Optimization Theory and Applications 100, 499–513 (1999). https://doi.org/10.1023/A:1022630321574
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DOI: https://doi.org/10.1023/A:1022630321574