Summary
Stochastic approximation methods, e.g. stochastic gradient procedures, can be accelerated considerably by using deterministic (feasible) descent directions or more exact gradient estimations at certain iteration points. A further improvement of the convergence behavior is attained by an adaptive selection of the step sizes. Given a certain optimality criterion, a recursive procedure for the computation of optimal step sizes is derived. Several properties of the optimal step sizes are shown, and the convergence rate of the resulting optimal semi-stochastic approximation algorithm is calculated. Furthermore, the initial behavior of the optimal procedure is examined.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Block H D (1957) Estimates of error for two modifications of the Robbins-Monro stochastic approximation process. Ann Math Statist 28:1003–1010
Dvoretzky A (1956) On stochastic approximation. Proceedings 3rd Berkeley Symposium on Mathematical Statistics and Probability, Vol. 1, University of California Press, pp 39–55
Ermoliev Y, Gaivoronski A (1983) Stochastic Quasigradient Methods and their Implementation. IIASA Working Paper, Laxenburg
Markl J (1985) Recursive Estimation as an Optimally Controlled Process. Kybernetika 21:272–286
Marti K (1980) Solving stochastic linear programs by semi-stochastic approximation algorithms. In: Kall P, Prekopa A (eds) Recent results in stochastic programming. Lecture Notes in Economics and Mathematical Systems, Vol 179:191–213, Springer, Berlin Heidelberg New York
Marti K (1980) Random search in optimization as a stochastic decision process (adaptive random search). Methods of Operations Research 36:223–234
Marti K (1985) Computation of descent directions in stochastic optimization problems with invariant distributions. ZAMM 65:355–378
Marti K (1986) On accelerations of stochastic gradient methods by using more exact gradient estimations. Methods of Operations Research 53:327–336
Marti K (1986) Computation of Descent Directions and Stationary (Efficient) Points in Stochastic Optimization Problems Having Discrete Distributions. Mitteilungen aus dem Forschungsschwerpunkt Simulation und Optimierung deterministischer und stocha-stischer dynamischer Systeme, UNIBw München
Marti K, Fuchs E (1986) Rates of Convergence of Semi-Stochastic Approximation Procedures for Solving Stochastic Optimization Problems. Optimization 17:243–265
Tsypkin Y Z (1973) Foundations of the Theory of Learning Systems. Academic Press, New York, London
Wasan M T (1969) Stochastic Approximation. Cambridge University Press
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1987 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Marti, K. (1987). Optimally Controlled Semi-Stochastic Approximation Procedures. In: Opitz, O., Rauhut, B. (eds) Ökonomie und Mathematik. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-72672-9_21
Download citation
DOI: https://doi.org/10.1007/978-3-642-72672-9_21
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-72673-6
Online ISBN: 978-3-642-72672-9
eBook Packages: Springer Book Archive