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Equivalent necessary and sufficient conditions on noise sequences for stochastic approximation algorithms

Part of: Numerical approximation and computational geometry Sequential methods

Published online by Cambridge University Press:  01 July 2016

I-Jeng Wang ,
Edwin K. P. Chong  and
Sanjeev R. Kulkarni
I-Jeng Wang*
Affiliation:
Purdue University
Edwin K. P. Chong*
Affiliation:
Purdue University
Sanjeev R. Kulkarni*
Affiliation:
Purdue University
*
∗ Postal address: School of Electrical and Computer Engineering, Purdue University, West Lafayette, IN 47907-1285, USA.
∗ Postal address: School of Electrical and Computer Engineering, Purdue University, West Lafayette, IN 47907-1285, USA.
∗∗ Postal address: Department of Electrical Engineering, Princeton University, Princeton, NJ 08544, USA.
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Abstract

We consider stochastic approximation algorithms on a general Hilbert space, and study four conditions on noise sequences for their analysis: Kushner and Clark's condition, Chen's condition, a decomposition condition, and Kulkarni and Horn's condition. We discuss various properties of these conditions. In our main result we show that the four conditions are all equivalent, and are both necessary and sufficient for convergence of stochastic approximation algorithms under appropriate assumptions.

Keywords

STOCHASTIC APPROXIMATIONCONVERGENCE: EQUIVALENT NECESSARY AND SUFFICIENT CONDITIONSNOISE SEQUENCES

MSC classification

Primary: 62L20: Stochastic approximation
Secondary: 62L12: Sequential estimation 65D99: None of the above, but in this section
Type
General Applied Probablity
Information
Advances in Applied Probability , Volume 28 , Issue 3 , September 1996 , pp. 784 - 801
Copyright
Copyright © Applied Probability Trust 1996 

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Footnotes

This research was supported in part by a Purdue Research Foundation Fellowship, and by the National Science Foundation through grant ECS-9410313.

This research was supported in part by the National Science Foundation under grant IRI-9457645 and by the Army Research Office under grant DAAL03-92-G-0320.

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