# Geometric structure in stochastic approximation

Geometric structure in stochastic approximation LetJ be the zero set of the gradientfx of a functionf∶Rn→R. Under fairly general conditions the stochastic approximation algorithm ensuresd(f(xk), f(J))→0, ask→∞. First of all, the paper considers this problem: Under what conditions the convergence\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$d(f(x_k ),f(J))\mathop \to \limits_{k \to \infty } 0$$\end{document} implies\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$d(x_k ,J)\mathop \to \limits_{k \to \infty } 0$$\end{document}. It is shown that such implication takes place iffx is continuous andf(J) is nowhere dense. Secondly, an intensified version of Sard's theorem has been proved, which itself is interesting. As a particular case, it provides two independent sufficient conditions as answers to the previous question: Iff is aC1 function and either i)J is a compact set or ii) for any bounded setB, f−1(B) is bounded, thef(J) is nowhere dense. Finally, some tools in algebraic geometry are used to prove thatf(J) is a finite set iff is a polynomial. Hencef(J) is nowhere dense in the polynomial case. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png "Acta Mathematicae Applicatae Sinica, English Series" Springer Journals

# Geometric structure in stochastic approximation

, Volume 17 (1) – Jan 1, 2001
7 pages

/lp/springer-journals/geometric-structure-in-stochastic-approximation-mYFTMrVM0q
Publisher
Springer Journals
ISSN
0168-9673
eISSN
1618-3932
DOI
10.1007/bf02669684
Publisher site
See Article on Publisher Site

### Abstract

LetJ be the zero set of the gradientfx of a functionf∶Rn→R. Under fairly general conditions the stochastic approximation algorithm ensuresd(f(xk), f(J))→0, ask→∞. First of all, the paper considers this problem: Under what conditions the convergence\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$d(f(x_k ),f(J))\mathop \to \limits_{k \to \infty } 0$$\end{document} implies\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$d(x_k ,J)\mathop \to \limits_{k \to \infty } 0$$\end{document}. It is shown that such implication takes place iffx is continuous andf(J) is nowhere dense. Secondly, an intensified version of Sard's theorem has been proved, which itself is interesting. As a particular case, it provides two independent sufficient conditions as answers to the previous question: Iff is aC1 function and either i)J is a compact set or ii) for any bounded setB, f−1(B) is bounded, thef(J) is nowhere dense. Finally, some tools in algebraic geometry are used to prove thatf(J) is a finite set iff is a polynomial. Hencef(J) is nowhere dense in the polynomial case.

### Journal

"Acta Mathematicae Applicatae Sinica, English Series"Springer Journals

Published: Jan 1, 2001

Keywords: Stochastic approximation; regular value; intensified Sard’s theorem; irreducible algebraic variety

### References

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