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Geometric structure in stochastic approximation
Dai Zhan CHENG, Hong DU , Han Fu CHEN
Acta Mathematicae Applicatae Sinica(English Series)
2001, 17 (1):
53-59.
Let J be the zero set of the gradient f_x of a function f:R~n→R.Under fairly general conditions the stochastic approximation algorithm ensures d(f(x_k),f(J))→0,as k→∞.First of all,the paper considers this problem:Under what conditions the convergence d(f(x_k),f(J))→〖DD(X〗k→∞〖DD)〗0 implies d(x_k,J)→〖DD(X〗k→∞〖DD)〗0.It is shown that such implication takes place if f_x is continuous and f(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:If f is a C~1 function and either i)J is a compact set or ii) for any bounded set B,f~(-1)(B) is bounded,then f(J) is nowhere dense.Finally,some tools in algebraic geometry are used to prove that f(J) is a finite set if f is a polynomial.Hence f(J) is nowhere dense in the polynomial case.
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