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 implies. 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.