Passive stochastic approximation. (English. Russian original) Zbl 0719.62092

Nazin, A. V.; Polyak, B. T.; Tsybakov, A. B.

Autom. Remote Control 50, No. 11, 1563-1569 (1989); translation from Avtom. Telemekh. 1989, No. 11, 127-134 (1989).

Summary: A root of the equation \(f(x)=0\) is sought in the case that the values of f(x) are measured with a random error at random points whose choice cannot be controlled. The recursive Härdle-Nixdorf method [W. K. Härdle and R. Nixdorf, IEEE Trans. Inf. Theory IT-33, 367-372 (1987; Zbl 0636.62079)] of solution of this problem is studied. Its almost surely convergence and in the mean square is proved, and the convergence rate is estimated. A technique of selection of the optimal parameters of this method is described, and it is proved that this yields a lower bound (in order of magnitude) for the accuracy of any method of solution of the problem.

MSC:

62L20	Stochastic approximation
65C99	Probabilistic methods, stochastic differential equations

Keywords:

mean square convergence; passive stochastic approximation; random error at random points; almost surely convergence; convergence rate; selection of the optimal parameters

Citations:

Zbl 0636.62079