Finding fixed points of functions by stochastic approximation method

Abstract

In this paper, a stochastic approximation method is used to find a fixed point of a function observed with an additive error. The result is obtained under the assumptions of Gladyshev’s theorem on root finding problem. It is also assumed that the function is either a pseudo-contraction, or relaxed contraction, or hemi-contraction, or quasi-contraction, or generalized contraction. By using techniques based on super-martingales, it is shown that a modified Robbins-Monro process converges to the fixed point with probability one. The established theorem is less restrictive than a prior result by S. V. Komarov since this theorem imposes no special requirements to the studied function. Refs 4.