On the Stochastic Approximation Method of Robbins and Monro

Abstract

In their interesting and pioneering paper Robbins and Monro [1] give a method for "solving stochastically" the equation in $x: M(x) = \alpha$, where $M(x)$ is the (unknown) expected value at level $x$ of the response to a certain experiment. They raise the question whether their results, which are contained in their Theorems 1 and 2, are valid under a condition (their condition (4'), our condition (1) below) which is statistically plausible and is weaker than the condition which they require to prove their results. In the present paper this question is answered in the affirmative. They also ask whether their conditions (33), (34), and (35) (our conditions (25), (26) and (27) below) can be replaced by their condition (5") (our condition (28) below). A counterexample shows that this is impossible. However, it is possible to weaken conditions (25), (26) and (27) by replacing them by condition (3) (abc) below. Thus our results generalize those of [1]. The statistical significance of these results is described in [1].