## The Annals of Mathematical Statistics

- Ann. Math. Statist.
- Volume 22, Number 3 (1951), 400-407.

### A Stochastic Approximation Method

Herbert Robbins and Sutton Monro

#### Abstract

Let $M(x)$ denote the expected value at level $x$ of the response to a certain experiment. $M(x)$ is assumed to be a monotone function of $x$ but is unknown to the experimenter, and it is desired to find the solution $x = \theta$ of the equation $M(x) = \alpha$, where $\alpha$ is a given constant. We give a method for making successive experiments at levels $x_1,x_2,\cdots$ in such a way that $x_n$ will tend to $\theta$ in probability.

#### Article information

**Source**

Ann. Math. Statist. Volume 22, Number 3 (1951), 400-407.

**Dates**

First available in Project Euclid: 28 April 2007

**Permanent link to this document**

http://projecteuclid.org/euclid.aoms/1177729586

**Digital Object Identifier**

doi:10.1214/aoms/1177729586

**Mathematical Reviews number (MathSciNet)**

MR42668

**Zentralblatt MATH identifier**

0054.05901

**JSTOR**

links.jstor.org

#### Citation

Robbins, Herbert; Monro, Sutton. A Stochastic Approximation Method. Ann. Math. Statist. 22 (1951), no. 3, 400--407. doi:10.1214/aoms/1177729586. http://projecteuclid.org/euclid.aoms/1177729586.