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  "body": "\n## Definition\n\nLet $M \\colon \\R \\to \\R$ be an unknown function and let $\\alpha \\in \\R$ be a target value.\nAssume that at each $\\theta \\in \\R$ one can observe a [[random variable|0DKRR22AEQ]] $N(\\theta)$ with [[expectation|0WQNRCNY3M]] $E[N(\\theta)] = M(\\theta)$.\nLet $(a_n)_{n \\ge 1}$ be positive step sizes.\nThe *Robbins–Monro algorithm* is this [[stochastic approximation|ZPCPW4W1Y9]] scheme:\n$$\\theta_{n+1} = \\theta_n - a_n \\left( N_n - \\alpha \\right), \\qquad n \\ge 1.$$\nHere $\\theta_1$ is an initial guess.\nGiven the past, $N_n$ is an observation of $N(\\theta_n)$, made independently of the earlier observations.\nIts aim is a root $\\theta^*$ of the equation $M(\\theta) = \\alpha$.\n\n## Remarks\n\nRobbins and Monro proved this convergence result in 1951.\nAssume that $N(\\theta)$ is uniformly bounded, $M$ is nondecreasing, $M'(\\theta^*)$ exists and is positive, and the step sizes satisfy $\\sum_n a_n = \\infty$ and $\\sum_n a_n^2 \u003c \\infty$.\nThen $\\theta_n \\to \\theta^*$ in $L^2$, and hence [[in probability|0QA1HBD9Z4]].\n\nExample: let $N(\\theta) = \\theta - Z$ with $Z$ uniform on $[0, 2]$, so $M(\\theta) = \\theta - 1$, and let $\\alpha = 0$ and $a_n = 1/n$.\nThe step sizes satisfy both conditions, since $\\sum 1/n = \\infty$ and $\\sum 1/n^2 \u003c \\infty$, and the root is $\\theta^* = 1$.\nWith fresh draws $Z_n$, the recursion is $\\theta_{n+1} = \\theta_n - \\frac1n (\\theta_n - Z_n)$, so $\\theta_{n+1} = (Z_1 + \\cdots + Z_n)/n$, the sample mean, which tends to $1$ by the law of large numbers.\n\nNon-example: constant step sizes $a_n = c \u003e 0$ violate $\\sum a_n^2 \u003c \\infty$; in the example, $\\theta_{n+1} = (1 - c) \\theta_n + c Z_n$ for $0 \u003c c \u003c 1$ keeps fluctuating with conditional variance $c^2 \\operatorname{Var}(Z) \u003e 0$ at each step, so it does not converge.\n",
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