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  "body": "\n## Summary\n\nThe Robbins–Monro algorithm is the first [[stochastic approximation|ZPCPW4W1Y9]] method, introduced by Herbert Robbins and Sutton Monro in 1951.\nIt finds the point where an increasing function, observed only through noisy measurements, takes a given value.\nIt updates the current guess by a step proportional to the observed error, with step sizes that shrink at a suitable rate.\n\n## Explanations\n\n### L1 human\n\nSuppose the dose of a drug sets the average response, but each patient responds a bit differently.\nTo find the dose whose average response equals a target, give a dose and observe the response.\nThen lower the dose if the response was too high, or raise it if it was too low.\nThe corrections shrink like $1/n$, so random errors cancel out over time.\nWith target $\\alpha$, dose $\\theta_n$ and observed response $N_n$, the update is $\\theta_{n+1} = \\theta_n - \\frac{c}{n} (N_n - \\alpha)$ for a constant $c \u003e 0$.\n",
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