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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Stochastic approximation ： ウィキペディア英語版 Stochastic approximation Stochastic approximation methods are a family of iterative stochastic optimization algorithms that attempt to find zeroes or extrema of functions which cannot be computed directly, but only estimated via noisy observations. Mathematically, this refers to solving: :$\backslash min\_\backslash ;\; f(x)\; =\; \backslash min\_\backslash mathbb\; E()$ where the objective is to find the parameter $x\; \backslash in\; \backslash Theta$, which minimizes $f(x)$ for some unknown random variable, $\backslash xi$. Denoting $d$ as the dimension of the parameter $x$, we can assume that while the domain $\backslash Theta\; \backslash subset\; \backslash mathbb\; R^d$ is known, the objective function, $f(x)$, cannot be computed exactly, but instead approximated via simulation. This can be intuitively explained as follows. $f(x)$ is the original function we want to minimize. However, due to noise, $f(x)$ can not be evaluated exactly. This situation is modeled by the function $F(x,\backslash xi)$, where $\backslash xi$ represents the noise and is a random variable. Since $\backslash xi$ is a random variable, so is the value of $F(x,\backslash xi)$. The objective is then to minimize $f(x)$, but through evaluating $F(x,\backslash xi)$. A reasonable way to do this is to minimize the expectation of $F(x,\backslash xi)$, i.e., $\backslash mathbb\; E()$. The first, and prototypical, algorithms of this kind are the Robbins-Monro and Kiefer-Wolfowitz algorithms introduced respectively in 1951 and 1952. ==Robbins–Monro algorithm== The Robbins–Monro algorithm, introduced in 1951 by Herbert Robbins and Sutton Monro, presented a methodology for solving a root finding problem, where the function is represented as an expected value. Assume that we have a function $M(x)$, and a constant $\backslash alpha$, such that the equation $M(x)\; =\; \backslash alpha$ has a unique root at $x=\backslash theta$. It is assumed that while we cannot directly observe the function $M(x)$, we can instead obtain measurements of the random variable $N(x)$ where $\backslash mathbb\; E()\; =\; M(x)$. The structure of the algorithm is to then generate iterates of the form: ::$x\_-x\_n=a\_n(\backslash alpha-N(x\_n))$ Here, $a\_1,\; a\_2,\; \backslash dots$ is a sequence of positive step sizes. Robbins and Monro proved 〔, Theorem 2 that $x\_n$ converges in $L^2$ (and hence also in probability) to $\backslash theta$ provided that: * $N(x)$ is uniformly bounded, * $M(x)$ is nondecreasing, * $M\text{'}(\backslash theta)$ exists and is positive, and * The sequence $a\_n$ satisfies the following requirements: :: A particular sequence of steps which satisfy these conditions, and was suggested by Robbins–Monro, have the form: $a\_n=a/n$, for $a\; >\; 0$. Other series are possible but in order to average out the noise in $N(x)$, the above condition must be met. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Stochastic approximation」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.