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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 Gradient descent ： ウィキペディア英語版 Gradient descent Gradient descent is a first-order optimization algorithm. To find a local minimum of a function using gradient descent, one takes steps proportional to the ''negative'' of the gradient (or of the approximate gradient) of the function at the current point. If instead one takes steps proportional to the ''positive'' of the gradient, one approaches a local maximum of that function; the procedure is then known as gradient ascent. Gradient descent is also known as steepest descent, or the method of steepest descent. However, gradient descent should not be confused with the method of steepest descent for approximating integrals. ==Description== Gradient descent is based on the observation that if the multivariable function $F(\backslash mathbf)$ is defined and differentiable in a neighborhood of a point $\backslash mathbf$, then $F(\backslash mathbf)$ decreases ''fastest'' if one goes from $\backslash mathbf$ in the direction of the negative gradient of $F$ at $\backslash mathbf$, $-\backslash nabla\; F(\backslash mathbf)$. It follows that, if :$\backslash mathbf\; =\; \backslash mathbf-\backslash gamma\backslash nabla\; F(\backslash mathbf)$ for $\backslash gamma$ small enough, then $F(\backslash mathbf)\backslash geq\; F(\backslash mathbf)$. With this observation in mind, one starts with a guess $\backslash mathbf\_0$ for a local minimum of $F$, and considers the sequence $\backslash mathbf\_0,\; \backslash mathbf\_1,\; \backslash mathbf\_2,\; \backslash dots$ such that :$\backslash mathbf\_=\backslash mathbf\_n-\backslash gamma\_n\; \backslash nabla\; F(\backslash mathbf\_n),\backslash \; n\; \backslash ge\; 0.$ We have :$F(\backslash mathbf\_0)\backslash ge\; F(\backslash mathbf\_1)\backslash ge\; F(\backslash mathbf\_2)\backslash ge\; \backslash cdots,$ so hopefully the sequence $(\backslash mathbf\_n)$ converges to the desired local minimum. Note that the value of the ''step size'' $\backslash gamma$ is allowed to change at every iteration. With certain assumptions on the function $F$ (for example, $F$ convex and $\backslash nabla\; F$ Lipschitz) and particular choices of $\backslash gamma$ (e.g., chosen via a line search that satisfies the Wolfe conditions), convergence to a local minimum can be guaranteed. When the function $F$ is convex, all local minima are also global minima, so in this case gradient descent can converge to the global solution. This process is illustrated in the picture to the right. Here $F$ is assumed to be defined on the plane, and that its graph has a bowl shape. The blue curves are the contour lines, that is, the regions on which the value of $F$ is constant. A red arrow originating at a point shows the direction of the negative gradient at that point. Note that the (negative) gradient at a point is orthogonal to the contour line going through that point. We see that gradient ''descent'' leads us to the bottom of the bowl, that is, to the point where the value of the function $F$ is minimal. 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「Gradient descent」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.