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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 Random coordinate descent ： ウィキペディア英語版 Random coordinate descent Randomized (Block) Coordinate Descent Method is an optimization algorithm popularized by Nesterov (2010) and Richtárik and Takáč (2011). The first analysis of this method, when applied to the problem of minimizing a smooth convex function, was performed by Nesterov (2010). In Nesterov's analysis the method needs to be applied to a quadratic perturbation of the original function with an unknown scaling factor. Richtárik and Takáč (2011) give iteration complexity bounds which do not require this, i.e., the method is applied to the objective function directly. Furthermore, they generalize the setting to the problem of minimizing a composite function, i.e., sum of a smooth convex and a (possibly nonsmooth) convex block-separable function: $F(x)\; =\; f(x)\; +\; \backslash Psi(x),$ where $\backslash Psi(x)\; =\; \backslash sum\_^n\; \backslash Psi\_i(x^),$ $x\backslash in\; R^N$ is decomposed into $n$ blocks of variables/coordinates: $x\; =\; (x^,\backslash dots,x^)$ and $\backslash Psi\_1,\backslash dots,\; \backslash Psi\_n$ are (simple) convex functions. Example (block decomposition): If $x\; =\; (x\_1,x\_2,\backslash dots,x\_5)\; \backslash in\; R^5$ and $n\; =\; 3$, one may choose $x^\; =\; (x\_1,x\_3),\; x^\; =\; (x\_2,x\_5)$ and $x^\; =\; x\_4$. Example (block-separable regularizers): # $n=N;\; \backslash Psi(x)\; =\; \backslash |x\backslash |\_1\; =\; \backslash sum\_^n\; |x\_i|$ # $N\; =\; N\_1\; +\; N\_2\; +\; \backslash dots\; +\; N\_n;\; \backslash Psi(x)\; =\; \backslash sum\_^n\; \backslash |x^\backslash |\_2$, where $x^\backslash in\; R^$ and $\backslash |\backslash cdot\backslash |\_2$ is the standard Euclidean norm. ==Algorithm== Consider the optimization problem :$\backslash min\_\; f(x),$ where $f$ is a convex and smooth function. Smoothness: By smoothness we mean the following: we assume the gradient of $f$ is coordinate-wise Lipschitz continuous with constants $L\_1,\; L\_2,\; \backslash dots\; ,\; L\_n$. That is, we assume that :$|\backslash nabla\_i\; f(x\; +\; h\; e\_i)\; -\; \backslash nabla\_i\; f(x)|\; \backslash leq\; L\_i\; |h|,$ for all $x\; \backslash in\; R^n$ and $h\; \backslash in\; R$, where $\backslash nabla\_i$ denotes the partial derivative with respect to variable $x^$. Nesterov, and Richtarik and Takac showed that the following algorithm converges to the optimal point: Input: $x\_0\; \backslash in\; R^n$ //starting point Output: $x$ set x=x_0 for k=1,... do choose coordinate $i\backslash in\; \backslash$, uniformly at random update $x^\; =\; x^\; -\; \backslash frac1\; \backslash nabla\; f\_i(x)$ endfor; 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「Random coordinate 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.