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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 Proximal gradient methods for learning ： ウィキペディア英語版 Proximal gradient methods for learning Proximal gradient (forward backward splitting) methods for learning is an area of research in optimization and statistical learning theory which studies algorithms for a general class of convex regularization problems where the regularization penalty may not be differentiable. One such example is $\backslash ell\_1$ regularization (also known as Lasso) of the form :$\backslash min\_\; \backslash frac\backslash sum\_^n\; (y\_i-\; \backslash langle\; w,x\_i\backslash rangle)^2+\; \backslash lambda\; \backslash |w\backslash |\_1,\; \backslash quad\; \backslash text\; x\_i\backslash in\; \backslash mathbb^d\backslash text\; y\_i\backslash in\backslash mathbb.$ Proximal gradient methods offer a general framework for solving regularization problems from statistical learning theory with penalties that are tailored to a specific problem application. Such customized penalties can help to induce certain structure in problem solutions, such as ''sparsity'' (in the case of lasso) or ''group structure'' (in the case of group lasso). == Relevant background == Proximal gradient methods are applicable in a wide variety of scenarios for solving convex optimization problems of the form :$\backslash min\_$ is some set, typically a Hilbert space. The usual criterion of $x$ minimizes $F(x)+R(x)$ if and only if $\backslash nabla\; (F+R)(x)\; =\; 0$ in the convex, differentiable setting is now replaced by :$0\backslash in\; \backslash partial\; (F+R)(x),$ where $\backslash partial\; \backslash varphi$ denotes the subdifferential of a real-valued, convex function $\backslash varphi$. Given a convex function $\backslash varphi:\backslash mathcal\; \backslash to\; \backslash mathbb$ an important operator to consider is its proximity operator $\backslash operatorname\_:\backslash mathcal\backslash to\backslash mathcal$ defined by :$\backslash operatorname\_(u)\; =\; \backslash operatorname\backslash min\_\backslash |u-x\backslash |\_2^2,$ which is well-defined because of the strict convexity of the $\backslash ell\_2$ norm. The proximity operator can be seen as a generalization of a projection.〔〔 We see that the proximity operator is important because $x^$ * is a minimizer to the problem $\backslash min\_\_\backslash left(x^$ *-\gamma\nabla F(x^ *)\right), where $\backslash gamma>0$ is any positive real number.〔 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「Proximal gradient methods for learning」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.