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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 \ell_1 regularization (also known as Lasso) of the form :\min_ \frac\sum_^n (y_i- \langle w,x_i\rangle)^2+ \lambda \, w\, _1, \quad \text x_i\in \mathbb^d\text y_i\in\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 : \ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Optimization
Mathematical optimization (alternatively spelled ''optimisation'') or mathematical programming is the selection of a best element, with regard to some criterion, from some set of available alternatives. It is generally divided into two subfields: discrete optimization and continuous optimization. Optimization problems of sorts arise in all quantitative disciplines from computer science and engineering to operations research and economics, and the development of solution methods has been of interest in mathematics for centuries. In the more general approach, an optimization problem consists of maximizing or minimizing a real function by systematically choosing input values from within an allowed set and computing the value of the function. The generalization of optimization theory and techniques to other formulations constitutes a large area of applied mathematics. More generally, optimization includes finding "best available" values of some objective function given a def ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Convex Conjugate
In mathematics and mathematical optimization, the convex conjugate of a function is a generalization of the Legendre transformation which applies to non-convex functions. It is also known as Legendre–Fenchel transformation, Fenchel transformation, or Fenchel conjugate (after Adrien-Marie Legendre and Werner Fenchel). It allows in particular for a far reaching generalization of Lagrangian duality. Definition Let X be a real topological vector space and let X^ be the dual space to X. Denote by :\langle \cdot , \cdot \rangle : X^ \times X \to \mathbb the canonical dual pairing, which is defined by \left( x^*, x \right) \mapsto x^* (x). For a function f : X \to \mathbb \cup \ taking values on the extended real number line, its is the function :f^ : X^ \to \mathbb \cup \ whose value at x^* \in X^ is defined to be the supremum: :f^ \left( x^ \right) := \sup \left\, or, equivalently, in terms of the infimum: :f^ \left( x^ \right) := - \inf \left\. This definition can ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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First Order Methods
First or 1st is the ordinal form of the number one (#1). First or 1st may also refer to: *World record, specifically the first instance of a particular achievement Arts and media Music * 1$T, American rapper, singer-songwriter, DJ, and record producer Albums * ''1st'' (album), a 1983 album by Streets * ''1st'' (Rasmus EP), a 1995 EP by The Rasmus, frequently identified as a single * '' 1ST'', a 2021 album by SixTones * ''First'' (Baroness EP), an EP by Baroness * ''First'' (Ferlyn G EP), an EP by Ferlyn G * ''First'' (David Gates album), an album by David Gates * ''First'' (O'Bryan album), an album by O'Bryan * ''First'' (Raymond Lam album), an album by Raymond Lam * ''First'', an album by Denise Ho Songs * "First" (Cold War Kids song), a song by Cold War Kids * "First" (Lindsay Lohan song), a song by Lindsay Lohan * "First", a song by Everglow from ''Last Melody'' * "First", a song by Lauren Daigle * "First", a song by Niki & Gabi * "First", a song by Jonas Brot ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Statistical Learning Theory
Statistical learning theory is a framework for machine learning drawing from the fields of statistics and functional analysis. Statistical learning theory deals with the statistical inference problem of finding a predictive function based on data. Statistical learning theory has led to successful applications in fields such as computer vision, speech recognition, and bioinformatics. Introduction The goals of learning are understanding and prediction. Learning falls into many categories, including supervised learning, unsupervised learning, online learning, and reinforcement learning. From the perspective of statistical learning theory, supervised learning is best understood. Supervised learning involves learning from a training set of data. Every point in the training is an input-output pair, where the input maps to an output. The learning problem consists of inferring the function that maps between the input and the output, such that the learned function can be used to pre ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Convex Analysis
Convex analysis is the branch of mathematics devoted to the study of properties of convex functions and convex sets, often with applications in convex minimization, a subdomain of optimization theory. Convex sets A subset C \subseteq X of some vector space X is if it satisfies any of the following equivalent conditions: #If 0 \leq r \leq 1 is real and x, y \in C then r x + (1 - r) y \in C. #If 0 is a if holds for any real 0 is called if \operatorname f \neq \varnothing and f(x) > -\infty for x \in \operatorname f. Alternatively, this means that there exists some x in the domain of f at which f(x) \in \mathbb and f is also equal to -\infty. In words, a function is if its domain is not empty, it never takes on the value -\infty, and it also is not identically equal to +\infty. If f : \mathbb^n \to \infty, \infty/math> is a proper convex function then there exist some vector b \in \mathbb^n and some r \in \mathbb such that :f(x) \geq x \cdot b - r for every x whe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Directed Acyclic Graph In mathematics, particularly |