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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) + \Psi(x), where \Psi(x) = \sum_^n \Psi_i(x^), x\in R^N is decomposed into n blocks of variables/coordinates: x = (x^,\dots,x^) and \Psi_1,\dots, \Psi_n are (simple) convex functions. Example (block decomposition): If x = (x ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convex Function
In mathematics, a real-valued function is called convex if the line segment between any two points on the graph of a function, graph of the function lies above the graph between the two points. Equivalently, a function is convex if its epigraph (mathematics), epigraph (the set of points on or above the graph of the function) is a convex set. A twice-differentiable function of a single variable is convex if and only if its second derivative is nonnegative on its entire domain. Well-known examples of convex functions of a single variable include the quadratic function x^2 and the exponential function e^x. In simple terms, a convex function refers to a function whose graph is shaped like a cup \cup, while a concave function's graph is shaped like a cap \cap. Convex functions play an important role in many areas of mathematics. They are especially important in the study of optimization problems where they are distinguished by a number of convenient properties. For instance, a st ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convergence On Small Problem
Convergence may refer to: Arts and media Literature *''Convergence'' (book series), edited by Ruth Nanda Anshen * "Convergence" (comics), two separate story lines published by DC Comics: **A four-part crossover storyline that united the four Weirdoverse titles in 1997 **A 2015 crossover storyline spanning the DC Comics Multiverse * ''Convergence'' (journal), an academic journal that covers the fields of communications and media * ''Convergence'' (novel), by Charles Sheffield * ''Convergence'' (Cherryh novel), by C. J. Cherryh Music * ''Convergence'' (Front Line Assembly album), 1988 * ''Convergence'' (David Arkenstone and David Lanz album), 1996 * ''Convergence'' (Dave Douglas album), 1999 * ''Convergence'' (Warren Wolf album), 2016 Other media * ''Convergence'' (2015 film), an American horror-thriller film * ''Convergence'' (2019 film), a British drama film *''Convergence'', a 2021 Netflix film by Orlando von Einsiedel * ''Convergence'' (Pollock), a 1952 oil painting by Jackson ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coordinate Descent
Coordinate descent is an optimization algorithm that successively minimizes along coordinate directions to find the minimum of a function. At each iteration, the algorithm determines a coordinate or coordinate block via a coordinate selection rule, then exactly or inexactly minimizes over the corresponding coordinate hyperplane while fixing all other coordinates or coordinate blocks. A line search along the coordinate direction can be performed at the current iterate to determine the appropriate step size. Coordinate descent is applicable in both differentiable and derivative-free contexts. Description Coordinate descent is based on the idea that the minimization of a multivariable function F(\mathbf) can be achieved by minimizing it along one direction at a time, i.e., solving univariate (or at least much simpler) optimization problems in a loop. In the simplest case of ''cyclic coordinate descent'', one cyclically iterates through the directions, one at a time, minimizing the objec ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gradient Descent
In mathematics, gradient descent (also often called steepest descent) is a first-order iterative optimization algorithm for finding a local minimum of a differentiable function. The idea is to take repeated steps in the opposite direction of the gradient (or approximate gradient) of the function at the current point, because this is the direction of steepest descent. Conversely, stepping in the direction of the gradient will lead to a local maximum of that function; the procedure is then known as gradient ascent. Gradient descent is generally attributed to Augustin-Louis Cauchy, who first suggested it in 1847. Jacques Hadamard independently proposed a similar method in 1907. Its convergence properties for non-linear optimization problems were first studied by Haskell Curry in 1944, with the method becoming increasingly well-studied and used in the following decades. Description Gradient descent is based on the observation that if the multi-variable function F(\mathbf) is def ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical 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 maxima and minima, maximizing or minimizing a Function of a real variable, real function by systematically choosing Argument of a function, input values from within an allowed set and computing the Value (mathematics), value of the function. The generalization of optimization theory and techniques to other formulations constitutes a large area of applied mathematics. More generally, opti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
