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Adaptive Coordinate Descent
Adaptive coordinate descent is an improvement of the coordinate descent algorithm to non-separable Mathematical optimization, optimization by the use of adaptive encoding. The adaptive coordinate descent approach gradually builds a transformation of the coordinate system such that the new coordinates are as decorrelated as possible with respect to the objective function. The adaptive coordinate descent was shown to be competitive to the state-of-the-art evolutionary algorithms and has the following invariance properties: # Invariance with respect to monotonous transformations of the function (scaling) # Invariance with respect to orthogonal transformations of the search space (rotation). CMA-ES, CMA-like Adaptive Encoding Update (b) mostly based on principal component analysis (a) is used to extend the coordinate descent method (c) to the optimization of non-separable problems (d). The adaptation of an appropriate coordinate system allows adaptive coordinate descent to outperform ... [...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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algorithm
In mathematics and computer science, an algorithm () is a finite sequence of Rigour#Mathematics, mathematically rigorous instructions, typically used to solve a class of specific Computational problem, problems or to perform a computation. Algorithms are used as specifications for performing calculations and data processing. More advanced algorithms can use Conditional (computer programming), conditionals to divert the code execution through various routes (referred to as automated decision-making) and deduce valid inferences (referred to as automated reasoning). In contrast, a Heuristic (computer science), heuristic is an approach to solving problems without well-defined correct or optimal results.David A. Grossman, Ophir Frieder, ''Information Retrieval: Algorithms and Heuristics'', 2nd edition, 2004, For example, although social media recommender systems are commonly called "algorithms", they actually rely on heuristics as there is no truly "correct" recommendation. As an e ... [...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 criteria, from some set of available alternatives. It is generally divided into two subfields: discrete optimization and continuous optimization. Optimization problems 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. Optimization problems Opti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Adaptive Encoding
Adaptation, in biology, is the process or trait by which organisms or population better match their environment Adaptation may also refer to: Arts * Adaptation (arts), a transfer of a work of art from one medium to another ** Film adaptation, a story from another work, adapted into a film ** Literary adaptation, a story from a literary source, adapted into another work ** Novelization, the adaptation of another work into a novel ** Theatrical adaptation, a story from another work, adapted into a play * ''Adaptation'' (film), a 2002 film by Spike Jonze * "Adaptation" (''The Walking Dead''), a television episode *''Adaptation'', a 2012 novel by Malinda Lo *"Adaptation", a song by the Weekend from his 2013 album ''Kiss Land'' Biology and medicine * Adaptation (eye), the eye's adjustment to light ** Chromatic adaptation, visual systems' adjustments to changes in illumination for preservation of colors ** Prism adaptation, sensory-motor adjustments after the visual field has been ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Parallel Problem Solving From Nature
Parallel Problem Solving from Nature, or PPSN, is a research conference focusing on the topic of natural computing. Other conferences in the area include the ACM Genetic and Evolutionary Computation Conference (GECCO), the IEEE Congress on Evolutionary Computation (CEC) and EvoStar (Evo*). In 2020 PPSN got a CORE rank of A, corresponding to an ''"excellent conference, and highly respected in a discipline area"''. History The idea behind PPSN emerged around 1989-1990 when Bernard Manderick, Reinhard Männer, Heinz Mühlenbein, and Hans-Paul Schwefel, realised they shared a common field of study that was not covered by the conferences on Operations Research, Physics, or Computer Science they attended regularly. The field of Genetic Algorithms had already been established in the form of the ICGA conference in 1985, but the "fathers" of PPSN wanted a wider focus, with algorithms that included problem solving, parallel computing and the use of natural metaphors (such as Darwin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Evolutionary Algorithms
Evolutionary algorithms (EA) reproduce essential elements of the biological evolution in a computer algorithm in order to solve "difficult" problems, at least Approximation, approximately, for which no exact or satisfactory solution methods are known. They belong to the class of Metaheuristic, metaheuristics and are a subset of Population Based Bio-Inspired Algorithms, population based bio-inspired algorithms and evolutionary computation, which itself are part of the field of computational intelligence. The mechanisms of biological evolution that an EA mainly imitates are reproduction, mutation, genetic recombination, recombination and natural selection, selection. Candidate solutions to the optimization problem play the role of individuals in a population, and the fitness function determines the quality of the solutions (see also loss function). Evolution of the population then takes place after the repeated application of the above operators. Evolutionary algorithms often perfor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orthogonal Transformation
In linear algebra, an orthogonal transformation is a linear transformation ''T'' : ''V'' → ''V'' on a real inner product space ''V'', that preserves the inner product. That is, for each pair of elements of ''V'', we have : \langle u,v \rangle = \langle Tu,Tv \rangle \, . Since the lengths of vectors and the angles between them are defined through the inner product, orthogonal transformations preserve lengths of vectors and angles between them. In particular, orthogonal transformations map orthonormal bases to orthonormal bases. Orthogonal transformations are injective: if Tv = 0 then 0 = \langle Tv,Tv \rangle = \langle v,v \rangle, hence v = 0, so the kernel of T is trivial. Orthogonal transformations in two- or three- dimensional Euclidean space are stiff rotations, reflections, or combinations of a rotation and a reflection (also known as improper rotations). Reflections are transformations that reverse the direction front to back, orthogonal to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
CMA-ES
Covariance matrix adaptation evolution strategy (CMA-ES) is a particular kind of strategy for numerical optimization. evolution strategy, Evolution strategies (ES) are stochastic, Derivative-free optimization, derivative-free methods for numerical optimization of non-Linear map, linear or non-Convex function, convex continuous optimization problems. They belong to the class of evolutionary algorithms and evolutionary computation. An evolutionary algorithm is broadly based on the principle of biological evolution, namely the repeated interplay of variation (via recombination and mutation) and selection: in each generation (iteration) new individuals (candidate solutions, denoted as x) are generated by variation of the current parental individuals, usually in a stochastic way. Then, some individuals are selected to become the parents in the next generation based on their fitness or objective function value f(x). Like this, individuals with better and better f-values are generated ove ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Principal Component Analysis
Principal component analysis (PCA) is a linear dimensionality reduction technique with applications in exploratory data analysis, visualization and data preprocessing. The data is linearly transformed onto a new coordinate system such that the directions (principal components) capturing the largest variation in the data can be easily identified. The principal components of a collection of points in a real coordinate space are a sequence of p unit vectors, where the i-th vector is the direction of a line that best fits the data while being orthogonal to the first i-1 vectors. Here, a best-fitting line is defined as one that minimizes the average squared perpendicular distance from the points to the line. These directions (i.e., principal components) constitute an orthonormal basis in which different individual dimensions of the data are linearly uncorrelated. Many studies use the first two principal components in order to plot the data in two dimensions and to visually identi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Adaptive Coordinate Descent Illustration
Adaptation, in biology, is the process or trait by which organisms or population better match their environment Adaptation may also refer to: Arts * Adaptation (arts), a transfer of a work of art from one medium to another ** Film adaptation, a story from another work, adapted into a film ** Literary adaptation, a story from a literary source, adapted into another work ** Novelization, the adaptation of another work into a novel ** Theatrical adaptation, a story from another work, adapted into a play * ''Adaptation'' (film), a 2002 film by Spike Jonze * "Adaptation" (''The Walking Dead''), a television episode *''Adaptation'', a 2012 novel by Malinda Lo *"Adaptation", a song by the Weekend from his 2013 album ''Kiss Land'' Biology and medicine * Adaptation (eye), the eye's adjustment to light ** Chromatic adaptation, visual systems' adjustments to changes in illumination for preservation of colors ** Prism adaptation, sensory-motor adjustments after the visual field has been ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rosenbrock Function
In mathematical optimization, the Rosenbrock function is a non-convex function, introduced by Howard H. Rosenbrock in 1960, which is used as a performance test problem for optimization algorithms. It is also known as Rosenbrock's valley or Rosenbrock's banana function. The global minimum is inside a long, narrow, parabolic-shaped flat valley. To find the valley is trivial. To converge to the global minimum, however, is difficult. The function is defined by :f(x,y) = (a-x)^2 + b(y-x^2)^2 It has a global minimum at (x,y)=(a,a^2), where f(x,y)=0. Usually, these parameters are set such that a = 1 and b = 100. Only in the trivial case where a=0 the function is symmetric and the minimum is at the origin. Multidimensional generalizations Two variants are commonly encountered. One is the sum of N/2 uncoupled 2D Rosenbrock problems, and is defined only for even Ns: : f(\mathbf) = f(x_1, x_2, \dots, x_N) = \sum_^ \left 00(x_^2 - x_)^2 + (x_ - 1)^2 \right This variant has predi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rosenbrock2D
Rosenbrock is a surname. Notable people with the surname include: * Eddie Rosenbrock (1908–1978), Australian rules footballer *Howard Harry Rosenbrock Howard Harry Rosenbrock (16 December 1920 – 21 October 2010) was a leading figure in control theory and control engineering. He was born in Ilford, England in 1920, graduated in 1941 from University College London with a 1st class honors degree ... (1920–2010), English control theorist and engineer * Peter Rosenbrock (1939–2005), Australian rules footballer {{surname ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
