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AdaBoost
AdaBoost, short for ''Adaptive Boosting'', is a statistical classification meta-algorithm formulated by Yoav Freund and Robert Schapire in 1995, who won the 2003 Gödel Prize for their work. It can be used in conjunction with many other types of learning algorithms to improve performance. The output of the other learning algorithms ('weak learners') is combined into a weighted sum that represents the final output of the boosted classifier. Usually, AdaBoost is presented for binary classification, although it can be generalized to multiple classes or bounded intervals on the real line. AdaBoost is adaptive in the sense that subsequent weak learners are tweaked in favor of those instances misclassified by previous classifiers. In some problems it can be less susceptible to the overfitting problem than other learning algorithms. The individual learners can be weak, but as long as the performance of each one is slightly better than random guessing, the final model can be proven to c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Boosting (meta-algorithm)
In machine learning, boosting is an ensemble meta-algorithm for primarily reducing bias, and also variance in supervised learning, and a family of machine learning algorithms that convert weak learners to strong ones. Boosting is based on the question posed by Kearns and Valiant (1988, 1989):Michael Kearns(1988)''Thoughts on Hypothesis Boosting'' Unpublished manuscript (Machine Learning class project, December 1988) "Can a set of weak learners create a single strong learner?" A weak learner is defined to be a classifier that is only slightly correlated with the true classification (it can label examples better than random guessing). In contrast, a strong learner is a classifier that is arbitrarily well-correlated with the true classification. Robert Schapire's affirmative answer in a 1990 paper to the question of Kearns and Valiant has had significant ramifications in machine learning and statistics, most notably leading to the development of boosting. When first introduced, ... [...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] |
CoBoosting
CoBoost is a semi-supervised training algorithm proposed by Collins and Singer in 1999. The original application for the algorithm was the task of Named Entity Classification using very weak learners.Michael Collins and Yoram Singer, Unsupervised Models for Named Entity Classification. Proceedings of the 1999 Joint SIGDAT Conference on Empirical Methods in Natural Language Processing and Very Large Corpora, pp. 100-110, 1999. It can be used for performing semi-supervised learning in cases in which there exist redundancy in features. It may be seen as a combination of co-training and boosting. Each example is available in two views (subsections of the feature set), and boosting is applied iteratively in alternation with each view using predicted labels produced in the alternate view on the previous iteration. CoBoosting is not a valid boosting algorithm in the PAC learning sense. Motivation CoBoosting was an attempt by Collins and Singer to improve on previous attempts to leverag ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bootstrap Aggregating
Bootstrap aggregating, also called bagging (from bootstrap aggregating), is a machine learning ensemble meta-algorithm designed to improve the stability and accuracy of machine learning algorithms used in statistical classification and regression. It also reduces variance and helps to avoid overfitting. Although it is usually applied to decision tree methods, it can be used with any type of method. Bagging is a special case of the model averaging approach. Description of the technique Given a standard training set D of size ''n'', bagging generates ''m'' new training sets D_i, each of size ''n′'', by sampling from ''D'' uniformly and with replacement. By sampling with replacement, some observations may be repeated in each D_i. If ''n ′''=''n'', then for large ''n'' the set D_i is expected to have the fraction (1 - 1/'' e'') (≈63.2%) of the unique examples of ''D'', the rest being duplicates. This kind of sample is known as a bootstrap sample. Sampling with replacement ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Programming
Linear programming (LP), also called linear optimization, is a method to achieve the best outcome (such as maximum profit or lowest cost) in a mathematical model whose requirements are represented by linear function#As a polynomial function, linear relationships. Linear programming is a special case of mathematical programming (also known as mathematical optimization). More formally, linear programming is a technique for the mathematical optimization, optimization of a linear objective function, subject to linear equality and linear inequality Constraint (mathematics), constraints. Its feasible region is a convex polytope, which is a set defined as the intersection (mathematics), intersection of finitely many Half-space (geometry), half spaces, each of which is defined by a linear inequality. Its objective function is a real number, real-valued affine function, affine (linear) function defined on this polyhedron. A linear programming algorithm finds a point in the polytope where ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
LPBoost
Linear Programming Boosting (LPBoost) is a supervised classifier from the boosting family of classifiers. LPBoost maximizes a ''margin'' between training samples of different classes and hence also belongs to the class of margin-maximizing supervised classification algorithms. Consider a classification function : f: \mathcal \to \, which classifies samples from a space \mathcal into one of two classes, labelled 1 and -1, respectively. LPBoost is an algorithm to ''learn'' such a classification function given a set of training examples with known class labels. LPBoost is a machine learning technique and especially suited for applications of joint classification and feature selection in structured domains. LPBoost overview As in all boosting classifiers, the final classification function is of the form :f(\boldsymbol) = \sum_^ \alpha_j h_j(\boldsymbol), where \alpha_j are non-negative weightings for ''weak'' classifiers h_j: \mathcal \to \. Each individual weak classifier h ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Overfitting
mathematical modeling, overfitting is "the production of an analysis that corresponds too closely or exactly to a particular set of data, and may therefore fail to fit to additional data or predict future observations reliably". An overfitted model is a mathematical model that contains more parameters than can be justified by the data. The essence of overfitting is to have unknowingly extracted some of the residual variation (i.e., the noise) as if that variation represented underlying model structure. Underfitting occurs when a mathematical model cannot adequately capture the underlying structure of the data. An under-fitted model is a model where some parameters or terms that would appear in a correctly specified model are missing. Under-fitting would occur, for example, when fitting a linear model to non-linear data. Such a model will tend to have poor predictive performance. The possibility of over-fitting exists because the criterion used for selecting the model is no ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Early Stopping
In machine learning, early stopping is a form of regularization used to avoid overfitting when training a learner with an iterative method, such as gradient descent. Such methods update the learner so as to make it better fit the training data with each iteration. Up to a point, this improves the learner's performance on data outside of the training set. Past that point, however, improving the learner's fit to the training data comes at the expense of increased generalization error. Early stopping rules provide guidance as to how many iterations can be run before the learner begins to over-fit. Early stopping rules have been employed in many different machine learning methods, with varying amounts of theoretical foundation. Background This section presents some of the basic machine-learning concepts required for a description of early stopping methods. Overfitting Machine learning algorithms train a model based on a finite set of training data. During this training, the model is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Numerical Instability
In the mathematics, mathematical subfield of numerical analysis, numerical stability is a generally desirable property of numerical algorithms. The precise definition of stability depends on the context. One is numerical linear algebra and the other is algorithms for solving ordinary and partial differential equations by discrete approximation. In numerical linear algebra, the principal concern is instabilities caused by proximity to singularities of various kinds, such as very small or nearly colliding eigenvalues. On the other hand, in numerical algorithms for differential equations the concern is the growth of round-off errors and/or small fluctuations in initial data which might cause a large deviation of final answer from the exact solution. Some numerical algorithms may damp out the small fluctuations (errors) in the input data; others might magnify such errors. Calculations that can be proven not to magnify approximation errors are called ''numerically stable''. One of th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Newton–Raphson
In numerical analysis, Newton's method, also known as the Newton–Raphson method, named after Isaac Newton and Joseph Raphson, is a root-finding algorithm which produces successively better approximations to the roots (or zeroes) of a real-valued function. The most basic version starts with a single-variable function defined for a real variable , the function's derivative , and an initial guess for a root of . If the function satisfies sufficient assumptions and the initial guess is close, then :x_ = x_0 - \frac is a better approximation of the root than . Geometrically, is the intersection of the -axis and the tangent of the graph of at : that is, the improved guess is the unique root of the linear approximation at the initial point. The process is repeated as :x_ = x_n - \frac until a sufficiently precise value is reached. This algorithm is first in the class of Householder's methods, succeeded by Halley's method. The method can also be extended to complex functions an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Logistic Regression
In statistics, the logistic model (or logit model) is a statistical model that models the probability of an event taking place by having the log-odds for the event be a linear function (calculus), linear combination of one or more independent variables. In regression analysis, logistic regression (or logit regression) is estimation theory, estimating the parameters of a logistic model (the coefficients in the linear combination). Formally, in binary logistic regression there is a single binary variable, binary dependent variable, coded by an indicator variable, where the two values are labeled "0" and "1", while the independent variables can each be a binary variable (two classes, coded by an indicator variable) or a continuous variable (any real value). The corresponding probability of the value labeled "1" can vary between 0 (certainly the value "0") and 1 (certainly the value "1"), hence the labeling; the function that converts log-odds to probability is the logistic function, h ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Logit
In statistics, the logit ( ) function is the quantile function associated with the standard logistic distribution. It has many uses in data analysis and machine learning, especially in data transformations. Mathematically, the logit is the inverse of the standard logistic function \sigma(x) = 1/(1+e^), so the logit is defined as :\operatorname p = \sigma^(p) = \ln \frac \quad \text \quad p \in (0,1). Because of this, the logit is also called the log-odds since it is equal to the logarithm of the odds \frac where is a probability. Thus, the logit is a type of function that maps probability values from (0, 1) to real numbers in (-\infty, +\infty), akin to the probit function. Definition If is a probability, then is the corresponding odds; the of the probability is the logarithm of the odds, i.e.: :\operatorname(p)=\ln\left( \frac \right) =\ln(p)-\ln(1-p)=-\ln\left( \frac-1\right)=2\operatorname(2p-1) The base of the logarithm function used is of little importance in t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
