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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 types of learning algorithm to improve performance. The output of multiple ''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 of real values. AdaBoost is adaptive in the sense that subsequent weak learners (models) are adjusted in favor of instances misclassified by previous models. In some problems, it can be less susceptible to overfitting 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 converge to a strong learner. Although AdaBo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Boosting (meta-algorithm)
In machine learning (ML), boosting is an ensemble metaheuristic for primarily reducing bias (as opposed to variance). It can also improve the stability and accuracy of ML classification and regression algorithms. Hence, it is prevalent in supervised learning for converting weak learners to strong learners. The concept of 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based on the question posed by Kearns and Valiant (1988, 1989):Michael Kearns(1988)''Thoughts on Hypothesis Boosting'' Unpublished manuscript (Machine Learning class project, December 1988) "Can a set of weak learners create a single strong learner?"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 as a classifier that is only slightly correlated with the true classification. A strong learner is a classifier that is arbitrarily well-correlated with the true classification. Robert Schapire answered the question in the affirmative in a paper published in 1990. This has had significant ramifications in machine learning and statistics, most notably leading to the developm ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Huber Loss Function
In statistics, the Huber loss is a loss function used in robust regression, that is less sensitive to outliers in data than the squared error loss. A variant for classification is also sometimes used. Definition The Huber loss function describes the penalty incurred by an estimation procedure . Huber (1964) defines the loss function piecewise by L_\delta (a) = \begin \frac & \text , a, \le \delta, \\ pt \delta \cdot \left(, a, - \frac\delta\right), & \text \end This function is quadratic for small values of , and linear for large values, with equal values and slopes of the different sections at the two points where , a, = \delta. The variable often refers to the residuals, that is to the difference between the observed and predicted values a = y - f(x), so the former can be expanded to L_\delta(y, f(x)) = \begin \frac ^2 & \text \left, y - f(x)\ \le \delta, \\ pt \delta\ \cdot \left(\left, y - f(x)\ - \frac\delta\right), & \text \end ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
CoBoosting
CoBoost is a semi-supervised training algorithm proposed by Collins and Singer in 1999.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. The original application for the algorithm was the task of named-entity recognition using very weak learners, but it can be used for performing semi-supervised learning in cases where data features may be redundant. 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 leverage re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bootstrap Aggregating
Bootstrap aggregating, also called bagging (from bootstrap aggregating) or bootstrapping, is a machine learning (ML) ensemble meta-algorithm designed to improve the stability and accuracy of ML classification and regression algorithms. It also reduces variance and 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 ensemble 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 samples of D, the rest being duplicates. This kind of sample is known as a bootstrap sample. Sampling with replacement ensures each bootstrap is independent from its peers, as it do ... [...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 and objective 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 polytope. A linear programming algorithm finds a point in the po ... [...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 thus also belongs to the class of margin classifier 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 for learning such a classification function, given a set of training examples with known class labels. LPBoost is a machine learning technique especially suited for 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_j may be just a little bit better than random, bu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Early Stopping
In machine learning, early stopping is a form of Regularization (mathematics), regularization used to avoid overfitting when training a model with an iterative method, such as gradient descent. Such methods update the model to make it better fit the Training, validation, and test data sets, training data with each iteration. Up to a point, this improves the model's performance on data outside of the training set (e.g., the validation set). Past that point, however, improving the model'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 tra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Numerical Instability
In the 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 important context is numerical linear algebra, and another 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 o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Newton–Raphson
In numerical analysis, the Newton–Raphson method, also known simply as Newton's 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 real-valued function , its derivative , and an initial guess for a root of . If satisfies certain assumptions and the initial guess is close, then x_ = x_0 - \frac is a better approximation of the root than . Geometrically, is the x-intercept of the tangent of the graph of at : that is, the improved guess, , is the unique root of the linear approximation of at the initial guess, . The process is repeated as x_ = x_n - \frac until a sufficiently precise value is reached. The number of correct digits roughly doubles with each step. This algorithm is first in the class of Householder's methods, and was succeeded by Halley's method. The method can also be extended to comp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Logistic Regression
In statistics, a logistic model (or logit model) is a statistical model that models the logit, log-odds of an event as a linear function (calculus), linear combination of one or more independent variables. In regression analysis, logistic regression (or logit regression) estimation theory, estimates the parameters of a logistic model (the coefficients in the linear or non linear combinations). 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, hence the name. The unit of measurement for the ... [...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 transformation (statistics), data transformations. Mathematically, the logit is the inverse function, inverse of the logistic function, 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, 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\operatornam ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Newton's Method
In numerical analysis, the Newton–Raphson method, also known simply as Newton's 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 real-valued function , its derivative , and an initial guess for a root of . If satisfies certain assumptions and the initial guess is close, then x_ = x_0 - \frac is a better approximation of the root than . Geometrically, is the x-intercept of the tangent of the graph of at : that is, the improved guess, , is the unique root of the linear approximation of at the initial guess, . The process is repeated as x_ = x_n - \frac until a sufficiently precise value is reached. The number of correct digits roughly doubles with each step. This algorithm is first in the class of Householder's methods, and was succeeded by Halley's method. The method can also be extended t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
