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76
An Empirical Comparison of Voting Classification Algorithms: Bagging, Boosting, and Variants
 MACHINE LEARNING
, 1999
"... Methods for voting classification algorithms, such as Bagging and AdaBoost, have been shown to be very successful in improving the accuracy of certain classifiers for artificial and realworld datasets. We review these algorithms and describe a large empirical study comparing several variants in co ..."
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Cited by 550 (2 self)
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Methods for voting classification algorithms, such as Bagging and AdaBoost, have been shown to be very successful in improving the accuracy of certain classifiers for artificial and realworld datasets. We review these algorithms and describe a large empirical study comparing several variants in conjunction with a decision tree inducer (three variants) and a NaiveBayes inducer.
The purpose of the study is to improve our understanding of why and
when these algorithms, which use perturbation, reweighting, and
combination techniques, affect classification error. We provide a
bias and variance decomposition of the error to show how different
methods and variants influence these two terms. This allowed us to
determine that Bagging reduced variance of unstable methods, while
boosting methods (AdaBoost and Arcx4) reduced both the bias and
variance of unstable methods but increased the variance for NaiveBayes,
which was very stable. We observed that Arcx4 behaves differently
than AdaBoost if reweighting is used instead of resampling,
indicating a fundamental difference. Voting variants, some of which
are introduced in this paper, include: pruning versus no pruning,
use of probabilistic estimates, weight perturbations (Wagging), and
backfitting of data. We found that Bagging improves when
probabilistic estimates in conjunction with nopruning are used, as
well as when the data was backfit. We measure tree sizes and show
an interesting positive correlation between the increase in the
average tree size in AdaBoost trials and its success in reducing the
error. We compare the meansquared error of voting methods to
nonvoting methods and show that the voting methods lead to large
and significant reductions in the meansquared errors. Practical
problems that arise in implementing boosting algorithms are
explored, including numerical instabilities and underflows. We use
scatterplots that graphically show how AdaBoost reweights instances,
emphasizing not only "hard" areas but also outliers and noise.
An Efficient Boosting Algorithm for Combining Preferences
, 1999
"... The problem of combining preferences arises in several applications, such as combining the results of different search engines. This work describes an efficient algorithm for combining multiple preferences. We first give a formal framework for the problem. We then describe and analyze a new boosting ..."
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Cited by 538 (18 self)
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The problem of combining preferences arises in several applications, such as combining the results of different search engines. This work describes an efficient algorithm for combining multiple preferences. We first give a formal framework for the problem. We then describe and analyze a new boosting algorithm for combining preferences called RankBoost. We also describe an efficient implementation of the algorithm for certain natural cases. We discuss two experiments we carried out to assess the performance of RankBoost. In the first experiment, we used the algorithm to combine different WWW search strategies, each of which is a query expansion for a given domain. For this task, we compare the performance of RankBoost to the individual search strategies. The second experiment is a collaborativefiltering task for making movie recommendations. Here, we present results comparing RankBoost to nearestneighbor and regression algorithms.
Discriminative Reranking for Natural Language Parsing
, 2005
"... This article considers approaches which rerank the output of an existing probabilistic parser. The base parser produces a set of candidate parses for each input sentence, with associated probabilities that define an initial ranking of these parses. A second model then attempts to improve upon this i ..."
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Cited by 272 (9 self)
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This article considers approaches which rerank the output of an existing probabilistic parser. The base parser produces a set of candidate parses for each input sentence, with associated probabilities that define an initial ranking of these parses. A second model then attempts to improve upon this initial ranking, using additional features of the tree as evidence. The strength of our approach is that it allows a tree to be represented as an arbitrary set of features, without concerns about how these features interact or overlap and without the need to define a derivation or a generative model which takes these features into account. We introduce a new method for the reranking task, based on the boosting approach to ranking problems described in Freund et al. (1998). We apply the boosting method to parsing the Wall Street Journal treebank. The method combined the loglikelihood under a baseline model (that of Collins [1999]) with evidence from an additional 500,000 features over parse trees that were not included in the original model. The new model achieved 89.75 % Fmeasure, a 13 % relative decrease in Fmeasure error over the baseline model’s score of 88.2%. The article also introduces a new algorithm for the boosting approach which takes advantage of the sparsity of the feature space in the parsing data. Experiments show significant efficiency gains for the new algorithm over the obvious implementation of the boosting approach. We argue that the method is an appealing alternative—in terms of both simplicity and efficiency—to work on feature selection methods within loglinear (maximumentropy) models. Although the experiments in this article are on natural language parsing (NLP), the approach should be applicable to many other NLP problems which are naturally framed as ranking tasks, for example, speech recognition, machine translation, or natural language generation.
Popular ensemble methods: an empirical study
 Journal of Artificial Intelligence Research
, 1999
"... An ensemble consists of a set of individually trained classifiers (such as neural networks or decision trees) whose predictions are combined when classifying novel instances. Previous research has shown that an ensemble is often more accurate than any of the single classifiers in the ensemble. Baggi ..."
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Cited by 192 (3 self)
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An ensemble consists of a set of individually trained classifiers (such as neural networks or decision trees) whose predictions are combined when classifying novel instances. Previous research has shown that an ensemble is often more accurate than any of the single classifiers in the ensemble. Bagging (Breiman, 1996c) and Boosting (Freund & Schapire, 1996; Schapire, 1990) are two relatively new but popular methods for producing ensembles. In this paper we evaluate these methods on 23 data sets using both neural networks and decision trees as our classification algorithm. Our results clearly indicate a number of conclusions. First, while Bagging is almost always more accurate than a single classifier, it is sometimes much less accurate than Boosting. On the other hand, Boosting can create ensembles that are less accurate than a single classifier – especially when using neural networks. Analysis indicates that the performance of the Boosting methods is dependent on the characteristics of the data set being examined. In fact, further results show that Boosting ensembles may overfit noisy data sets, thus decreasing its performance. Finally, consistent with previous studies, our work suggests that most of the gain in an ensemble’s performance comes in the first few classifiers combined; however, relatively large gains can be seen up to 25 classifiers when Boosting decision trees. 1.
Boosting with the L_2Loss: Regression and Classification
, 2001
"... This paper investigates a variant of boosting, L 2 Boost, which is constructed from a functional gradient descent algorithm with the L 2 loss function. Based on an explicit stagewise re tting expression of L 2 Boost, the case of (symmetric) linear weak learners is studied in detail in both regressi ..."
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Cited by 123 (16 self)
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This paper investigates a variant of boosting, L 2 Boost, which is constructed from a functional gradient descent algorithm with the L 2 loss function. Based on an explicit stagewise re tting expression of L 2 Boost, the case of (symmetric) linear weak learners is studied in detail in both regression and twoclass classification. In particular, with the boosting iteration m working as the smoothing or regularization parameter, a new exponential biasvariance trade off is found with the variance (complexity) term bounded as m tends to infinity. When the weak learner is a smoothing spline, an optimal rate of convergence result holds for both regression and twoclass classification. And this boosted smoothing spline adapts to higher order, unknown smoothness. Moreover, a simple expansion of the 01 loss function is derived to reveal the importance of the decision boundary, bias reduction, and impossibility of an additive biasvariance decomposition in classification. Finally, simulation and real data set results are obtained to demonstrate the attractiveness of L 2 Boost, particularly with a novel componentwise cubic smoothing spline as an effective and practical weak learner.
Maximum Entropy Discrimination
, 1999
"... We present a general framework for discriminative estimation based on the maximum entropy principle and its extensions. All calculations involve distributions over structures and/or parameters rather than specific settings and reduce to relative entropy projections. This holds even when the data is ..."
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Cited by 122 (20 self)
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We present a general framework for discriminative estimation based on the maximum entropy principle and its extensions. All calculations involve distributions over structures and/or parameters rather than specific settings and reduce to relative entropy projections. This holds even when the data is not separable within the chosen parametric class, in the context of anomaly detection rather than classification, or when the labels in the training set are uncertain or incomplete. Support vector machines are naturally subsumed under this class and we provide several extensions. We are also able to estimate exactly and efficiently discriminative distributions over tree structures of classconditional models within this framework. Preliminary experimental results are indicative of the potential in these techniques.
Empirical margin distributions and bounding the generalization error of combined classifiers
 Ann. Statist
, 2002
"... Dedicated to A.V. Skorohod on his seventieth birthday We prove new probabilistic upper bounds on generalization error of complex classifiers that are combinations of simple classifiers. Such combinations could be implemented by neural networks or by voting methods of combining the classifiers, such ..."
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Cited by 119 (8 self)
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Dedicated to A.V. Skorohod on his seventieth birthday We prove new probabilistic upper bounds on generalization error of complex classifiers that are combinations of simple classifiers. Such combinations could be implemented by neural networks or by voting methods of combining the classifiers, such as boosting and bagging. The bounds are in terms of the empirical distribution of the margin of the combined classifier. They are based on the methods of the theory of Gaussian and empirical processes (comparison inequalities, symmetrization method, concentration inequalities) and they improve previous results of Bartlett (1998) on bounding the generalization error of neural networks in terms of ℓ1norms of the weights of neurons and of Schapire, Freund, Bartlett and Lee (1998) on bounding the generalization error of boosting. We also obtain rates of convergence in Lévy distance of empirical margin distribution to the true margin distribution uniformly over the classes of classifiers and prove the optimality of these rates.
Statistical Behavior and Consistency of Classification Methods based on Convex Risk Minimization
, 2001
"... We study how close the optimal Bayes error rate can be approximately reached using a classification algorithm that computes a classifier by minimizing a convex upper bound of the classification error function. The measurement of closeness is characterized by the loss function used in the estimation. ..."
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Cited by 111 (6 self)
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We study how close the optimal Bayes error rate can be approximately reached using a classification algorithm that computes a classifier by minimizing a convex upper bound of the classification error function. The measurement of closeness is characterized by the loss function used in the estimation. We show that such a classification scheme can be generally regarded as a (non maximumlikelihood) conditional inclass probability estimate, and we use this analysis to compare various convex loss functions that have appeared in the literature. Furthermore, the theoretical insight allows us to design good loss functions with desirable properties. Another aspect of our analysis is to demonstrate the consistency of certain classification methods using convex risk minimization.
Linear programming boosting via column generation
 Machine Learning
, 2002
"... 1 Introduction Recent papers [20] have shown that boosting, arcing, and related ensemble methods (hereafter summarized asboosting) can be viewed as margin maximization in function space. By changing the cost function, different ..."
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Cited by 102 (3 self)
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1 Introduction Recent papers [20] have shown that boosting, arcing, and related ensemble methods (hereafter summarized asboosting) can be viewed as margin maximization in function space. By changing the cost function, different
A New Approximate Maximal Margin Classification Algorithm
 JOURNAL OF MACHINE LEARNING RESEARCH
, 2001
"... A new incremental learning algorithm is described which approximates the maximal margin hyperplane w.r.t. norm p 2 for a set of linearly separable data. Our algorithm, called alma p (Approximate Large Margin algorithm w.r.t. norm p), takes O (p 1) 2 2 corrections to separate the data wi ..."
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Cited by 90 (5 self)
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A new incremental learning algorithm is described which approximates the maximal margin hyperplane w.r.t. norm p 2 for a set of linearly separable data. Our algorithm, called alma p (Approximate Large Margin algorithm w.r.t. norm p), takes O (p 1) 2 2 corrections to separate the data with pnorm margin larger than (1 ) , where is the (normalized) pnorm margin of the data. alma p avoids quadratic (or higherorder) programming methods. It is very easy to implement and is as fast as online algorithms, such as Rosenblatt's Perceptron algorithm. We performed extensive experiments on both realworld and artificial datasets. We compared alma 2 (i.e., alma p with p = 2) to standard Support vector Machines (SVM) and to two incremental algorithms: the Perceptron algorithm and Li and Long's ROMMA. The accuracy levels achieved by alma 2 are superior to those achieved by the Perceptron algorithm and ROMMA, but slightly inferior to SVM's. On the other hand, alma 2 is quite faster and easier to implement than standard SVM training algorithms. When learning sparse target vectors, alma p with p > 2 largely outperforms Perceptronlike algorithms, such as alma 2 .