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Induction of Decision Trees
 Mach. Learn
, 1986
"... systems Abstract. The technology for building knowledgebased systems by inductive inference from examples has been demonstrated successfully in several practical applications. This paper summarizes an approach to synthesizing decision trees that has been used in a variety of systems, and it describ ..."
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Cited by 3335 (4 self)
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systems Abstract. The technology for building knowledgebased systems by inductive inference from examples has been demonstrated successfully in several practical applications. This paper summarizes an approach to synthesizing decision trees that has been used in a variety of systems, and it describes one such system, ID3, in detail. Results from recent studies show ways in which the methodology can be modified to deal with information that is noisy and/or incomplete. A reported shortcoming of the basic algorithm is discussed and two means of overcoming it are compared. The paper concludes with illustrations of current research directions. 1.
A DecisionTheoretic Generalization of onLine Learning and an Application to Boosting
, 1997
"... In the first part of the paper we consider the problem of dynamically apportioning resources among a set of options in a worstcase online framework. The model we study can be interpreted as a broad, abstract extension of the wellstudied online prediction model to a general decisiontheoretic set ..."
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Cited by 2307 (59 self)
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In the first part of the paper we consider the problem of dynamically apportioning resources among a set of options in a worstcase online framework. The model we study can be interpreted as a broad, abstract extension of the wellstudied online prediction model to a general decisiontheoretic setting. We show that the multiplicative weightupdate rule of Littlestone and Warmuth [20] can be adapted to this model yielding bounds that are slightly weaker in some cases, but applicable to a considerably more general class of learning problems. We show how the resulting learning algorithm can be applied to a variety of problems, including gambling, multipleoutcome prediction, repeated games and prediction of points in R n . In the second part of the paper we apply the multiplicative weightupdate technique to derive a new boosting algorithm. This boosting algorithm does not require any prior knowledge about the performance of the weak learning algorithm. We also study generalizations of...
Random forests
 Machine Learning
, 2001
"... Abstract. Random forests are a combination of tree predictors such that each tree depends on the values of a random vector sampled independently and with the same distribution for all trees in the forest. The generalization error for forests converges a.s. to a limit as the number of trees in the fo ..."
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Cited by 1382 (2 self)
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Abstract. Random forests are a combination of tree predictors such that each tree depends on the values of a random vector sampled independently and with the same distribution for all trees in the forest. The generalization error for forests converges a.s. to a limit as the number of trees in the forest becomes large. The generalization error of a forest of tree classifiers depends on the strength of the individual trees in the forest and the correlation between them. Using a random selection of features to split each node yields error rates that compare favorably to Adaboost (Y. Freund & R. Schapire, Machine Learning: Proceedings of the Thirteenth International conference, ∗∗∗, 148–156), but are more robust with respect to noise. Internal estimates monitor error, strength, and correlation and these are used to show the response to increasing the number of features used in the splitting. Internal estimates are also used to measure variable importance. These ideas are also applicable to regression.
Additive Logistic Regression: a Statistical View of Boosting
 Annals of Statistics
, 1998
"... Boosting (Freund & Schapire 1996, Schapire & Singer 1998) is one of the most important recent developments in classification methodology. The performance of many classification algorithms can often be dramatically improved by sequentially applying them to reweighted versions of the input data, and t ..."
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Cited by 1217 (21 self)
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Boosting (Freund & Schapire 1996, Schapire & Singer 1998) is one of the most important recent developments in classification methodology. The performance of many classification algorithms can often be dramatically improved by sequentially applying them to reweighted versions of the input data, and taking a weighted majority vote of the sequence of classifiers thereby produced. We show that this seemingly mysterious phenomenon can be understood in terms of well known statistical principles, namely additive modeling and maximum likelihood. For the twoclass problem, boosting can be viewed as an approximation to additive modeling on the logistic scale using maximum Bernoulli likelihood as a criterion. We develop more direct approximations and show that they exhibit nearly identical results to boosting. Direct multiclass generalizations based on multinomial likelihood are derived that exhibit performance comparable to other recently proposed multiclass generalizations of boosting in most...
On combining classifiers
 IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE
, 1998
"... We develop a common theoretical framework for combining classifiers which use distinct pattern representations and show that many existing schemes can be considered as special cases of compound classification where all the pattern representations are used jointly to make a decision. An experimental ..."
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Cited by 988 (29 self)
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We develop a common theoretical framework for combining classifiers which use distinct pattern representations and show that many existing schemes can be considered as special cases of compound classification where all the pattern representations are used jointly to make a decision. An experimental comparison of various classifier combination schemes demonstrates that the combination rule developed under the most restrictive assumptions—the sum rule—outperforms other classifier combinations schemes. A sensitivity analysis of the various schemes to estimation errors is carried out to show that this finding can be justified theoretically.
Boosting the margin: A new explanation for the effectiveness of voting methods
 In Proceedings International Conference on Machine Learning
, 1997
"... Abstract. One of the surprising recurring phenomena observed in experiments with boosting is that the test error of the generated classifier usually does not increase as its size becomes very large, and often is observed to decrease even after the training error reaches zero. In this paper, we show ..."
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Cited by 721 (52 self)
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Abstract. One of the surprising recurring phenomena observed in experiments with boosting is that the test error of the generated classifier usually does not increase as its size becomes very large, and often is observed to decrease even after the training error reaches zero. In this paper, we show that this phenomenon is related to the distribution of margins of the training examples with respect to the generated voting classification rule, where the margin of an example is simply the difference between the number of correct votes and the maximum number of votes received by any incorrect label. We show that techniques used in the analysis of Vapnik’s support vector classifiers and of neural networks with small weights can be applied to voting methods to relate the margin distribution to the test error. We also show theoretically and experimentally that boosting is especially effective at increasing the margins of the training examples. Finally, we compare our explanation to those based on the biasvariance decomposition. 1
Improved Boosting Algorithms Using Confidencerated Predictions
 MACHINE LEARNING
, 1999
"... We describe several improvements to Freund and Schapire’s AdaBoost boosting algorithm, particularly in a setting in which hypotheses may assign confidences to each of their predictions. We give a simplified analysis of AdaBoost in this setting, and we show how this analysis can be used to find impr ..."
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Cited by 698 (26 self)
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We describe several improvements to Freund and Schapire’s AdaBoost boosting algorithm, particularly in a setting in which hypotheses may assign confidences to each of their predictions. We give a simplified analysis of AdaBoost in this setting, and we show how this analysis can be used to find improved parameter settings as well as a refined criterion for training weak hypotheses. We give a specific method for assigning confidences to the predictions of decision trees, a method closely related to one used by Quinlan. This method also suggests a technique for growing decision trees which turns out to be identical to one proposed by Kearns and Mansour. We focus next on how to apply the new boosting algorithms to multiclass classification problems, particularly to the multilabel case in which each example may belong to more than one class. We give two boosting methods for this problem, plus a third method based on output coding. One of these leads to a new method for handling the singlelabel case which is simpler but as effective as techniques suggested by Freund and Schapire. Finally, we give some experimental results comparing a few of the algorithms discussed in this paper.
Greedy Function Approximation: A Gradient Boosting Machine
 Annals of Statistics
, 2000
"... Function approximation is viewed from the perspective of numerical optimization in function space, rather than parameter space. A connection is made between stagewise additive expansions and steepest{descent minimization. A general gradient{descent \boosting" paradigm is developed for additive ex ..."
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Cited by 563 (12 self)
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Function approximation is viewed from the perspective of numerical optimization in function space, rather than parameter space. A connection is made between stagewise additive expansions and steepest{descent minimization. A general gradient{descent \boosting" paradigm is developed for additive expansions based on any tting criterion. Specic algorithms are presented for least{squares, least{absolute{deviation, and Huber{M loss functions for regression, and multi{class logistic likelihood for classication. Special enhancements are derived for the particular case where the individual additive components are regression trees, and tools for interpreting such \TreeBoost" models are presented. Gradient boosting of regression trees produces competitive, highly robust, interpretable procedures for both regression and classication, especially appropriate for mining less than clean data. Connections between this approach and the boosting methods of Freund and Shapire 1996, and Frie...
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 539 (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.