Tree ensembles are flexible predictive models that can capture relevant variables and to some extent their interactions in a compact and interpretable manner. Most algorithms for obtaining tree ensembles are based on versions of boosting or Random Forest. Previous work showed that boosting algorithms exhibit a cyclic behavior of selecting the same tree again and again due to the way the loss is optimized. At the same time, Random Forest is not based on loss optimization and obtains a more complex and less interpretable model. In this paper we present a novel method for obtaining compact tree ensembles by growing a large pool of trees in parallel with many independent boosting threads and then selecting a small subset and updating their leaf weights by loss optimization. We allow for the trees in the initial pool to have different depths which further helps with generalization. Experiments on real datasets show that the obtained model has usually a smaller loss than boosting, which is also reflected in a lower misclassification error on the test set.
There has been a large amount of work on different versions of boosting, which can be used for generating tree ensembles. Different versions of boosting minimize different loss functions, starting from Adaboost @cite_23 for the exponential loss, Logitboost @cite_2 for the logistic loss, and Gradientboost @cite_10 for any differentiable loss. Other examples include Floatboost @cite_9 and Robust Logitboost @cite_4 .
@cite_4 @cite_9 @cite_23 @cite_2 @cite_10
1647002864 2140274257 1988790447 2024046085 1678356000
Logitboost is an influential boosting algorithm for classification. In this paper, we develop robust logitboost to provide an explicit formulation of tree-split criterion for building weak learners (regression trees) for logitboost. This formulation leads to a numerically stable implementation of logitboost. We then propose abc-logitboost for multi-class classification, by combining robust logitboost with the prior work of abc-boost. Previously, abc-boost was implemented as abc-mart using the mart algorithm. Our extensive experiments on multi-class classification compare four algorithms: mart, abcmart, (robust) logitboost, and abc-logitboost, and demonstrate the superiority of abc-logitboost. Comparisons with other learning methods including SVM and deep learning are also available through prior publications. A novel learning procedure, called FloatBoost, is proposed for learning a boosted classifier for achieving the minimum error rate. FloatBoost learning uses a backtrack mechanism after each iteration of AdaBoost learning to minimize the error rate directly, rather than minimizing an exponential function of the margin as in the traditional AdaBoost algorithms. A second contribution of the paper is a novel statistical model for learning best weak classifiers using a stagewise approximation of the posterior probability. These novel techniques lead to a classifier which requires fewer weak classifiers than AdaBoost yet achieves lower error rates in both training and testing, as demonstrated by extensive experiments. Applied to face detection, the FloatBoost learning method, together with a proposed detector pyramid architecture, leads to the first real-time multiview face detection system reported. In the first part of the paper we consider the problem of dynamically apportioning resources among a set of options in a worst-case on-line framework. The model we study can be interpreted as a broad, abstract extension of the well-studied on-line prediction model to a general decision-theoretic setting. We show that the multiplicative weight-update Littlestone?Warmuth rule 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, multiple-outcome prediction, repeated games, and prediction of points in Rn. In the second part of the paper we apply the multiplicative weight-update 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 the new boosting algorithm to the problem of learning functions whose range, rather than being binary, is an arbitrary finite set or a bounded segment of the real line. Boosting is one of the most important recent developments in classification methodology. Boosting works by sequentially applying a classification algorithm to reweighted versions of the training data and then taking a weighted majority vote of the sequence of classifiers thus produced. For many classification algorithms, this simple strategy results in dramatic improvements in performance. 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 two-class 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 situations, and far superior in some. We suggest a minor modification to boosting that can reduce computation, often by factors of 10 to 50. Finally, we apply these insights to produce an alternative formulation of boosting decision trees. This approach, based on best-first truncated tree induction, often leads to better performance, and can provide interpretable descriptions of the aggregate decision rule. It is also much faster computationally, making it more suitable to large-scale data mining applications. Function estimation 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 fitting criterion. Specific algorithms are presented for least-squares, least absolute deviation, and Huber-M loss functions for regression, and multiclass logistic likelihood for classification. 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 classification, especially appropriate for mining less than clean data. Connections between this approach and the boosting methods of Freund and Shapire and Friedman, Hastie and Tibshirani are discussed.