Two-Stage Bagging Pruning for Reducing the Ensemble Size and Improving the Classification Performance

Ensemble methods, such as the traditional bagging algorithm, can usually improve the performance of a single classifier. However, they usually require large storage space as well as relatively time-consuming predictions. Many approaches were developed to reduce the ensemble size and improve the classification performance by pruning the traditional bagging algorithms. In this article, we proposed a two-stage strategy to prune the traditional bagging algorithm by combining two simple approaches: accuracy-based pruning (AP) and distance-based pruning (DP). These two methods, as well as their two combinations, “AP+DP” and “DP+AP” as the two-stage pruning strategy, were all examined. Comparing with the single pruning methods, we found that the two-stage pruning methods can furthermore reduce the ensemble size and improve the classification. “AP+DP” method generally performs better than the “DP+AP” method when using four base classifiers: decision tree, Gaussian naive Bayes, K-nearest neighbor, and logistic regression. Moreover, as compared to the traditional bagging, the two-stage method “AP+DP” improved the classification accuracy by 0.88%, 4.06%, 1.26%, and 0.96%, respectively, averaged over 28 datasets under the four base classifiers. It was also observed that “AP+DP” outperformed other three existing algorithms Brag, Nice, and TB assessed on 8 common datasets. In summary, the proposed two-stage pruning methods are simple and promising approaches, which can both reduce the ensemble size and improve the classification accuracy

Mixed-Integer Convex Nonlinear Optimization with Gradient-Boosted Trees Embedded

Decision trees usefully represent sparse, high dimensional and noisy data. Having learned a function from this data, we may want to thereafter integrate the function into a larger decision-making problem, e.g., for picking the best chemical process catalyst. We study a large-scale, industrially-relevant mixed-integer nonlinear nonconvex optimization problem involving both gradient-boosted trees and penalty functions mitigating risk. This mixed-integer optimization problem with convex penalty terms broadly applies to optimizing pre-trained regression tree models. Decision makers may wish to optimize discrete models to repurpose legacy predictive models, or they may wish to optimize a discrete model that particularly well-represents a data set. We develop several heuristic methods to find feasible solutions, and an exact, branch-and-bound algorithm leveraging structural properties of the gradient-boosted trees and penalty functions. We computationally test our methods on concrete mixture design instance and a chemical catalysis industrial instance

An analysis of ensemble pruning techniques based on ordered aggregation

Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works. G. Martínez-Muñoz, D. Hernández-Lobato and A. Suárez, "An analysis of ensemble pruning techniques based on ordered aggregation", IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 31, no. 2, pp. 245-249, February 2009Several pruning strategies that can be used to reduce the size and increase the accuracy of bagging ensembles are analyzed. These heuristics select subsets of complementary classifiers that, when combined, can perform better than the whole ensemble. The pruning methods investigated are based on modifying the order of aggregation of classifiers in the ensemble. In the original bagging algorithm, the order of aggregation is left unspecified. When this order is random, the generalization error typically decreases as the number of classifiers in the ensemble increases. If an appropriate ordering for the aggregation process is devised, the generalization error reaches a minimum at intermediate numbers of classifiers. This minimum lies below the asymptotic error of bagging. Pruned ensembles are obtained by retaining a fraction of the classifiers in the ordered ensemble. The performance of these pruned ensembles is evaluated in several benchmark classification tasks under different training conditions. The results of this empirical investigation show that ordered aggregation can be used for the efficient generation of pruned ensembles that are competitive, in terms of performance and robustness of classification, with computationally more costly methods that directly select optimal or near-optimal subensembles.The authors acknowledge support form the Spanish Ministerio de Educación y Ciencia under Project TIN2007-66862-C02-0

Confidence in prediction: an approach for dynamic weighted ensemble.

Combining classifiers in an ensemble is beneficial in achieving better prediction than using a single classifier. Furthermore, each classifier can be associated with a weight in the aggregation to boost the performance of the ensemble system. In this work, we propose a novel dynamic weighted ensemble method. Based on the observation that each classifier provides a different level of confidence in its prediction, we propose to encode the level of confidence of a classifier by associating with each classifier a credibility threshold, computed from the entire training set by minimizing the entropy loss function with the mini-batch gradient descent method. On each test sample, we measure the confidence of each classifier’s output and then compare it to the credibility threshold to determine whether a classifier should be attended in the aggregation. If the condition is satisfied, the confidence level and credibility threshold are used to compute the weight of contribution of the classifier in the aggregation. By this way, we are not only considering the presence but also the contribution of each classifier based on the confidence in its prediction on each test sample. The experiments conducted on a number of datasets show that the proposed method is better than some benchmark algorithms including a non-weighted ensemble method, two dynamic ensemble selection methods, and two Boosting methods

Convolutional Neural Fabrics

Despite the success of CNNs, selecting the optimal architecture for a given task remains an open problem. Instead of aiming to select a single optimal architecture, we propose a "fabric" that embeds an exponentially large number of architectures. The fabric consists of a 3D trellis that connects response maps at different layers, scales, and channels with a sparse homogeneous local connectivity pattern. The only hyper-parameters of a fabric are the number of channels and layers. While individual architectures can be recovered as paths, the fabric can in addition ensemble all embedded architectures together, sharing their weights where their paths overlap. Parameters can be learned using standard methods based on back-propagation, at a cost that scales linearly in the fabric size. We present benchmark results competitive with the state of the art for image classification on MNIST and CIFAR10, and for semantic segmentation on the Part Labels dataset.Comment: Corrected typos (In proceedings of NIPS16

Using boosting to prune bagging ensembles

This is the author’s version of a work that was accepted for publication in Pattern Recognition Letters. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Pattern Recognition Letters 28.1 (2007): 156 – 165, DOI: 10.1016/j.patrec.2006.06.018Boosting is used to determine the order in which classifiers are aggregated in a bagging ensemble. Early stopping in the aggregation of the classifiers in the ordered bagging ensemble allows the identification of subensembles that require less memory for storage, classify faster and can improve the generalization accuracy of the original bagging ensemble. In all the classification problems investigated pruned ensembles with 20 % of the original classifiers show statistically significant improvements over bagging. In problems where boosting is superior to bagging, these improvements are not sufficient to reach the accuracy of the corresponding boosting ensembles. However, ensemble pruning preserves the performance of bagging in noisy classification tasks, where boosting often has larger generalization errors. Therefore, pruned bagging should generally be preferred to complete bagging and, if no information about the level of noise is available, it is a robust alternative to AdaBoost.The authors acknowledge financial support from the Spanish Dirección General de Investigación, project TIN2004-07676-C02-02