Bagging and Boosting Algorithms for Support Vector Machine Classifiers

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1 Bagging and Boosting Algorithms for Support Vector Machine Classifiers Noritaka SHIGEI and Hiromi MIYAJIMA Dept. of Electrical and Electronics Engineering, Kagoshima University , Korimoto, Kagoshima , JAPAN Abstract: The performance of support vector machines (SVMs) greatly depends on the used values for hyperparameters. The tuning of hyper-parameters is a time-consuming task especially when the amount of data is large. In this paper, in order to overcome this difficulty, ensemble learning methods based on bagging and boosting are proposed. The proposed bagging methods reduce the computation time while keeping a reasonable accuracy. The proposed boosting method improves the accuracy of a conventional SVM classifier. The effectiveness of the proposed methods are demonstrated by numerical simulations. Key Words: Support vector machines, ensemble learning, bagging, boosting, computation time, accuracy 1 Introduction Support vector machines (SVMs) are known as classifiers that can achieve high accuracy and have good generalization abilities[2, 6]. The performance of SVM classifiers greatly depends on the used values for hyper-parameters. The tuning of hyper-parameters is often performed in a grid-search fashion[10]. However, the tuning task is time-consuming especially when the amount of data is large. On the other hand, ensemble learning has attracted attention in machine learning. Ensemble learning is an approach that aims to obtain a better solution by using some weak learners[1, 3, 5, 4, 12], where a weak learner means learning machine that consists of fewer parameters or is constructed from fewer training data. Bagging[3] and boosting[1] are known as ensemble learning algorithms. Some works on ensemble learning for SVM have been made[7, 11, 13]. Bagging with relatively large number of weak learners has been discussed in [7]. A naive bagging implementation has been considered in [11]. Infinite ensemble learning has been studied in [13]. In this paper, ensemble learning for SVM is studied in order to reduce the computation time for hyperparameter tuning and to achive better accuracy without increasing the computation time. We propose two bagging methods and one boosting method. The bagging methods reduce the computation time while keeping a reasonable accuracy. The boosting method improves the accuracy of a conventional SVM classifier. The effectiveness of the proposed methods are demonstrated by numerical simulations. 2 Classification and Support Vector Machine 2.1 Classification Problem Let X IR s be a s-dimensional input vector space. Assuming that X consists of two subspaces X and X + such that X = X X + and X X + =, where X and X + correspond to classes negative and positive, respectively. Then, the classification problem is to construct a classifier h(x) that determines whether a given input x Xbelongs to negative or positive. { 1 ;x is classified as negative, h(x) = (1) 1 ;x is classified as positive. Let (x,y) be a labeled data, where x Xand if x X, y = 1 and if x X +, y = 1. Let D = {(x, 1), (x +, 1) x X, x + X + } be the set of labeled data. In general, only a subset D of D is available to construct a classifier. The subset D and a labeled data (x,y) D are called the training data set and a training data, respectively. Let X = {x X x X} and X + = {x X + x X}. Then, the training data set is defined as D = {(x, 1), (x +, 1) x X, x + X + }. A set of labeled data D Dsuch that D D = is also called a set of test data. The classification error of h(x) for a labeled data set D Ddefined as follows: E = D y h(x) (2) (x,y) D ISSN: ISBN:

2 It is relatively easy to minimize the classification error for the learning data set D. It is important to minimize the error not only for the learning data set but also for the test data set. Lower classification error for the test data set means higher generalization ability of the classifier. 2.2 Linear SVM Let us consider a separating hyperplane f(x) =0is given by the following equation. f(x) =w x + b, (3) where the weight vector w corresponds to the normal vector of the hyperplane and the bias b corresponds to the length of perpendicular line from the original point to the hyperplane. A linear discriminant is given by the following equation. h(x) =sgn(f(x)), (4) where sgn(u) =1for u 0 and sgn(u) = 1 for u < 0. If there exists a pair of w and b such that (x,y) D: sgn(f(x)) = y, then the data set D is said to be linearly separable. In order to maximize the generalization ability of classifier, the linear SVM algorithm maximizes the margin M = 1 w under the constraint (x,y) D : y(w x + b) 1. Such a w for D = {(x 1,y 1 ),, (x N,y N )} is obtained by solving the following dual problem with Lagrange multipliers λ 1,,λ N. Maximize λ i 1 2 subject to λ i 0, j=1 y i y j λ i λ j x i x j (5) y i λ i =0, (6) where w = N λ iy i x i. The bias b is also calculated as b = y s w x s, where x s is an arbitrary support vector such that (x s,y s ) D and λ s Nonlinear SVM and Soft-Margin SVM Most of classification problems existing in the real world are not linearly separable. Such a problem can be transformed into a linearly separable one by mapping the original input space onto a higher feature space. Let φ(x) be a mapping function from the input space into a feature space. Then, for nonlinear SVM, φ(x) is used as x in Eqs.(3), (4) and (5). However, the direct calculation of φ(x) is very expensive in computational cost. This difficulty can be avoided by introducing a kernel function K(x i, x j )[2]. K(x i, x j )=φ(x i ) φ(x j ) (7) If K(x i, x j ) satisfies Mercer s theorem, we may calculate directly K(x i, x j ) without calculating explicitly φ(x). A nonlinear discriminant can be written as follows: ( N ) h(x) =sgn λ i y i K(x i, x)+b, (8) where λ i is a Lagrange multiplier. The Lagrange multipliers λ 1,, λ N are obtained by solving the following dual problem. Maximize λ i 1 2 subject to λ i 0, j=1 y i y j λ i λ j K(x i, x j ) (9) y i λ i =0. (10) Further, a training data set may contain some outliers. A soft-margin SVM allows some training samples to be misclassified to a certain extent. The tradeoff between maximizing the margin and the training error is controlled by a regularization parameter C. A 1-norm soft-margin SVM is obtained by solving Eq.(9) with the following condition. subject to 0 λ i C, 3 Proposed Methods y i λ i =0. (11) In this section, we propose two types of ensemble learning algorithms for SVM. The first one, which is based on bagging[3], is effective for reducing the computation time. The second one, which is based on boosting[1], is effective for improving the classification accuracy. 3.1 Bagging for SVM The learning time of SVM increases with the number of learning data. In the bagging method, we aim to reduce the learning time by using less learning data for each sub-learner. In the conventional bagging, the training data set for a sub-learner is created by sampling with replacement from the given training data set. In this case, the created data set is a multiset, ISSN: ISBN:

3 which may contain multiple elements corresponding to a labeled data. However, this approach is not effective for SVM because the training set D with x i = x j is same as the training set D = D \{(x j,y j )}. Let λ i and λ j be the obtained Lagrange multipliers for x i and x j in D. Let λ i be the obtained multiplier for x i in D. Then, λ i = λ j and 2λ i = λ i. The conventional approach will reduce the number of actual training data and waste the computation time. In order to avoid this inefficiency, we consider two types of bagging algorithms. The first type of bagging divides the given training data set into M equal subsets. The algorithm is as follows. Algorithm Bagging 1 Input: D: the training data set. M: the number of sub-learners, where M is an odd number. Step 1: Create M training data sets D 1, D 2,, D M by randomly partitioning the set D into M subsets, where D = M D i, D i D j = for any i j, D i = D /M for i {1,,M 1} and D M = D M 1 D i. Step 2: For each m {1, 2,, M}, solve the problem Eq.9) for the data D m. Let the obtained classifier denote h m (x). After the algorithm completes, we have M classifiers h 1 (x), h 2 (x),, h M (x). The ensemble output is decided by majority vote as follows: ( M ) h(x) =sgn h m (x). (12) m=1 In Bagging 1, the size of the subset D m decreases directly with M. This property will degrades the accuracy of ensemble output for a large M. The second bagging algorithm provides the method where the subset size D m is independent of the number of sub-learners M. The algorithm is as follows. Algorithm bagging 2 Input: D: the training data set. M: the number of sub-learners, where M is an odd number. N: the number of training data for a sub-learner, where N D. Step 1: (1-1) m 1. (1-2) Create a training data set D m by randomly sampling N times without replacement from D. (1-3) If m<mthen m m +1and go to (1-2). Otherwise go to Step 2. Step 2: For each m {1, 2,, M}, solve the problem Eq.9) for the data D m. Let the obtained classifier denote h m (x). Like Bagging 1, the ensemble output of Bagging 2 is calculated by Eq.(12). 3.2 Filter-Based Boosting for SVM The boosting based on filter sequentially creates three classifiers h f1, h f2 and h f3. The classifier h f2 has a high probability of correctly classifying the data misclassified by h f1. The classifier h f3 has a high probability of correctly classifying the data misclassified by either h f1 or h f2. Thanks to the different properties of h f1, h f2 and h f3, the majority vote of them is expected to achive a higher classification accuracy. Algorithm Filter-Based Boosting for SVM Input: D: training data set. η: the ratio of training data reduction. Step 1: Solve the problem Eq.(5) for the data D. Let the obtained classifier denote h f1. Step 2: Let D miss = {(x,y) D h f1 (x)y 1} and D correct = {(x,y) D h f1 (x)y =1}. (2-1) D f2 D miss. (2-2) Select (1 γ) D correct data from D correct by randomly sampling without replacement. Append the selected (1 γ) D correct data to D f2. Step 3: Solve the problem Eq.(5) for the data D f2. Let the obtained classifier denote h f2. Step 4: Let D f3 = {(x,y) D h f1 (x) h f2 (x)}. Solve the problem Eq.(5) for the data D f3. Let the obtained classifier denote h f3. Like bagging methods, the ensemble output of the boosting method is calculated by Eq.(12). 4 Numerical Simulation In the simulation, the proposed Bagging 1, Bagging 2, Boosting methods and a conventional single SVM method are evaluated for four types of real-world data sets: a1a[8], splice[14], svmguide1[10] and w1a[8]. The specifications of the data sets are summarized in Table 1. All the methods adopt the 1-norm soft-margin SVM model given by Eqs.(9), (11) and an RBF kernel function K(x i, x j )=exp( γ x i x j 2 ). For every method, in order to find an effective value pair of C and γ, a grid-search 5-fold cross-validation is performed. Note that, for ensemble methods Bagging 1, Bagging 2 and Boosting, the grid-search is performed for each sub-learner. The accuracy and the computation time are calculated as averages over 10 simulation runs, where each simulation run is performed with the different order of the ISSN: ISBN:

4 Table 1: The Specifications of benchmark problems. Problem # of features # of training data # of testing data a1a 123 1,605 30,956 splice 60 1,000 2,175 svmguide1 4 3,089 4,000 w1a 300 2,477 47, (a) For a1a (b) For splice (c) For svmguide (d) For w1a Figure 1: The accuracy rate versus the computation time for single SVM and bagging methods. same training patterns. The simulation program is implemented by using LIBSVM 2.88[9] and runs on a GNU/Linux PC with kernel, Intel Core2 Quad CPU Q6600 (2.40GHz) and 4GB of RAM. In the first simulation, Bagging 1, Bagging 2 and conventional methods are evaluated in terms of the accuracy versus the computation time. For every method, the grid-search is performed for C = 2 5, 2 3, 2 1,, 2 15 and γ = 2 3, 2 1, 2 1,, Other used simulation conditions are as follows: M = 3, 5, 7 for Bagging methods and η = 0.60, 0.70, 0.80, 0.90 for Bagging 2. The simulation results are shown in Fig. 1. The following tendencies are observed in the results. For a1a, Bagging 1 shows a good performance in terms of the accuracy versus the computation time, because Bagging 1 for M =7is approximately 8 times faster than single SVM. For every training data set, the points plotted by Bagging 1 and a single SVM seem to be on the same curve. For a1a, Bagging 2 achieves a better performance as M increases; Bagging 2 achieves almost same accuracy as single SVM. On the other ISSN: ISBN:

5 Table 2: The accuracy rate (%) and its standard deviation for single SVM and Boosting methods. Method grid-search a1a splice svmguide1 w1a (i) ± ± ± ±0.027 (ii) ± ± ± ±0.036 Boosting (i) ± ± ± ±0.049 (ii) ± ± ± ±0.003 hand, for other problems splice, svmguide1 and w1a, the tendency is opposite. Although both of bagging methods outperform the accuracy of single SVM, they provide a good trade-off between the accuracy and the computation cost. For example, for a1a, Bagging 2 with M =7achieves approximately 99% accuracy of single SVM while running approximately twice faster than single SVM. In the second simulation, Boosting and conventional methods are compared in terms of the accuracy and stability. In this simulation, we adopt two types of the grid point sets for grid-search: (i) C =2 5,2 3, 2 1,, and γ = 2 3, 2 1, 2 1, and (ii) C = 2 1, 2 2, 2 3, 2 4 and γ =2 11, 2 9, 2 7,, 2 1. Table 2 shows the simulation results on the accuracy and its standard deviation. The following tendencies are observed in the results. Boosting with grid-search (ii) achieves the best accuracy for a1a and splice and near best accuracy with a small standard deviation for w1a. When using grid-search (i), Boosting outperforms the accuracy of single SVM for splice, svmguide1 and w1a. As an overall tendency, Boosting shows smaller standard deviation than single SVM. The simulation result shows that Boosting can achive a better accuracy than single SVM and has a good stability. 5 Conclusion This paper has presented ensemble learning algorithms for SVM: two bagging methods and one boosting method. The effectiveness of the proposed methods is demonstrated by numerical simulations on realworld data sets. The proposed Bagging methods reduce the computation time while keeping a reasonable accuracy. The proposed boosting method improves the accuracy of a conventional single SVM classifier. In our future works, we will consider the relation between ensemble learning and regularization parameter C and develop more effective algorithms. Acknowledgements: The work was partially supported by KAKENHI ( ). References: [1] R.E. Schapire, The strength of weak learnability, Machine Learning 5, 1990, pp [2] B.E. Boser, I.M. Guyon and V.N. Vapnik, A training algorithm for optimal margin classifiers, Proc. of the 5th Annual ACM Workshop on Computational Learning Theory, ACM Press, 1992, pp [3] L. Breiman, Bagging Predictors, Machine Learning 24, 1996, pp [4] Y. Freund and R.E. Schapire, A decision-theoretic generalization of on-line learning and an application to boosting, Journal of Computer and System Sciences 55, 1997, pp [5] R.E. Schapire, A brief introduction to boosting, Proc. 16th Int. Joint Conf. on Artificial Intelligence, 1999, pp [6] N. Cristianini and J. Shawe Taylor, An Introduction to Support Vector Machines, Cambridge University Press, [7] T. Evgeniou, L. Perez-Breva, M. Pontil and T. Poggio, Bounds on the Generalization Performance of Kernel Machines Ensembles, Proc. of ICML 2000, 2000, pp [8] J. Platt, Fast training of support vector machines using sequential minimal optimization, Advances in Kernel Methods Support Vector Learning, MIT Press, 1999, pp [9] C.-C. Chang and C.-J. Lin, LIBSVM: a library for support vector machines, 2000, Software available at cjlin/libsvm [10] C.-W. Hsu, C.-C. Chang and C.-J. Lin, A practical guide to support vector classification, 2003, available at cjlin/papers/guide/guide.pdf [11] H.-C. Kim, S. Pang, H.-M. Je, D. Kim and S.Y. Bang, Constructing support vector machine ensemble, Pattern Recognition 36, 2003, pp [12] S. Miyoshi, K. Hara and M. Okada, Analysis of ensemble learning using simple perceptron based on online learning theory, Physical Review E 71, 2005, pp ISSN: ISBN:

6 [13] H.-T. Lin and L. Li, Support vector machinery for infinite ensemble learning, Journal of Machine Learning Research 9, 2008, pp [14] A. Asuncion and D.J. Newman, UCI Machine Learning Repository, mlearn/mlrepository.html ISSN: ISBN:

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