{"title": "Boosting on Manifolds: Adaptive Regularization of Base Classifiers", "book": "Advances in Neural Information Processing Systems", "page_first": 665, "page_last": 672, "abstract": null, "full_text": " Boosting on manifolds: adaptive regularization\n of base classifiers\n\n\n\n Balazs Kegl and Ligen Wang\n Department of Computer Science and Operations Research, University of Montreal\n CP 6128 succ. Centre-Ville, Montreal, Canada H3C 3J7\n {kegl|wanglige}@iro.umontreal.ca\n\n\n\n Abstract\n\n In this paper we propose to combine two powerful ideas, boosting and\n manifold learning. On the one hand, we improve ADABOOST by incor-\n porating knowledge on the structure of the data into base classifier design\n and selection. On the other hand, we use ADABOOST's efficient learn-\n ing mechanism to significantly improve supervised and semi-supervised\n algorithms proposed in the context of manifold learning. Beside the spe-\n cific manifold-based penalization, the resulting algorithm also accommo-\n dates the boosting of a large family of regularized learning algorithms.\n\n\n\n1 Introduction\n\nADABOOST [1] is one of the machine learning algorithms that have revolutionized pattern\nrecognition technology in the last decade. The algorithm constructs a weighted linear com-\nbination of simple base classifiers in an iterative fashion. One of the remarkable properties\nof ADABOOST is that it is relatively immune to overfitting even after the training error has\nbeen driven to zero. However, it is now a common knowledge that ADABOOST can overfit\nif it is run long enough. The phenomenon is particularly pronounced on noisy data, so most\nof the effort to regularize ADABOOST has been devoted to make it tolerant to outliers by\neither \"softening\" the exponential cost function (e.g., [2]) or by explicitly detecting outliers\nand limiting their influence on the final classifier [3].\n\nIn this paper we propose a different approach based on complexity regularization. Rather\nthan focusing on possibly noisy data points, we attempt to achieve regularization by fa-\nvoring base classifiers that are smooth in a certain sense. The situation that motivated\nthe algorithm is not when the data is noisy, rather when it has a certain structure that is\nignored by ordinary ADABOOST. Consider, for example, the case when the data set is em-\nbedded in a high-dimensional space but concentrated around a low dimensional manifold\n(Figure 1(a)). ADABOOST will compare base classifiers based on solely their weighted\nerrors so, implicitly, it will consider every base classifier having the same (usually low)\ncomplexity. On the other hand, intuitively, we may hope to achieve better generalization if\nwe prefer base classifiers that \"cut through\" sparse regions to base classifiers that cut into\n\"natural\" clusters or cut the manifold several times. To formalize this intuition, we use the\ngraph Laplacian regularizer proposed in connection to manifold learning [4] and spectral\nclustering [5] (Section 3). For binary base classifiers, this penalty is proportional to the\nnumber of edges of the neighborhood graph that the classifier cuts (Figure 1(b)).\n\n\f\n (a) (b)\n\n\n\n\n\nFigure 1: (a) Given the data, the vertical stump has a lower \"effective\" complexity than\nthe horizontal stump. (b) The graph Laplacian penalty is proportional to the number of\nseparated neighbors.\n\n\n\nTo incorporate this adaptive penalization of base classifiers into ADABOOST, we will turn\nto the marginal ADABOOST algorithm [6] also known as arc-gv [7]. This algorithm can\nbe interpreted as ADABOOST with an L1 weight decay on the base classifier coefficients\nwith a weight decay coefficient . The algorithm has been used to maximize the hard\nmargin on the data [7, 6] and also for regularization [3]. The coefficient is adaptive\nin all these applications: in [7] and [6] it depends on the hard margin and the weighted\nerror, respectively, whereas in [3] it is different for every training point and it quantifies the\n\"noisiness\" of the points. The idea of this paper is to make dependent on the individual\nbase classifiers, in particular, to set to the regularization penalty of the base classifier.\nFirst, with this choice, the objective of base learning becomes standard regularized error\nminimization so the proposed algorithm accommodates the boosting of a large family of\nregularized learning algorithms. Second, the coefficients of the base classifiers are lowered\nproportionally with their complexity, which can be interpreted as an adaptive weight decay.\nThe formulation can be also justified by theoretical arguments which are sketched after the\nformal description of the algorithm in Section 2.\n\nExperimental results (Section 4) show that the regularized algorithm can improve general-\nization. Even when the improvement is not significant, the difference between the training\nerror and the test error decreases significantly and the final classifier is much sparser than\nADABOOST's solution, both of which indicate reduced overfitting. Since the Laplacian\npenalty can be computed without knowing the labels, the algorithm can also be used for\nsemi-supervised learning. Experiments in this context show that algorithm besignificantly\nthe semi-supervised algorithm proposed in [4].\n\n\n2 The REGBOOST algorithm\n\nFor the formal description, let the training data be Dn = (x1, y1), . . . , (xn, yn) where\ndata points (xi, yi) are from the set Rd {-1, 1}. The algorithm maintains a weight distri-\nbution w(t) = w(t)\n 1 , . . . , w(t)\n n over the data points. The weights are initialized uniformly\nin line 1 (Figure 2), and are updated in each iteration in line 10. We suppose that we are\ngiven a base learner algorithm BASE Dn, w, P () that, in each iteration t, returns a base\nclassifier h(t) coming from a subset of H = h : Rd {-1, 1} . In ADABOOST, the\ngoal of the base classifier is to minimize the weighted error\n\n n\n\n = (t)(h) = w(t)I {h(x\n i i) = yi} , 12\n i=1\n\n 1The indicator function I{A} is 1 if its argument A is true and 0 otherwise.\n 2We will omit the iteration index (t) and the argument (h) where it does not cause confusion.\n\n\f\n REGBOOST Dn, BASE(, , ), P (), , T\n\n 1 w (1/n, . . . , 1/n)\n\n 2 for t 1 to T\n\n 3 h(t) BASE Dn, w(t), P ()\n\n n\n\n 4 (t) w(t)h(t)(x\n i i)yi edge\n i=1\n\n\n 5 (t) 2P (h(t)) edge offset\n\n 1 1 + (t) 1 - (t)\n 6 (t) ln base coefficient\n 2 1 - (t) 1 + (t)\n\n 7 if (t) 0 base error (1 - (t))/2\n\n 8 return f (t-1)() = t-1 (j)h(j)()\n j=1\n\n 9 for i 1 to n\n\n exp -(t)h(t)(xi)yi\n 10 w(t+1) w(t)\n i i n w(t) exp - (t)h(t)(x\n j=1 j j )yj\n\n 11 return f (T )() = T (t)h(t)()\n t=1\n\n\nFigure 2: The pseudocode of the REGBOOST algorithm with binary base classifiers. Dn\nis the training data, BASE is the base learner, P is the penalty functional, is the penalty\ncoefficient, and T is the number of iterations.\n\n\nwhich is equivalent to maximizing the edge = 1 - 2 = n w(t)h(x\n i=1 i i)yi. The goal of\nREGBOOST's base learner is to minimize the penalized cost\n\n 1 1\n R1(h) = (h) + P (h) = - ( - ), (1)\n 2 2\n\nwhere P : H R is an arbitrary penalty functional or regularization operator, provided\nto REGBOOST and to the base learner, is the penalty coefficient, and = 2P (h) is the\nedge offset. Intuitively, the edge quantifies by how much h is better than a random guess,\nwhile the edge offset indicates by how much h(t) must be better than a random guess.\nThis means that for complex base classifiers (with large penalties), we require a better\nbase classification than for simple classifiers. The main advantage of R1 is that it has the\nform of conventional regularized error minimization, so it accommodates the boosting of\nall learning algorithms that minimize an error functional of this form (e.g., neural networks\nwith weight decay). However, the minimization of R1 is suboptimal from boosting's point\nof view.3 If computationally possible, the base learner should minimize\n\n 1 - 1+ 1- 1 + 1+ 1 - 1-\n R2(h) = 2 = . (2)\n 1 + 1 - 1 + 1 - \n\n 3This statement along with the formulae for R1, R2, and (t) are explained formally after Theo-\nrem 1.\n\n\f\nAfter computing the edge and the edge offset in lines 4 and 5, the algorithm sets the coef-\nficient (t) of the base classifier h(t) to\n 1 1 + (t) 1 1 + (t)\n (t) = ln - ln . (3)\n 2 1 - (t) 2 1 - (t)\n\nIn line 11, the algorithm returns the weighted average of the base classifiers f (T )() =\n T (t)h(t)() as the combined classifier, and uses the sign of f (T )(x) to classify x.\n t=1\nThe algorithm must terminate if (t) 0 which is equivalent to (t) (t) and to (t) \n(1-(t))/2.4 In this case, the algorithm returns the actual combined classifier in line 8. This\nmeans that either the capacity of the set of base classifiers is too small ((t) is small), or the\npenalty is too high ((t) is high), so we cannot find a new base classifier that would improve\nthe combined classifier. Note that the algorithm is formally equivalent to ADABOOST if\n(t) 0 and to marginal ADABOOST if (t) is constant.\n\nFor the analysis of the algorithm, we first define the unnormalized margin achieved by f (T )\non (xi, yi) as\n i = f (T )(xi)yi,\nand the (normalized) margin as\n T (t)h(t)(x\n i t=1 i)yi\n i = = , (4)\n T\n 1 (t)\n t=1\nwhere 1 = T (t) is the L\n t=1 1 norm of the coefficient vector. Let the average penalty\nor margin offset be defined as the average edge offset\n T\n (t)(t)\n = t=1 . (5)\n T (t)\n t=1\nThe following theorem upper bounds the marginal training error\n n\n 1\n L()(f (T )) = I \n n i < \n (6)\n i=1\n\nachieved by the combined classifier f (T ) that REGBOOST outputs.\n\nTheorem 1 Let (t) = 2P (h(t)), let \n and L()(f (T )) be as defined in (5) and (6), re-\nspectively. Let w(t) be the weight of training point (x\n i i, yi) after the tth iteration (updated\nin line 10 in Figure 2), and let (t) be the weight of the base regressor h(t)() (computed in\nline 6 in Figure 2). Then\n T n T\n\n L()(f (T )) e(t)(t) w(t)e-(t)h(t)(xi)yi = E(t) (t), h(t) . (7)\n i\n t=1 i=1 t=1\n\n\nProof. The proof is an extension of the proof of Theorem 5 in [8].\n n T T\n 1\n L()(f (T )) = I \n (t) - (t)h(t)(x\n n i)yi 0 (8)\n i=1 t=1 t=1\n n\n 1 \n e PT (t)-PT (t)h(t)(\n t=1 t=1 xi)yi (9)\n n i=1\n T n n\n \n = e PT (t)\n t=1 w(t)e-(t)h(t)(xj)yj w(T +1). (10)\n j i\n t=1 j=1 i=1\n\n 4Strictly speaking, (t) = 0 could be allowed but in this case the (t) would remain 0 forever so\nit makes no sense to continue.\n\n\f\nIn (8) we used the definitions (6) and (4), the inequality (9) holds since ex I{x 0},\nand we obtained (10) by recursively applying line 10 in Figure 2. The theorem follows by\nthe definition (5) and since n w(T +1) = 1.\n i=1 i\n\nFirst note that Theorem 1 explains the base objectives (1) and (2) and the base coefficient\n(3). The goal of REGBOOST is the greedy minimization of the exponential bound in (7),\nthat is, in each iteration we attempt to minimize E(t) (, h). Given h(t), E(t) , h(t) is\nminimized by (3), and with this choice for (t), R2(h) = E(t) (t), h , so the base learner\nshould attempt to minimize R2(h). If this is computationally impossible, we follow Mason\net al.'s functional gradient descent approach [2], that is, we find h(t) by maximizing the\n\nnegative gradient - E(t)(,h) in = 0. Since - E(t)(,h) = - , this criterion is\n =0\nequivalent to the minimization of R1(h).5\n\nTheorem 1 also suggests various interpretations of REGBOOST which indicate why it\nwould indeed achieve regularization. First, by (9) it can be seen that REGBOOST directly\nminimizes n\n 1 exp -\n n i + \n 1 ,\n i=1\nwhich can be interpreted as an exponential cost on the unnormalized margin with an L1\nweight decay. The weight decay coefficient \n is proportional to the average complexity\nof the base classifiers. Second, Theorem 1 also indicates that REGBOOST indirectly min-\nimizes the marginal error L()(f (T )) (6) where the margin parameter \n , again, is moving\nadaptively with the average complexity of the base classifiers. This explanation is sup-\nported by theoretical results that bound the generalization error in terms of the marginal\nerror (e.g., Theorem 2 in [8]). The third explanation is based on results that show that the\ndifference between the marginal error and the generalization error can be upper bounded in\nterms of the complexity of the base classifier class H (e.g., Theorem 4 in [9]). By imposing\na non-zero penalty on the base classifiers, we can reduce the pool of admissible functions\nto those of which the edge is larger than the edge offset . Although the theoretical\nresults do not apply directly, they support the empirical evidence (Section 4) that indicate\nthat the reduction of the pool of admissible base classifiers and the sparsity of the combined\nclassifier play an important role in decreasing the generalization error.\n\nFinally note that the algorithm can be easily extended to real-valued base classifiers along\nthe lines of [10] and to regression by using the algorithm proposed in [11]. If base clas-\nsifiers come from the set {h : Rd R}, we can only use the base objective R1(h) (1),\nand the analytical solution (3) for the base coefficients (t) must be replaced by a simple\nnumerical minimization (line search) of E(t) , h(t) .6 In the case of regression, the bi-\nnary cost function I {h(x) = y} should be replaced by an appropriate regression cost (e.g.,\nquadratic), and the final regressor should be the weighted median of the base regressors\ninstead of their weighted average.\n\n\n3 The graph Laplacian regularizer\n\nThe algorithm can be used with any regularized base learner that optimizes a penalized\ncost of the form (1). In this paper we apply a smoothness functional based on the graph\n\n 5Note that if is constant (ADABOOST or marginal ADABOOST), the minimization of R1(h) and\nR2(h) leads to the same solution, namely, to the base classifier that minimizes the weighted error .\nThis is no more the case if depends on h.\n 6As a side remark, note that applying a non-zero (even constant) penalty would provide an\nalternative solution to the singularity problem ((t) = ) in the abstaining base classifier model of\n[10].\n\n\f\nLaplacian operator, proposed in a similar context by [4]. The advantage of this penalty is\nthat it is relatively simple to compute for enumerable base classifiers (e.g., decision stumps\nor decision trees) and that it suits applications where the data exhibits a low dimensional\nmanifold structure.\n\nFormally, let G = (V, E) be the neighborhood graph of the training set where the vertex\nset V = {x1, . . . , xn} is identical to the set of observations, and the edge set E contains\npairs of \"neighboring\" vertices (xi, xj) such that either xi - xj < r or xi (xj) is among\nthe k nearest neighbors of xj (xi) where r or k is fixed. This graph plays a crucial role\nin several recently developed dimensionality reduction methods since it approximates the\nnatural topology of the data if it is confined to a low-dimensional smooth manifold in the\nembedding space. To penalize base classifiers that cut through dense regions, we use the\nsmoothness functional\n\n n n\n 1 2\n PL(h) = h(x W\n 2| i) - h(xj ) ij ,\n W| i=1 j=i+1\n\nwhere W is the adjacency matrix of G, that is, Wij = I (xi, xj) E , and 2|W| =\n2 n n W\n i=1 j=1 ij is a normalizing factor so that 0 PL(h) 1.7 For binary base\nclassifiers, PL(h) is proportional to the number of separated neighbors, that is, the number\nof connected pairs that are classified differently by h. Let the diagonal matrix D defined\nby Dii = n W\n j=1 ij , and let L = D - W be the graph Laplacian of G. Then it is easy to\nsee that\n\n n\n\n 2|W|PL(h) = hLhT = h, Lh = i h, ei ,\n j=1\n\n\nwhere h = h(x1), . . . , h(xn) , and ei and i are the (normalized) eigenvectors and eigen-\nvalues of L, that is, Lei = iei, ei = 1. Since L is positive definite, all the eigenvalues\nare non-negative. The eigenvectors with the smallest eigenvalues can be considered as the\n\"smoothest\" functions on the neighborhood graph. Based on this observation, [4] proposed\nto learn a linear combination of a small number of the eigenvectors with the smallest eigen-\nvalues. One problem of this approach is that the out-of-sample extension of the obtained\nclassifier is non-trivial since the base functions are only known at the data points that par-\nticipated in forming the neighborhood graph, so it can only be used in a semi-supervised\nsettings (when unlabeled test points are known before the learning). Our approach is based\non the same intuition, but instead of looking for a linear combination of the eigenvectors,\nwe form a linear combination of known base functions and penalize them according to their\nsmoothness on the underlying manifold. So, beside semi-supervised learning (explored in\nSection 4), our algorithm can also be used to classify out-of-sample test observations.\n\nThe penalty functional can also be justified from the point of view of spectral clustering\n[5]. The eigenvectors of L with the smallest eigenvalues8 represent \"natural\" clusters in\nthe data set, so PL(h) is small if h is aligned with these eigenvectors, and PL(h) is large if\nh splits the corresponding clusters.\n\n\n 7Another variant (that we did not explore in this paper) is to weight edges decreasingly with their\nlengths.\n 8Starting from the second smallest; the smallest is 0 and it corresponds to the constant func-\ntion. Also note that spectral clustering usually uses the eigenvectors of the normalized Laplacian\ne\nL = D-1/2LD-1/2. Nevertheless, if the neighborhood graph is constructed by connecting a fixed\nnumber of nearest neighbors, Dii is approximately constant, so the eigenvectors of L and e\n L are\napproximately equal.\n\n\f\n4 Experiments\n\nIn this section we present experimental results on four UCI benchmark datasets. The re-\nsults are preliminary in the sense that we only validated the penalty coefficient , and did\nnot optimize the number of neighbors (set to k = 8) and the weighting scheme of the edges\nof the neighborhood graph (Wij = 0 or 1). We used decision stumps as base classifiers,\n10-fold cross validation for estimating errors, and 5-fold cross validation for determining .\nThe results (Figure 3(a)-(d) and Table 1) show that the REGBOOST consistently improves\ngeneralization. Although the improvement is within the standard deviation, the difference\nbetween the test and the training error decreases significantly in two of the four experi-\nments, which indicates reduced overfitting. The final classifier is also significantly sparser\nafter 1000 iterations (last two columns of Table 1). To measure how the penalty affects the\nbase classifier pool, in each iteration we calculated the number of admissible base classi-\nfiers relative to the total number of stumps considered by ADABOOST. Figure 3(e) shows\nthat, as expected, REGBOOST traverses only a (sometimes quite small) subset of the base\nclassifier space.\n\n (a) (b) (c)\n ionosphere breast cancer sonar\n\n 0.25 0.09 0.6\n training error (AdaBoost) training error (AdaBoost) training error (AdaBoost)\n test error (AdaBoost) test error (AdaBoost) test error (AdaBoost)\n training error (RegBoost) 0.08 training error (RegBoost) training error (RegBoost)\n test error (RegBoost) test error (RegBoost) 0.5 test error (RegBoost)\n 0.2 0.07\n\n\n 0.06 0.4\n\n 0.15\n 0.05\n 0.3\n 0.04\n 0.1\n\n 0.03 0.2\n\n\n 0.02\n 0.05\n 0.1\n 0.01\n\n\n 0 0 0\n 1 10 100 1000 1 10 100 1000 1 10 100 1000\n\n t t t\n\n pima indians diabetes rate of admissible stumps semi-supervised ionosphere\n\n 0.35 1 0.2\n training error (AdaBoost) ionosphere training error (AdaBoost)\n test error (AdaBoost) breast cancer test error (AdaBoost)\n training error (RegBoost) 0.9 sonar 0.18 training error (RegBoost)\n test error (RegBoost) pima indians diabetes test error (RegBoost)\n 0.3 0.16\n 0.8\n\n 0.14\n 0.7\n\n 0.25 0.12\n 0.6\n 0.1\n 0.5\n 0.2 0.08\n\n 0.4\n 0.06\n\n 0.3\n 0.15 0.04\n\n 0.2 0.02\n\n\n 0.1 0.1 0\n 1 10 100 1000 1 10 100 1000 1 10 100 1000\n\n t t t\n (d) (e) (f)\nFigure 3: Learning curves. Test and training errors for the (a) ionosphere, (b) breast\ncancer, (c) sonar, and (d) Pima Indians diabetes data sets. (e) Rate of admissible stumps.\n(f) Test and training errors for the ionosphere data set with 100 labeled and 251 unlabeled\ndata points.\n\n\n data set training error test error # of stumps\n ADAB REGB ADAB REGB ADAB REGB\n ionosphere 0% 0% 9.14% (7.1) 7.7% (6.0) 182 114\n breast cancer 0% 2.44% 5.29% (3.5) 3.82% (3.7) 58 30\n sonar 0% 0% 32.5% (19.8) 29.8% (18.8) 234 199\n Pima Indians 10.9% 16.0% 25.3% (5.3) 23.3% (6.8) 175 91\n\n Table 1: Errors rates and number of base classifiers after 1000 iterations.\n\nSince the Laplacian penalty can be computed without knowing the labels, the algorithm\ncan also be used for semi-supervised learning. Figure 3(f) shows the results when only a\nsubset of the training points are labeled. In this case, REGBOOST can use the combined\ndata set to calculate the penalty, whereas both algorithms can use only the labeled points\n\n\f\nto determine the base errors. Figure 3(f) indicates that REGBOOST has a clear advantage\nhere. REGBOOST is also far better than the semi-supervised algorithm proposed in [12]\n(their best test error using the same settings is 18%).\n\n\n5 Conclusion\n\nIn this paper we proposed to combine two powerful ideas, boosting and manifold learn-\ning. The algorithm can be used to boost any regularized base learner. Experimental results\nindicate that REGBOOST slightly improves ADABOOST by incorporating knowledge on\nthe structure of the data into base classifier selection. REGBOOST also significantly im-\nproves a recently proposed semi-supervised algorithm based on the same regularizer. In\nthe immediate future our goal is to conduct a larger scale experimental study in which\nwe optimize all the parameters of the algorithm, and compare it not only to ADABOOST,\nbut also to marginal ADABOOST, that is, REGBOOST with a constant penalty . Marginal\nADABOOST might exhibit a similar behavior on the supervised task (sparsity, reduced num-\nber of admissible base classifiers), however, it can not be used to semi-supervised learning.\nWe also plan to experiment with other penalties which are computationally less costly than\nthe Laplacian penalty.\n\n\nReferences\n\n [1] Y. Freund and R. E. Schapire, \"A decision-theoretic generalization of on-line learning and an\n application to boosting,\" Journal of Computer and System Sciences, vol. 55, pp. 119139, 1997.\n\n [2] L. Mason, P. Bartlett, J. Baxter, and M. Frean, \"Boosting algorithms as gradient descent,\" in\n Advances in Neural Information Processing Systems. 2000, vol. 12, pp. 512518, The MIT\n Press.\n\n [3] G. 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Lugosi, \"Data-dependent margin-based generalization\n bounds for classification,\" Journal of Machine Learning Research, pp. 7398, 2002.\n\n[10] R. E. Schapire and Y. Singer, \"Improved boosting algorithms using confidence-rated predic-\n tions,\" Machine Learning, vol. 37, no. 3, pp. 297336, 1999.\n\n[11] B. Kegl, \"Robust regression by boosting the median,\" in Proceedings of the 16th Conference\n on Computational Learning Theory, Washington, D.C., 2003, pp. 258272.\n\n[12] M. Belkin, I. Matveeva, and P. Niyogi, \"Regression and regularization on large graphs,\" in\n Proceedings of the 17th Conference on Computational Learning Theory, 2004.\n\n\f\n", "award": [], "sourceid": 2613, "authors": [{"given_name": "Ligen", "family_name": "Wang", "institution": null}, {"given_name": "Bal\u00e1zs", "family_name": "K\u00e9gl", "institution": null}]}