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Feature Selection in Nonlinear Kernel Classification Workshop on Optimization-Based Data Mining Techniques with Applications IEEE International Conference on Data Mining Omaha, Nebraska, October 28, 2007 Olvi Mangasarian & Edward Wild University of Wisconsin Madison However, data is nonlinearly separable using only the feature x2 Best linear classifier that uses only 1 feature selects the feature x1 Example x2 + + + + + + _ _ Data is nonlinearly separable: In general nonlinear kernels use both x1 and x2 + + _ _ _ _ _ _ + ++ + + + + + x1 Feature selection in nonlinear classification is important Outline Minimize the number of input space features selected by a nonlinear kernel classifier Start with a standard 1-norm nonlinear support vector machine (SVM) Add 0-1 diagonal matrix to suppress or keep features Leads to a nonlinear mixed-integer program Introduce algorithm to obtain a good local solution to the resulting mixed-integer program Evaluate algorithm on two public datasets from the UCI repository and synthetic NDCC data Linear kernel: (K(A, B))ij = (AB)ij = AiB j = K(Ai, B j) ¢ ¢ kernel, parameter :(K(A, B)) = exp(-||A -B || ) SupportGaussian Vector SVMs Machines ¢ x 2 Rn SVM defined by parameters u and threshold of the nonlinear surface A contains all data points {+…+} ½ A+ {…} ½ A e is a vector of ones K(A, A0)u· e e ij _ __ 0 j 2 K(A+, A0)u ¸ e +e + ++ _ _ i + + + + + + ++ Minimize e y (hinge loss or plus _ + + + Minimize e s (||u|| at+ function or max{•, 0}) to fit _ + solution) __to reduce + data K(x , A )u = overfitting + _ _ K(x , A )u = _ __ _ Slack variable y ¸ 0 _ _ _ K(x , A )u = 1 _ allows points to be on the _ wrong side of the _ _ bounding surface _ 0 0 1 0 0 0 0 0 0 To suppress features, add the number of features present (e0Ee) to the objective with weight ¸ 0 As is increased, more features will be removed from the classifier Reduced Start with Feature Full SVM SVM Replace A with AE, where E is a diagonal n £ n matrix Eii 2 {1, 0},present in All with features are i =the 1, …, n kernel matrix K(A, A0) If Eii is 0 the ith feature is removed Reduced Feature SVM (RFSVM) 1) Initialize diagonal matrix E randomly 2) For fixed 0-1 values E, solve the SVM linear program to obtain (u, , y, s) 3)Fix (u, , s) and sweep through E repeatedly as follows: For each component of E replace 1 by 0 and conversely provided the change decreases the overall objective function by more than tol 4)Go to (3) if a change was made in the last sweep, otherwise continue to (5) 5)Solve the SVM linear program with the new matrix E. If the objective decrease is less than tol, stop, otherwise go to (3) RFSVM Convergence (for tol = 0) Objective function value converges Each step decreases the objective Objective is bounded below by 0 Limit of the objective function value is attained at any accumulation point of the sequence of iterates Accumulation point is a “local minimum solution” Continuous variables are optimal for the fixed integer variables Changing any single integer variable will not decrease the objective Experimental Results Classification accuracy versus number of features used Compare our RFSVM to Relief and RFE (Recursive Feature Elimination) Results given on two public datasets from the UCI repository Ability of RFSVM to handle problems with up to 1000 features tested on synthetic NDCC datasets Set feature selection parameter = 1 Relief and RFE Relief Kira and Rendell, 1992 Filter method: feature selection is a preprocessing procedure Features are selected as relevant if they tend to have different feature values for points in different classes RFE (Recursive Feature Elimination) Guyon, Weston, Barnhill, and Vapnik, 2002 Wrapper method: feature selection is based on classification Features are selected as relevant if removing them causes a large change in the margin of an SVM Ionosphere Dataset 34 SVM with 351 Points in RNonlinear Cross-validation accuracy If the appropriate value of is selected, RFSVM can obtain higher accuracy using fewer features than SVM1 no feature selection Even for feature selection = 0, some Note that parameter accuracy decreases features be removed slightly until aboutmay 10 features when remain, and thenremoving decreasesthem more the hinge loss sharply asdecreases they are removed Linear 1-norm SVM Number of features used Points are generated from normal distributions centered at vertices of 1-norm cubes Dataset is not linearly separable Normally Distributed Clusters on Cubes Dataset (Thompson, 2006) Each point is vs. the SVM without Feature Selection (NKSVM1) RFSVM average ontest NDCC onsetNDCC DataData withwith 20 True 100 True Features Features and Varying and correctness over Numbers 1000 Irrelevant of Irrelevant Features Features 10 datasets with 200 training, 200 tuning, and 1000 When 480 testing points irrelevant Average Accuracy on 1000 Test Points features are added, the 0.70 RFSVM accuracy of RFSVM is 0.53 NKSVM1 45% higher than that of NKSVM1 Conclusion New rigorous formulation with precise objective for feature selection in nonlinear SVM classifiers Obtain a local solution to the resulting mixed-integer program Alternate between a linear program to compute continuous variables and successive sweeps to update the integer variables Efficiently learns accurate nonlinear classifiers with reduced numbers of features Handles problems with 1000 features, 900 of which are irrelevant Questions? Websites with links to papers and talks http://www.cs.wisc.edu/~olvi http://www.cs.wisc.edu/~wildt NDCC generator http://www.cs.wisc.edu/dmi/svm/ndcc/ Running Time on the Ionosphere Dataset Averages 5.7 sweeps through the integer variables Averages 3.4 linear programs 75% of the time consumed in objective function evaluations 15% of time consumed in solving linear programs Complete experiment (1960 runs) took 1 hour 3 GHz Pentium 4 Written in MATLAB CPLEX 9.0 used to solve the linear programs Gaussian kernel written in C Sonar Dataset 208 Points in R60 Cross-validation accuracy Number of features used Related Work Approaches that use specialized kernels Weston, Mukherjee, Chapelle, Pontil, Poggio, and Vapnik, 2000: structural risk minimization Gold, Holub, and Sollich, 2005: Bayesian interpretation Zhang, 2006: smoothing spline ANOVA kernels Margin-based approach Frölich and Zell, 2004: remove features if there is little change to the margin if they are removed Other approaches which combine feature selection with basis reduction Bi, Bennett, Embrechts, Breneman, and Song, 2003 Avidan, 2004 Future Work Datasets with more features Reduce the number of objective function evaluations Limit the number of integer cycles Other ways to update the integer variables Application to regression problems Automatic choice of Algorithm Global solution to nonlinear mixed-integer program cannot be found efficiently Requires solving 2n linear programs For fixed values of the integer diagonal matrix, the problem is reduced to an ordinary SVM linear program Solution strategy: alternate optimization of continuous and integer variables: For fixed values of E, solve a linear program for (u, , y, s) For fixed values of (u, , s), sweep through the components of E and make updates which decrease the objective function Notation Data points represented as rows of an m £ n matrix A Data labels of +1 or -1 are given as elements of an m £ m diagonal matrix D Example XOR: 4 points in R2 Points (0, 1) , (1, 0) have label +1 Points (0, 0) , (1, 1) have label 1 Kernel K(A, B) : Rm£n £ Rn£k ! Rm£k Linear kernel: (K(A, B))ij = (AB)ij = AiB¢j = K(Ai, B¢j) Gaussian kernel, parameter :(K(A, B))ij = exp(-||Ai0 - B¢j||2) Methodology UCI Datasets To reduce running time, 1/11 of each dataset was used as a tuning set to select and the kernel parameter Remaining 10/11 used for 10-fold cross validation Procedure repeated 5 times for each dataset with different random choice of tuning set each time NDCC Generate multiple datasets with 200 training, 200 tuning, and 1000 testing points