1. Feature Selection in Nonlinear Kernel Classification Olvi Mangasarian & Edward Wild University of Wisconsin Madison Workshop on Optimization-Based Data Mining Techniques with Applications IEEE International Conference on Data Mining Omaha, Nebraska, October 28, 2007

2. Example +++ + +++ + +++ + +++ + ___ _ ___ _ x1x1 x2x2 However, data is nonlinearly separable using only the feature x 2 Best linear classifier that uses only 1 feature selects the feature x 1 Feature selection in nonlinear classification is important Data is nonlinearly separable: In general nonlinear kernels use both x 1 and x 2

3. 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

4. K(x 0, A 0 )u =  1  Support Vector Machines K(A +, A 0 )u ¸ e  +e K(A , A 0 )u · e   e + _ _ _ _ _ _ _ + + + + + + + + _ _ _ _ _ _ _ _ _ _ _ + + + + + + + + + _ _ _ _ _ K(x 0, A 0 )u =  K(x 0, A 0 )u =  Slack variable y ¸ 0 allows points to be on the wrong side of the bounding surface  x 2 R n  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 SVMs Minimize e 0 s (||u|| 1 at solution) to reduce overfitting Minimize e 0 y (hinge loss or plus function or max{, 0}) to fit data Linear kernel: (K(A, B)) ij = (AB) ij = A i B ¢ j = K(A i, B ¢ j ) Gaussian kernel, parameter  (K(A, B)) ij = exp(-  ||A i 0 -B ¢ j || 2 )

5. Reduced Feature SVM If E ii is 0 the i th feature is removed  To suppress features, add the number of features present (e 0 Ee) to the objective with weight  ¸ 0  As  is increased, more features will be removed from the classifier All features are present in the kernel matrix K(A, A 0 ) Replace A with AE, where E is a diagonal n £ n matrix with E ii 2 {1, 0}, i = 1, …, n Start with Full SVM

6. 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)

7. 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

8. 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

9. 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

10. Ionosphere Dataset 351 Points in R 34 Nonlinear SVM with no feature selection Linear 1-norm SVM Even for feature selection parameter  = 0, some features may be removed when removing them decreases the hinge loss Note that accuracy decreases slightly until about 10 features remain, and then decreases more sharply as they are removed  Number of features used Cross-validation accuracy If the appropriate value of  is selected, RFSVM can obtain higher accuracy using fewer features than SVM1

11. Normally Distributed Clusters on Cubes Dataset (Thompson, 2006)  Points are generated from normal distributions centered at vertices of 1-norm cubes  Dataset is not linearly separable

12. RFSVM vs. SVM without Feature Selection (NKSVM1) on NDCC Data with 100 True Features and 1000 Irrelevant Features RFSVM vs. SVM without Feature Selection (NKSVM1) on NDCC Data with 20 True Features and Varying Numbers of Irrelevant Features Each point is the average test set correctness over 10 datasets with 200 training, 200 tuning, and 1000 testing points When 480 irrelevant features are added, the accuracy of RFSVM is 45% higher than that of NKSVM1 0.53 NKSVM1 0.70 RFSVM Average Accuracy on 1000 Test Points

13. 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

14. 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/

15. 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

16. Sonar Dataset 208 Points in R 60  Number of features used Cross-validation accuracy

17. 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

18. 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 

19. Algorithm  Global solution to nonlinear mixed-integer program cannot be found efficiently  Requires solving 2 n 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

20. 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 R 2  Points (0, 1), (1, 0) have label +1  Points (0, 0), (1, 1) have label  1  Kernel K(A, B) : R m £ n £ R n £ k ! R m £ k  Linear kernel: (K(A, B)) ij = (AB) ij = A i B ¢ j = K(A i, B ¢ j )  Gaussian kernel, parameter  (K(A, B)) ij = exp(-  ||A i 0 - B ¢ j || 2 )

21. 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