1. Nov 23rd, 2001Copyright © 2001, 2003, Andrew W. Moore Support Vector Machines Andrew W. Moore Professor School of Computer Science Carnegie Mellon University www.cs.cmu.edu/~awm awm@cs.cmu.edu 412-268-7599 Note to other teachers and users of these slides. Andrew would be delighted if you found this source material useful in giving your own lectures. Feel free to use these slides verbatim, or to modify them to fit your own needs. PowerPoint originals are available. If you make use of a significant portion of these slides in your own lecture, please include this message, or the following link to the source repository of Andrew’s tutorials: http://www.cs.cmu.edu/~awm/tutorials. Comments and corrections gratefully received. http://www.cs.cmu.edu/~awm/tutorials Slides Modified for Comp537, Spring, 2006, HKUST

2. Support Vector Machines: Slide 2 Copyright © 2001, 2003, Andrew W. Moore History SVM is a classifier derived from statistical learning theory by Vapnik and Chervonenkis SVMs introduced by Boser, Guyon, Vapnik in COLT-92 Initially popularized in the NIPS community, now an important and active field of all Machine Learning research. Special issues of Machine Learning Journal, and Journal of Machine Learning Research.

3. Support Vector Machines: Slide 3 Copyright © 2001, 2003, Andrew W. Moore Roadmap Hard-Margin Linear Classifier Maximize Margin Support Vector Quadratic Programming Soft-Margin Linear Classifier Maximize Margin Support Vector Quadratic Programming Non-Linear Separable Problem XOR Transform to Non-Linear by Kernels Reference

4. Support Vector Machines: Slide 4 Copyright © 2001, 2003, Andrew W. Moore Linear Classifiers f x  y est denotes +1 denotes -1 f(x,w,b) = sign(w. x - b) How would you classify this data?

5. Support Vector Machines: Slide 5 Copyright © 2001, 2003, Andrew W. Moore Linear Classifiers f x  y est denotes +1 denotes -1 f(x,w,b) = sign(w. x - b) How would you classify this data?

6. Support Vector Machines: Slide 6 Copyright © 2001, 2003, Andrew W. Moore Linear Classifiers f x  y est denotes +1 denotes -1 f(x,w,b) = sign(w. x - b) How would you classify this data?

7. Support Vector Machines: Slide 7 Copyright © 2001, 2003, Andrew W. Moore Linear Classifiers f x  y est denotes +1 denotes -1 f(x,w,b) = sign(w. x - b) How would you classify this data?

8. Support Vector Machines: Slide 8 Copyright © 2001, 2003, Andrew W. Moore Linear Classifiers f x  y est denotes +1 denotes -1 f(x,w,b) = sign(w. x - b) Any of these would be fine....but which is best?

9. Support Vector Machines: Slide 9 Copyright © 2001, 2003, Andrew W. Moore Classifier Margin f x  y est denotes +1 denotes -1 f(x,w,b) = sign(w. x - b) Define the margin of a linear classifier as the width that the boundary could be increased by before hitting a datapoint.

10. Support Vector Machines: Slide 10 Copyright © 2001, 2003, Andrew W. Moore Maximum Margin f x  y est denotes +1 denotes -1 f(x,w,b) = sign(w. x - b) The maximum margin linear classifier is the linear classifier with the, um, maximum margin. This is the simplest kind of SVM (Called an LSVM) Linear SVM

11. Support Vector Machines: Slide 11 Copyright © 2001, 2003, Andrew W. Moore Maximum Margin f x  y est denotes +1 denotes -1 f(x,w,b) = sign(w. x - b) The maximum margin linear classifier is the linear classifier with the, um, maximum margin. This is the simplest kind of SVM (Called an LSVM) Support Vectors are those datapoints that the margin pushes up against Linear SVM

12. Support Vector Machines: Slide 12 Copyright © 2001, 2003, Andrew W. Moore Why Maximum Margin? denotes +1 denotes -1 f(x,w,b) = sign(w. x - b) The maximum margin linear classifier is the linear classifier with the, um, maximum margin. This is the simplest kind of SVM (Called an LSVM) Support Vectors are those datapoints that the margin pushes up against 1.Intuitively this feels safest. 2.Empirically it works very well. 3.If we’ve made a small error in the location of the boundary (it’s been jolted in its perpendicular direction) this gives us least chance of causing a misclassification. 4.LOOCV is easy since the model is immune to removal of any non- support-vector datapoints. 5.There’s some theory (using VC dimension) that is related to (but not the same as) the proposition that this is a good thing.

13. Support Vector Machines: Slide 13 Copyright © 2001, 2003, Andrew W. Moore Estimate the Margin What is the distance expression for a point x to a line wx+b= 0? denotes +1 denotes -1 x wx +b = 0 X – Vector W – Normal Vector b – Scale Value W

14. Support Vector Machines: Slide 14 Copyright © 2001, 2003, Andrew W. Moore Estimate the Margin What is the expression for margin? denotes +1 denotes -1 wx +b = 0 Margin

15. Support Vector Machines: Slide 15 Copyright © 2001, 2003, Andrew W. Moore Maximize Margin denotes +1 denotes -1 wx +b = 0 Margin

16. Support Vector Machines: Slide 16 Copyright © 2001, 2003, Andrew W. Moore Maximize Margin denotes +1 denotes -1 wx +b = 0 Margin Min-max problem  game problem WX i +b ≥ 0 iff y i =1 WX i +b ≤ 0 iff y i =-1 yi( WX i +b) ≥ 0 ≥0

17. Support Vector Machines: Slide 17 Copyright © 2001, 2003, Andrew W. Moore Maximize Margin denotes +1 denotes -1 wx +b = 0 Margin Strategy: WX i +b≥0 iff y i =1 WX i +b ≤ 0 iff y i =-1 yi( WX i +b) ≥ 0 wx +b = 0 α (wx +b) = 0 where α ≠0

18. Support Vector Machines: Slide 18 Copyright © 2001, 2003, Andrew W. Moore Maximize Margin How does it come ? We have Thus,

19. Support Vector Machines: Slide 19 Copyright © 2001, 2003, Andrew W. Moore Maximum Margin Linear Classifier How to solve it?

20. Support Vector Machines: Slide 20 Copyright © 2001, 2003, Andrew W. Moore Learning via Quadratic Programming QP is a well-studied class of optimization algorithms to maximize a quadratic function of some real-valued variables subject to linear constraints. Detail solution of Quadratic Programming Convex Optimization Stephen P. Boyd Online Edition, Free for Downloading

21. Support Vector Machines: Slide 21 Copyright © 2001, 2003, Andrew W. Moore Quadratic Programming Find And subject to n additional linear inequality constraints e additional linear equality constraints Quadratic criterion Subject to

22. Support Vector Machines: Slide 22 Copyright © 2001, 2003, Andrew W. Moore Quadratic Programming for the Linear Classifier

23. Support Vector Machines: Slide 23 Copyright © 2001, 2003, Andrew W. Moore Online Demo Popular Tools - LibSVM

24. Support Vector Machines: Slide 24 Copyright © 2001, 2003, Andrew W. Moore Roadmap Hard-Margin Linear Classifier Maximize Margin Support Vector Quadratic Programming Soft-Margin Linear Classifier Maximize Margin Support Vector Quadratic Programming Non-Linear Separable Problem XOR Transform to Non-Linear by Kernels Reference

25. Support Vector Machines: Slide 25 Copyright © 2001, 2003, Andrew W. Moore Uh-oh! denotes +1 denotes -1 This is going to be a problem! What should we do?

26. Support Vector Machines: Slide 26 Copyright © 2001, 2003, Andrew W. Moore Uh-oh! denotes +1 denotes -1 This is going to be a problem! What should we do? Idea 1: Find minimum w.w, while minimizing number of training set errors. Problemette: Two things to minimize makes for an ill-defined optimization

27. Support Vector Machines: Slide 27 Copyright © 2001, 2003, Andrew W. Moore Uh-oh! denotes +1 denotes -1 This is going to be a problem! What should we do? Idea 1.1: Minimize w.w + C (#train errors) There’s a serious practical problem that’s about to make us reject this approach. Can you guess what it is? Tradeoff parameter

28. Support Vector Machines: Slide 28 Copyright © 2001, 2003, Andrew W. Moore Uh-oh! denotes +1 denotes -1 This is going to be a problem! What should we do? Idea 1.1: Minimize w.w + C (#train errors) There’s a serious practical problem that’s about to make us reject this approach. Can you guess what it is? Tradeoff parameter Can’t be expressed as a Quadratic Programming problem. Solving it may be too slow. (Also, doesn’t distinguish between disastrous errors and near misses) So… any other ideas?

29. Support Vector Machines: Slide 29 Copyright © 2001, 2003, Andrew W. Moore Uh-oh! denotes +1 denotes -1 This is going to be a problem! What should we do? Idea 2.0: Minimize w.w + C (distance of error points to their correct place)

30. Support Vector Machines: Slide 30 Copyright © 2001, 2003, Andrew W. Moore Support Vector Machine (SVM) for Noisy Data Any problem with the above formulism? denotes +1 denotes -1

31. Support Vector Machines: Slide 31 Copyright © 2001, 2003, Andrew W. Moore Support Vector Machine (SVM) for Noisy Data Balance the trade off between margin and classification errors denotes +1 denotes -1

32. Support Vector Machines: Slide 32 Copyright © 2001, 2003, Andrew W. Moore Support Vector Machine for Noisy Data How do we determine the appropriate value for c ?

33. Support Vector Machines: Slide 33 Copyright © 2001, 2003, Andrew W. Moore The Dual Form of QP Maximize where Subject to these constraints: Then define: Then classify with: f(x,w,b) = sign(w. x - b)

34. Support Vector Machines: Slide 34 Copyright © 2001, 2003, Andrew W. Moore The Dual Form of QP Maximize where Subject to these constraints: Then define:

35. Support Vector Machines: Slide 35 Copyright © 2001, 2003, Andrew W. Moore An Equivalent QP Maximize where Subject to these constraints: Then define: Datapoints with  k > 0 will be the support vectors..so this sum only needs to be over the support vectors.

36. Support Vector Machines: Slide 36 Copyright © 2001, 2003, Andrew W. Moore Support Vectors denotes +1 denotes -1 Support Vectors Decision boundary is determined only by those support vectors !  i = 0 for non-support vectors  i  0 for support vectors

37. Support Vector Machines: Slide 37 Copyright © 2001, 2003, Andrew W. Moore The Dual Form of QP Maximize where Subject to these constraints: Then define: Then classify with: f(x,w,b) = sign(w. x - b) How to determine b ?

38. Support Vector Machines: Slide 38 Copyright © 2001, 2003, Andrew W. Moore An Equivalent QP: Determine b A linear programming problem ! Fix w

39. Support Vector Machines: Slide 39 Copyright © 2001, 2003, Andrew W. Moore An Equivalent QP Maximize where Subject to these constraints: Then define: Then classify with: f(x,w,b) = sign(w. x - b) Datapoints with  k > 0 will be the support vectors..so this sum only needs to be over the support vectors. Why did I tell you about this equivalent QP? It’s a formulation that QP packages can optimize more quickly Because of further jaw- dropping developments you’re about to learn.

40. Support Vector Machines: Slide 40 Copyright © 2001, 2003, Andrew W. Moore Online Demo Parameter c is used to control the fitness Noise

41. Support Vector Machines: Slide 41 Copyright © 2001, 2003, Andrew W. Moore Roadmap Hard-Margin Linear Classifier (Clean Data) Maximize Margin Support Vector Quadratic Programming Soft-Margin Linear Classifier (Noisy Data) Maximize Margin Support Vector Quadratic Programming Non-Linear Separable Problem XOR Transform to Non-Linear by Kernels Reference

42. Support Vector Machines: Slide 42 Copyright © 2001, 2003, Andrew W. Moore Feature Transformation ? The problem is non-linear Find some trick to transform the input Linear separable after Feature Transformation What Features should we use ? XOR Problem Basic Idea :

43. Support Vector Machines: Slide 43 Copyright © 2001, 2003, Andrew W. Moore Suppose we’re in 1-dimension What would SVMs do with this data? x=0

44. Support Vector Machines: Slide 44 Copyright © 2001, 2003, Andrew W. Moore Suppose we’re in 1-dimension Not a big surprise Positive “plane” Negative “plane” x=0

45. Support Vector Machines: Slide 45 Copyright © 2001, 2003, Andrew W. Moore Harder 1-dimensional dataset That’s wiped the smirk off SVM’s face. What can be done about this? x=0

46. Support Vector Machines: Slide 46 Copyright © 2001, 2003, Andrew W. Moore Harder 1-dimensional dataset x=0 Map the data from low-dimensional space to high-dimensional space Let’s permit them here too

47. Support Vector Machines: Slide 47 Copyright © 2001, 2003, Andrew W. Moore Harder 1-dimensional dataset Map the data from low-dimensional space to high-dimensional space Let’s permit them here too x=0 Feature Enumeration

48. Support Vector Machines: Slide 48 Copyright © 2001, 2003, Andrew W. Moore Non-linear SVMs: Feature spaces General idea: the original input space can always be mapped to some higher-dimensional feature space where the training set is separable: Φ: x → φ(x)

49. Support Vector Machines: Slide 49 Copyright © 2001, 2003, Andrew W. Moore Online Demo Polynomial features for the XOR problem

50. Support Vector Machines: Slide 50 Copyright © 2001, 2003, Andrew W. Moore Online Demo But……Is it the best margin Intuitively?

51. Support Vector Machines: Slide 51 Copyright © 2001, 2003, Andrew W. Moore Online Demo Why not something like this ?

52. Support Vector Machines: Slide 52 Copyright © 2001, 2003, Andrew W. Moore Online Demo Or something like this ? Could We ? A More Symmetric Boundary

53. Support Vector Machines: Slide 53 Copyright © 2001, 2003, Andrew W. Moore Degree of Polynomial Features X^1X^2X^3 X^4X^5X^6

54. Support Vector Machines: Slide 54 Copyright © 2001, 2003, Andrew W. Moore Towards Infinite Dimensions of Features Enuermate polynomial features of all degrees ? Taylor Expension of exponential function zk = ( radial basis functions of xk )

55. Support Vector Machines: Slide 55 Copyright © 2001, 2003, Andrew W. Moore Online Demo “Radius basis functions” for the XOR problem

56. Support Vector Machines: Slide 56 Copyright © 2001, 2003, Andrew W. Moore Efficiency Problem in Computing Feature Feature space Mapping Example: all 2 degree Monomials 9 Multipllication 3 Multipllication kernel trick This use of kernel function to avoid carrying out Φ(x) explicitly is known as the kernel trick

57. Support Vector Machines: Slide 57 Copyright © 2001, 2003, Andrew W. Moore Common SVM basis functions z k = ( polynomial terms of x k of degree 1 to q ) z k = ( radial basis functions of x k ) z k = ( sigmoid functions of x k )

58. Support Vector Machines: Slide 58 Copyright © 2001, 2003, Andrew W. Moore Online Demo “Radius Basis Function” (Gaussian Kernel) Could solve complicated Non- Linear Problems γ and c control the complexity of decision boundary

59. Support Vector Machines: Slide 59 Copyright © 2001, 2003, Andrew W. Moore How to Control the Complexity Bob got up and found that breakfast was ready Level-1 His Child (Underfitting) Level-2 His Wife (Reasonble) Level-3 The Alien (Overfitting) Which reasoning below is the most probable?

60. Support Vector Machines: Slide 60 Copyright © 2001, 2003, Andrew W. Moore How to Control the Complexity SVM is powerful to approximate any training data The complexity affects the performance on new data SVM supports parameters for controlling the complexity SVM does not tell you how to set these parameters Determine the Parameters by Cross-Validation UnderfittingOverfitting complexity

61. Support Vector Machines: Slide 61 Copyright © 2001, 2003, Andrew W. Moore General Condition for Predictivity in Learning Theory Tomaso Poggio, Ryan Rifkin, Sayan Mukherjee and Partha Niyogi. General Condition for Predictivity in Learning Theory. Nature. Vol 428, March, 2004.

62. Support Vector Machines: Slide 62 Copyright © 2001, 2003, Andrew W. Moore Recall The MDL principle…… MDL stands for minimum description length The description length is defined as: Space required to described a theory + Space required to described the theory’s mistakes In our case the theory is the classifier and the mistakes are the errors on the training data Aim: we want a classifier with minimal DL MDL principle is a model selection criterion

63. Support Vector Machines: Slide 63 Copyright © 2001, 2003, Andrew W. Moore Support Vector Machine (SVM) for Noisy Data Balance the trade off between margin and classification errors denotes +1 denotes -1 Describe the TheoryDescribe the Mistake

64. Support Vector Machines: Slide 64 Copyright © 2001, 2003, Andrew W. Moore SVM Performance Anecdotally they work very very well indeed. Example: They are currently the best-known classifier on a well-studied hand-written-character recognition benchmark Another Example: Andrew knows several reliable people doing practical real-world work who claim that SVMs have saved them when their other favorite classifiers did poorly. There is a lot of excitement and religious fervor about SVMs as of 2001. Despite this, some practitioners are a little skeptical.

65. Support Vector Machines: Slide 65 Copyright © 2001, 2003, Andrew W. Moore References An excellent tutorial on VC-dimension and Support Vector Machines: C.J.C. Burges. A tutorial on support vector machines for pattern recognition. Data Mining and Knowledge Discovery, 2(2):955-974, 1998. http://citeseer.nj.nec.com/burges98tutorial.html The VC/SRM/SVM Bible: (Not for beginners including myself) Statistical Learning Theory by Vladimir Vapnik, Wiley- Interscience; 1998 Software: SVM-light, http://svmlight.joachims.org/, LibSVM, http://www.csie.ntu.edu.tw/~cjlin/libsvm/ SMO in Wekahttp://svmlight.joachims.org/http://www.csie.ntu.edu.tw/~cjlin/libsvm/

66. Nov 23rd, 2001Copyright © 2001, 2003, Andrew W. Moore Support Vector Regression

67. Support Vector Machines: Slide 67 Copyright © 2001, 2003, Andrew W. Moore Roadmap Squared-Loss Linear Regression Little Noise Large Noise Linear-Loss Function Support Vector Regression

68. Support Vector Machines: Slide 68 Copyright © 2001, 2003, Andrew W. Moore Linear Regression f x  y est f(x,w,b) = w. x - b How would you fit this data?

69. Support Vector Machines: Slide 69 Copyright © 2001, 2003, Andrew W. Moore Linear Regression f x  y est f(x,w,b) = w. x - b How would you fit this data?

70. Support Vector Machines: Slide 70 Copyright © 2001, 2003, Andrew W. Moore Linear Regression f x  y est f(x,w,b) = w. x - b How would you fit this data?

71. Support Vector Machines: Slide 71 Copyright © 2001, 2003, Andrew W. Moore Linear Regression f x  y est f(x,w,b) = w. x - b How would you fit this data?

72. Support Vector Machines: Slide 72 Copyright © 2001, 2003, Andrew W. Moore Linear Regression f x  y est f(x,w,b) = w. x - b Any of these would be fine....but which is best?

73. Support Vector Machines: Slide 73 Copyright © 2001, 2003, Andrew W. Moore Linear Regression f x  y est f(x,w,b) = w. x - b How to define the fitting error of a linear regression ?

74. Support Vector Machines: Slide 74 Copyright © 2001, 2003, Andrew W. Moore Linear Regression f x  y est f(x,w,b) = w. x - b How to define the fitting error of a linear regression ? Squared-Loss

75. Support Vector Machines: Slide 75 Copyright © 2001, 2003, Andrew W. Moore Online Demo http://www.math.csusb.edu/faculty/stanton/m262 /regress/regress.html

76. Support Vector Machines: Slide 76 Copyright © 2001, 2003, Andrew W. Moore Sensitive to Outliers Outlier

77. Support Vector Machines: Slide 77 Copyright © 2001, 2003, Andrew W. Moore Why ? Squared-Loss Function Fitting Error Grows Quadratically

78. Support Vector Machines: Slide 78 Copyright © 2001, 2003, Andrew W. Moore How about Linear-Loss ? Linear-Loss Function Fitting Error Grows Linearly

79. Support Vector Machines: Slide 79 Copyright © 2001, 2003, Andrew W. Moore Actually SVR uses the Loss Function below  -insensitive loss function 

80. Support Vector Machines: Slide 80 Copyright © 2001, 2003, Andrew W. Moore Epsilon Support Vector Regression (  -SVR) Given: a data set {x 1,..., x n } with target values {u 1,..., u n }, we want to do  -SVR The optimization problem is Similar to SVM, this can be solved as a quadratic programming problem

81. Support Vector Machines: Slide 81 Copyright © 2001, 2003, Andrew W. Moore Online Demo Less Sensitive to Outlier

82. Support Vector Machines: Slide 82 Copyright © 2001, 2003, Andrew W. Moore Again, Extend to Non-Linear Case Similar with SVM

83. Support Vector Machines: Slide 83 Copyright © 2001, 2003, Andrew W. Moore What We Learn Linear Classifier with Clean Data Linear Classifier with Noisy Data SVM for Noisy and Non-Linear Data Linear Regression with Clean Data Linear Regression with Noisy Data SVR for Noisy and Non-Linear Data General Condition for Predictivity in Learning Theory

84. Support Vector Machines: Slide 84 Copyright © 2001, 2003, Andrew W. Moore The End

85. Support Vector Machines: Slide 85 Copyright © 2001, 2003, Andrew W. Moore Saddle Point