Support Vector Machines

Transcription

1 Support Vector Machines Chih-Jen Lin Department of Computer Science National Taiwan University Talk at Machine Learning Summer School 2006, Taipei Chih-Jen Lin (National Taiwan Univ.) MLSS 2006, Taipei 1 / 98

2 Outline Basic concepts SVM primal/dual problems Training linear and nonlinear SVMs Parameter/kernel selection and practical issues Multi-class classification Discussion and conclusions Chih-Jen Lin (National Taiwan Univ.) MLSS 2006, Taipei 2 / 98

3 Outline Basic concepts Basic concepts SVM primal/dual problems Training linear and nonlinear SVMs Parameter/kernel selection and practical issues Multi-class classification Discussion and conclusions Chih-Jen Lin (National Taiwan Univ.) MLSS 2006, Taipei 3 / 98

4 Basic concepts Why SVM and Kernel Methods SVM: in many cases competitive with existing classification methods Relatively easy to use Kernel techniques: many extensions Regression, density estimation, kernel PCA, etc. Chih-Jen Lin (National Taiwan Univ.) MLSS 2006, Taipei 4 / 98

5 Basic concepts Support Vector Classification Training vectors : x i, i = 1,...,l Feature vectors. For example, A patient = [height, weight,...] Consider a simple case with two classes: Define an indicator vector y { 1 if xi in class 1 y i = 1 if x i in class 2, A hyperplane which separates all data Chih-Jen Lin (National Taiwan Univ.) MLSS 2006, Taipei 5 / 98

6 Basic concepts w T x + b = [ +1 ] 0 1 A separating hyperplane: w T x + b = 0 (w T x i ) + b > 0 if y i = 1 (w T x i ) + b < 0 if y i = 1 Decision function f (x) = sgn(w T x+b), x: test data Many possible choices of w and b Chih-Jen Lin (National Taiwan Univ.) MLSS 2006, Taipei 6 / 98

7 Maximal Margin Basic concepts Distance between w T x + b = 1 and 1: 2/ w = 2/ w T w A quadratic programming problem [Boser et al., 1992] min w,b 1 2 wt w subject to y i (w T x i + b) 1, i = 1,...,l. Chih-Jen Lin (National Taiwan Univ.) MLSS 2006, Taipei 7 / 98

8 Basic concepts Data May Not Be Linearly Separable An example: Allow training errors Higher dimensional ( maybe infinite ) feature space φ(x) = (φ 1 (x), φ 2 (x),...). Chih-Jen Lin (National Taiwan Univ.) MLSS 2006, Taipei 8 / 98

9 Basic concepts Standard SVM [Cortes and Vapnik, 1995] min w,b,ξ 1 2 wt w + C l i=1 subject to y i (w T φ(x i ) + b) 1 ξ i, ξ i 0, i = 1,...,l. Example: x R 3, φ(x) R 10 φ(x) = (1, 2x 1, 2x 2, 2x 3, x 2 1, x 2 2, x 2 3, 2x 1 x 2, 2x 1 x 3, 2x 2 x 3 ) ξ i Chih-Jen Lin (National Taiwan Univ.) MLSS 2006, Taipei 9 / 98

10 Basic concepts Finding the Decision Function w: maybe infinite variables The dual problem min α subject to 1 2 αt Qα e T α 0 α i C, i = 1,...,l y T α = 0, where Q ij = y i y j φ(x i ) T φ(x j ) and e = [1,...,1] T At optimum w = l i=1 α iy i φ(x i ) A finite problem: #variables = #training data Chih-Jen Lin (National Taiwan Univ.) MLSS 2006, Taipei 10 / 98

11 Kernel Tricks Basic concepts Q ij = y i y j φ(x i ) T φ(x j ) needs a closed form Example: x i R 3, φ(x i ) R 10 φ(x i ) = (1, 2(x i ) 1, 2(x i ) 2, 2(x i ) 3, (x i ) 2 1, (x i ) 2 2, (x i ) 2 3, 2(x i ) 1 (x i ) 2, 2(x i ) 1 (x i ) 3, 2(x i ) 2 (x i ) 3 ) Then φ(x i ) T φ(x j ) = (1 + x T i x j ) 2. Kernel: K(x,y) = φ(x) T φ(y); common kernels: e γ x i x j 2, (Radial Basis Function) (x T i x j /a + b) d (Polynomial kernel) Chih-Jen Lin (National Taiwan Univ.) MLSS 2006, Taipei 11 / 98

12 Basic concepts Can be inner product in infinite dimensional space Assume x R 1 and γ > 0. e γ x i x j 2 = e γ(x i x j ) 2 = e γx2 i +2γx ix j γxj 2 ( 2γx i x j (2γx ix j ) 2 =e γx2 i γx2 j =e γx2 i γx2 j where + (2γx ix j ) 3 + ) 1! 2! 3! ( 2γ 2γ (2γ) ! x i 1! x 2 (2γ) j+ xi 2 2 2! 2! (2γ) 3 (2γ) + xi 3 3 xj 3 + ) = φ(x i ) T φ(x j ), 3! 3! φ(x) = e γx2 [1, 2γ (2γ) 1! x, 2 (2γ) x 2 3 T, x 3, ]. 2! 3! x 2 j Chih-Jen Lin (National Taiwan Univ.) MLSS 2006, Taipei 12 / 98

13 More about Kernels Basic concepts How do we know kernels help to separate data? In R l, any l independent vectors linearly separable (x 1 ) T. w = (x l ) T [ ] +e e If K positive definite data linearly separable K = LL T. Transforming training points to independent vectors in R l Chih-Jen Lin (National Taiwan Univ.) MLSS 2006, Taipei 13 / 98

14 Basic concepts So what kind of kernel should I use? What kind of functions are valid kernels? How to decide kernel parameters? Will be discussed later Chih-Jen Lin (National Taiwan Univ.) MLSS 2006, Taipei 14 / 98

15 Decision function Basic concepts At optimum Decision function = = w = l i=1 α iy i φ(x i ) w T φ(x) + b l α i y i φ(x i ) T φ(x) + b i=1 l α i y i K(x i,x) + b i=1 Only φ(x i ) of α i > 0 used support vectors Chih-Jen Lin (National Taiwan Univ.) MLSS 2006, Taipei 15 / 98

16 Basic concepts Support Vectors: More Important Data Chih-Jen Lin (National Taiwan Univ.) MLSS 2006, Taipei 16 / 98

17 Basic concepts So we have roughly shown basic ideas of SVM A 3-D demonstration cjlin/libsvmtools/svmtoy3d Further references, for example, [Cristianini and Shawe-Taylor, 2000, Schölkopf and Smola, 2002] Also see discussion on kernel machines blackboard Chih-Jen Lin (National Taiwan Univ.) MLSS 2006, Taipei 17 / 98

18 Outline SVM primal/dual problems Basic concepts SVM primal/dual problems Training linear and nonlinear SVMs Parameter/kernel selection and practical issues Multi-class classification Discussion and conclusions Chih-Jen Lin (National Taiwan Univ.) MLSS 2006, Taipei 18 / 98

19 Deriving the Dual SVM primal/dual problems Consider the problem without ξ i min w,b subject to 1 2 wt w y i (w T φ(x i ) + b) 1, i = 1,...,l. Its dual min α 1 2 αt Qα e T α subject to 0 α i, i = 1,...,l, y T α = 0. Chih-Jen Lin (National Taiwan Univ.) MLSS 2006, Taipei 19 / 98

20 Lagrangian Dual SVM primal/dual problems where ( max min L(w, b, α)), α 0 w,b L(w, b, α) = 1 2 w 2 l ( α i yi (w T φ(x i ) + b) 1 ) i=1 Strong duality (be careful about this) ( min Primal = max min L(w, b, α)) α 0 w,b Chih-Jen Lin (National Taiwan Univ.) MLSS 2006, Taipei 20 / 98

21 SVM primal/dual problems Simplify the dual. When α is fixed, min L(w, b, α) = w,b { if l i=1 α iy i 0, 1 min w 2 wt w l i=1 α i[y i (w T φ(x i ) 1] if l i=1 α iy i = 0. If l i=1 α iy i 0, decrease in L(w, b, α) to b l α i y i i=1 Chih-Jen Lin (National Taiwan Univ.) MLSS 2006, Taipei 21 / 98

22 SVM primal/dual problems If l i=1 α iy i = 0, optimum of the strictly convex 1 2 wt w l i=1 α i[y i (w T φ(x i ) 1] happens when Thus, L(w, b, α) = 0. w w = l α i y i φ(x i ). i=1 Chih-Jen Lin (National Taiwan Univ.) MLSS 2006, Taipei 22 / 98

23 SVM primal/dual problems Note that w T w = = i,j ( l i=1 ) T ( l α i y i φ(x i ) j=1 α i α j y i y j φ(x i ) T φ(x j ) ) α j y j φ(x j ) max α 0 The dual is l α i 1 2 α i α j y i y j φ(x i ) T φ(x j ) i=1 i,j if l i=1 α iy i = 0, if l i=1 α iy i 0. Chih-Jen Lin (National Taiwan Univ.) MLSS 2006, Taipei 23 / 98

24 SVM primal/dual problems Lagrangian dual: max α 0 ( minw,b L(w, b, α) ) definitely not maximum of the dual Dual optimal solution not happen when. Dual simplified to max α R l l α i y i 0 i=1 l α i 1 2 i=1 l i=1 l α i α j y i y j φ(x i ) T φ(x j ) j=1 subject to y T α = 0, α i 0, i = 1,...,l. Chih-Jen Lin (National Taiwan Univ.) MLSS 2006, Taipei 24 / 98

25 SVM primal/dual problems More about Dual Problems After SVM is popular Quite a few people think that for any optimization problem Lagrangian dual exists and strong duality holds Wrong! We usually need Convex programming; Constraint qualification We have them SVM primal is convex; Linear constraints Chih-Jen Lin (National Taiwan Univ.) MLSS 2006, Taipei 25 / 98

26 SVM primal/dual problems Our problems may be infinite dimensional Can still use Lagrangian duality See a rigorous discussion in [Lin, 2001] Chih-Jen Lin (National Taiwan Univ.) MLSS 2006, Taipei 26 / 98

27 Outline Training linear and nonlinear SVMs Basic concepts SVM primal/dual problems Training linear and nonlinear SVMs Parameter/kernel selection and practical issues Multi-class classification Discussion and conclusions Chih-Jen Lin (National Taiwan Univ.) MLSS 2006, Taipei 27 / 98

28 Training Nonlinear SVMs Training linear and nonlinear SVMs If using kernels, we solve the dual min α subject to 1 2 αt Qα e T α 0 α i C, i = 1,...,l y T α = 0 Large dense quadratic programming Q ij 0, Q : an l by l fully dense matrix 30,000 training points: 30,000 variables: (30, /2) bytes = 3GB RAM to store Q: Traditional methods: Newton, Quasi Newton cannot be directly applied Chih-Jen Lin (National Taiwan Univ.) MLSS 2006, Taipei 28 / 98

29 Decomposition Methods Training linear and nonlinear SVMs Working on some variables each time (e.g., [Osuna et al., 1997, Joachims, 1998, Platt, 1998]) Similar to coordinate-wise minimization Working set B, N = {1,...,l}\B fixed Sub-problem at each iteration: subject to ] [ 1 [ min α T α B 2 B (α k N )T][ Q BB Q BN αb Q NB Q NN α k N ] [ e T B (e k N )T][ α B α k N ] 0 α t C, t B, y T Bα B = y T Nα k N Chih-Jen Lin (National Taiwan Univ.) MLSS 2006, Taipei 29 / 98

30 Avoid Memory Problems Training linear and nonlinear SVMs The new objective function 1 2 αt BQ BB α B + ( e B + Q BN α k N) T α B + constant B columns of Q needed Calculated when used Trade time for space Chih-Jen Lin (National Taiwan Univ.) MLSS 2006, Taipei 30 / 98

31 Does it Really Work? Training linear and nonlinear SVMs Compared to Newton, Quasi-Newton Slow convergence However, no need to have very accurate α ( l ) sgn α i y i K(x i,x) + b i=1 Prediction not affected much In some situations, # support vectors # training points Initial α 1 = 0, some elements never used Machine learning knowledge affects optimization Chih-Jen Lin (National Taiwan Univ.) MLSS 2006, Taipei 31 / 98

32 Training linear and nonlinear SVMs An example of training 50,000 instances using LIBSVM $./svm-train -m 200 -c 16 -g 4 22features optimization finished, #iter = Total nsv = 3370 time 5m1.456s On a Pentium M 1.4 GHz Laptop Calculating Q may have taken more than 5 minutes #SVs = 3,370 50,000 A good case where some remain at zero all the time Chih-Jen Lin (National Taiwan Univ.) MLSS 2006, Taipei 32 / 98

33 Training linear and nonlinear SVMs Issues of Decomposition Methods Working set size/selection Asymptotic convergence Finite termination & stopping conditions Convergence rate Numerical issues Optimization researchers are now also interested in these issues If interested in them, check my talk to optimization researchers in Rome last year: cjlin/talks/rome.pdf Chih-Jen Lin (National Taiwan Univ.) MLSS 2006, Taipei 33 / 98

34 Caching and Shrinking Training linear and nonlinear SVMs Speed up decomposition methods Caching [Joachims, 1998] Store recently used kernel columns in computer memory 100K Cache $ time./libsvm-2.81/svm-train -m 0.01 a4a s 40M Cache $ time./libsvm-2.81/svm-train -m 40 a4a 7.817s Chih-Jen Lin (National Taiwan Univ.) MLSS 2006, Taipei 34 / 98

35 Training linear and nonlinear SVMs Shrinking [Joachims, 1998] Some bounded elements remain until the end Heuristically resized to a smaller problem After certain iterations, most bounded elements identified and not changed [Lin, 2002] So caching and shrinking are useful Chih-Jen Lin (National Taiwan Univ.) MLSS 2006, Taipei 35 / 98

36 Caching: Issues Training linear and nonlinear SVMs A simple way: Store recently used columns What if in working set selection, deliberately select some indices in cache Goal: minimize the total number of columns calculated Difficult to connect algorithm and this goal Chih-Jen Lin (National Taiwan Univ.) MLSS 2006, Taipei 36 / 98

37 SVM doesn t Scale Up Training linear and nonlinear SVMs Yes, if you use kernels Training millions of data is time consuming But other nonlinear methods face the same problem e.g., kernel logistic regression Two possibilities 1 Linear SVMs: in some situations, can solve much larger problems 2 Approximation Chih-Jen Lin (National Taiwan Univ.) MLSS 2006, Taipei 37 / 98

38 Training Linear SVMs Training linear and nonlinear SVMs Linear kernel: min w,b,ξ 1 2 wt w + C l i=1 subject to y i (w T x i + b) 1 ξ i, ξ i 0. At optimum: ξ i = max ( 0, 1 y i (w T x i + b) ) ξ i Chih-Jen Lin (National Taiwan Univ.) MLSS 2006, Taipei 38 / 98

39 Training linear and nonlinear SVMs Remaining variables: w, b min w,b 1 2 wt w + C l max ( 0, 1 y i (w T x i + b) ) i=1 #variables = #features + 1 If #features small, easier to solve Chih-Jen Lin (National Taiwan Univ.) MLSS 2006, Taipei 39 / 98

40 Training linear and nonlinear SVMs Traditional optimization methods can be applied Training time similar to methods such as logistic regression What if #features and #instances both large? Very challenging Some language/document problems are of this type Chih-Jen Lin (National Taiwan Univ.) MLSS 2006, Taipei 40 / 98

41 Training linear and nonlinear SVMs Decomposition Methods for Linear SVMs Could we still solve the dual by decomposition methods? Even if #features small Slow convergence when C is large $bsvm-train -b 500 -c 500 -t 0 australian_scale optimization finished, #iter = obj = , rho = K ij = x T i x j, rank #features positive semi-definite only Still a research topic in understanding this Chih-Jen Lin (National Taiwan Univ.) MLSS 2006, Taipei 41 / 98

42 Training linear and nonlinear SVMs Decomposition Methods for Linear SVMs But no need to use large C C large enough, w the same [Keerthi and Lin, 2003] decision function the same Remember w = l α i y i x i R n, b R 1 i=1 # of 0 < α i < C n + 1 As C changes, optimal α share many elements at 0 and C Chih-Jen Lin (National Taiwan Univ.) MLSS 2006, Taipei 42 / 98

43 Training linear and nonlinear SVMs Decomposition Methods for Linear SVMs (Cont d) Warm start very effective [Kao et al., 2004] Starting from small C, faster convergence Using C = 1, 2, 4, 8,... $bsvm-train -c 500 -t 0 australian_scale optimization finished, #iter = So decomposition methods can still handle large linear SVMs Chih-Jen Lin (National Taiwan Univ.) MLSS 2006, Taipei 43 / 98

44 Approximations Training linear and nonlinear SVMs #instances large and using nonlinear kernels Difficult to solve the dual Subsampling Simple and often effective From this many more advanced techniques E.g., stratified subsampling Chih-Jen Lin (National Taiwan Univ.) MLSS 2006, Taipei 44 / 98

45 Approximations (Cont d) Training linear and nonlinear SVMs Incremental way: (e.g., [Syed et al., 1999]) Data 10 parts train 1st part SVs, train SVs + 2nd part,... Select good points first: KNN or heuristics e.g., [Bakır et al., 2005] Hierarchical settings (e.g., [Yu et al., 2003]) Clustering training data to several groups SVM models built for each group Chih-Jen Lin (National Taiwan Univ.) MLSS 2006, Taipei 45 / 98

46 Approximations (Cont d) Training linear and nonlinear SVMs Using only a subset to construct w w = i B α i y i φ(x i ). Put this into the primal min α B,b,ξ subject to 1 2 αt BQ BB α B + C l i=1 Q :,B α B + by e ξ ξ i Without considering ξ i, #variables = B + 1 Chih-Jen Lin (National Taiwan Univ.) MLSS 2006, Taipei 46 / 98

47 Approximations (Cont d) Training linear and nonlinear SVMs Selecting B: random [Lee and Mangasarian, 2001], incremental [Keerthi et al., 2006], and many other ways Chih-Jen Lin (National Taiwan Univ.) MLSS 2006, Taipei 47 / 98

48 Approximations (Cont d) Training linear and nonlinear SVMs All these approaches some simple but some sophisticated In machine learning, very often balance between simplification and performance Chih-Jen Lin (National Taiwan Univ.) MLSS 2006, Taipei 48 / 98

49 Outline Parameter/kernel selection and practical issues Basic concepts SVM primal/dual problems Training linear and nonlinear SVMs Parameter/kernel selection and practical issues Multi-class classification Discussion and conclusions Chih-Jen Lin (National Taiwan Univ.) MLSS 2006, Taipei 49 / 98

50 Parameter/kernel selection and practical issues Let s Try a Practical Example A problem from astroparticle physics 1 1:2.6173e+01 2: e+01 3: e-01 4: e :5.7073e+01 2: e+02 3: e-02 4: e :1.7259e+01 2: e+02 3: e-01 4: e :2.1779e+01 2: e+02 3: e-01 4: e :9.1339e+01 2: e+02 3: e-01 4: e :5.5375e+01 2: e+02 3: e-01 4: e :2.9562e+01 2: e+02 3: e-02 4: e+02 Training and testing sets available: 3,089 and 4,000 Chih-Jen Lin (National Taiwan Univ.) MLSS 2006, Taipei 50 / 98

51 Parameter/kernel selection and practical issues The Story Behind this Data Set User: I am using libsvm in a astroparticle physics application.. First, let me congratulate you to a really easy to use and nice package. Unfortunately, it gives me astonishingly bad results... OK. Please send us your data I am able to get 97% test accuracy. Is that good enough for you? User: You earned a copy of my PhD thesis Chih-Jen Lin (National Taiwan Univ.) MLSS 2006, Taipei 51 / 98

52 Training and Testing Training Parameter/kernel selection and practical issues $./svm-train train.1 optimization finished, #iter = 6131 nsv = 3053, nbsv = 724 Total nsv = 3053 Testing $./svm-predict test.1 train.1.model test.1.out Accuracy = % (2677/4000) nsv and nbsv: number of SVs and bounded SVs (α i = C). Chih-Jen Lin (National Taiwan Univ.) MLSS 2006, Taipei 52 / 98

53 Why this Fails Parameter/kernel selection and practical issues After training, nearly 100% support vectors Training and testing accuracy different $./svm-predict train.1 train.1.model o Accuracy = % (3082/3089) Most kernel elements: K ij = e x i x j 2 /4 { = 1 if i = j, 0 if i j. Some features in rather large ranges Chih-Jen Lin (National Taiwan Univ.) MLSS 2006, Taipei 53 / 98

54 Data Scaling Parameter/kernel selection and practical issues Without scaling Attributes in greater numeric ranges may dominate Example: and height gender x F x M x M y 1 = 0, y 2 = 1, y 3 = 1. Chih-Jen Lin (National Taiwan Univ.) MLSS 2006, Taipei 54 / 98

55 Parameter/kernel selection and practical issues The separating hyperplane almost vertical x 1 x 2 x 3 Strongly depends on the first attribute; but second may be also important Linearly scale the first to [0, 1] by: 1st attribute 150, Scaling generally helps, but not always Chih-Jen Lin (National Taiwan Univ.) MLSS 2006, Taipei 55 / 98

56 Parameter/kernel selection and practical issues Other ways for scaling Needed for k Nearest Neighbor, Neural networks as well unless the method is scale-invariant Chih-Jen Lin (National Taiwan Univ.) MLSS 2006, Taipei 56 / 98

57 Data Scaling: Same Factors Parameter/kernel selection and practical issues A common mistake $./svm-scale -l -1 -u 1 train.1 > train.1.scale $./svm-scale -l -1 -u 1 test.1 > test.1.scale Same factor on training and testing $./svm-scale -s range1 train.1 > train.1.scale $./svm-scale -r range1 test.1 > test.1.scale Chih-Jen Lin (National Taiwan Univ.) MLSS 2006, Taipei 57 / 98

58 After Data Scaling Parameter/kernel selection and practical issues Train scaled data and then prediction $./svm-train train.1.scale $./svm-predict test.1.scale train.1.scale.model test.1.predict Accuracy = 96.15% Training accuracy now is $./svm-predict train.1.scale train.1.scale.model Accuracy = % (2979/3089) Default parameter: C = 1, γ = 0.25 Chih-Jen Lin (National Taiwan Univ.) MLSS 2006, Taipei 58 / 98

59 Different Parameters Parameter/kernel selection and practical issues If we use C = 20, γ = 400 $./svm-train -c 20 -g 400 train.1.scale $./svm-predict train.1.scale train.1.scale.m Accuracy = 100% (3089/3089) 100% training accuracy but $./svm-predict test.1.scale train.1.scale.mo Accuracy = 82.7% (3308/4000) Very bad test accuracy Overfitting happens Chih-Jen Lin (National Taiwan Univ.) MLSS 2006, Taipei 59 / 98

60 Overfitting Parameter/kernel selection and practical issues In theory You can easily achieve 100% training accuracy This is useless When training and predicting a data, we should Avoid underfitting: small training error Avoid overfitting: small testing error Chih-Jen Lin (National Taiwan Univ.) MLSS 2006, Taipei 60 / 98

61 Parameter/kernel selection and practical issues and : training; and : testing Chih-Jen Lin (National Taiwan Univ.) MLSS 2006, Taipei 61 / 98

62 Parameter Selection Parameter/kernel selection and practical issues Is important Now parameters are C, kernel parameters Example: How to select them? So performance better? γ of e γ x i x j 2 a, b, d of (x T i x j /a + b) d Chih-Jen Lin (National Taiwan Univ.) MLSS 2006, Taipei 62 / 98

63 Parameter Selection (Cont d) Parameter/kernel selection and practical issues Also how to select kernels? e.g., RBF or polynomial Moreover, how to select methods? e.g., SVM or decision trees? Chih-Jen Lin (National Taiwan Univ.) MLSS 2006, Taipei 63 / 98

64 Performance Evaluation Parameter/kernel selection and practical issues l training data, x i R n, y i {+1, 1}, i = 1,...,l, a learning machine: x f (x, α), f (x, α) = 1 or 1. Different α: different machines The expected test error (generalized error) R(α) = y: class of x (i.e. 1 or -1) 1 y f (x, α) dp(x, y) 2 Chih-Jen Lin (National Taiwan Univ.) MLSS 2006, Taipei 64 / 98

65 Parameter/kernel selection and practical issues P(x, y) unknown, empirical risk (training error): R emp (α) = 1 2l l y i f (x i, α) i=1 Training errors not important; only test errors count 1 2 y i f (x i, α) : loss, choose 0 η 1, with probability at least 1 η: R(α) R emp (α) + another term A good classification method: minimize both terms at the same time Chih-Jen Lin (National Taiwan Univ.) MLSS 2006, Taipei 65 / 98

66 Parameter/kernel selection and practical issues But R emp (α) 0; another term large SVM: min w,b,ξ 1 2 wt w + C l i=1 subject to y i (w T φ(x i ) + b) 1 ξ i, ξ i 0, i = 1,. l i=1 ξ i related to training error w T w/2 relate to another term: called regularization term C: balance between the two ξ i Chih-Jen Lin (National Taiwan Univ.) MLSS 2006, Taipei 66 / 98

67 Parameter/kernel selection and practical issues Performance Evaluation (Cont d) In practice Available data training and validation Train the training Test the validation k-fold cross validation: Data randomly separated to k groups Each time k 1 as training and one as testing Chih-Jen Lin (National Taiwan Univ.) MLSS 2006, Taipei 67 / 98

68 Parameter/kernel selection and practical issues Using CV on training + validation Predict testing with the best parameters from CV Chih-Jen Lin (National Taiwan Univ.) MLSS 2006, Taipei 68 / 98

69 CV and Test Accuracy Parameter/kernel selection and practical issues If we select parameters so that CV is the highest, Does CV represent future test accuracy? Slightly different If we have enough parameters, we can achieve 100% CV as well e.g., more parameters than # of training data Available data with class labels training, validation, testing Chih-Jen Lin (National Taiwan Univ.) MLSS 2006, Taipei 69 / 98

70 Selecting Kernels Parameter/kernel selection and practical issues RBF, polynomial, or others? or even combinations Two situations: Too many kernels complicates the selection Design kernels suitable for target applications Chih-Jen Lin (National Taiwan Univ.) MLSS 2006, Taipei 70 / 98

71 Selecting Kernels (Cont d) Parameter/kernel selection and practical issues Contradicting but practically ok We have few general kernels RBF, polynomial, etc. somewhat related Beginners don t have many choices On the other hand researchers design many special ones e.g., string kernels Chih-Jen Lin (National Taiwan Univ.) MLSS 2006, Taipei 71 / 98

72 Selecting Kernels (Cont d) Parameter/kernel selection and practical issues For beginners, use RBF first Linear kernel: special case of RBF Performance of linear the same as RBF under certain parameters [Keerthi and Lin, 2003] Polynomial: numerical difficulties (< 1) d 0, (> 1) d More parameters than RBF Chih-Jen Lin (National Taiwan Univ.) MLSS 2006, Taipei 72 / 98

73 A Simple Procedure Parameter/kernel selection and practical issues 1 Conduct simple scaling on the data 2 Consider RBF kernel K(x,y) = e γ x y 2 3 Use cross-validation to find the best parameter C and γ 4 Use the best C and γ to train the whole training set 5 Test For beginners only, you can do a lot more Chih-Jen Lin (National Taiwan Univ.) MLSS 2006, Taipei 73 / 98

74 Parameter/kernel selection and practical issues Contour of Parameter Selection d lg(gamma) lg(c) Chih-Jen Lin (National Taiwan Univ.) MLSS 2006, Taipei 74 / 98

75 Parameter/kernel selection and practical issues The good region of parameters is quite large SVM is sensitive to parameters, but not that sensitive Sometimes default parameters work but it s good to select them if time is allowed Chih-Jen Lin (National Taiwan Univ.) MLSS 2006, Taipei 75 / 98

76 Efficient Parameter Selection Parameter/kernel selection and practical issues CV on grid points may be time consuming OK if one or two parameters But if more than two? E.g., feature scaling: K(x,y) = e n i=1 γ i(x i y i ) 2 Some features more important Still a challenging research issue Chih-Jen Lin (National Taiwan Univ.) MLSS 2006, Taipei 76 / 98

77 Parameter/kernel selection and practical issues Remember given parameters C and γ, we solve SVM to obtain optimal w or α Model a function of parameters min f (α(c, γ 1,...,γ n ), C, γ 1,...,γ n ) C,γ 1,...,γ n But usually non-convex The function from Bayesian frameworks (e.g., [Chu et al., 2003]) or smoothing CV bound CV(C, γ 1,...,γ n ) f (α(c, γ 1,...,γ n ), C, γ 1,...,γ n ) Chih-Jen Lin (National Taiwan Univ.) MLSS 2006, Taipei 77 / 98

78 Parameter/kernel selection and practical issues The minimization: Gradient-type methods or global optimization (e.g., genetic algorithms) The difficulty: Certainly more efforts than one single γ But performance may be just similar? Chih-Jen Lin (National Taiwan Univ.) MLSS 2006, Taipei 78 / 98

79 Kernel Combination Parameter/kernel selection and practical issues How about using where t 1 K 1 + t 2 K t r K r, t t r = 1 as the kernel Related to parameter selection t 1 e γ 1 x y + + t r e γ r x y If γ 1 good t 1 close to 1, others close to 0 Chih-Jen Lin (National Taiwan Univ.) MLSS 2006, Taipei 79 / 98

80 Parameter/kernel selection and practical issues [Lanckriet et al., 2004] form a convex f (α(t 1,...,t r ), t 1,...,t r ) when C is fixed Semi-definite programming problem But computational cost is also high Need more empirical studies Chih-Jen Lin (National Taiwan Univ.) MLSS 2006, Taipei 80 / 98

81 Design Kernels Parameter/kernel selection and practical issues Still a research issue e.g., in bioinformatics and vision, many new kernels But, should be careful if the function is a valid one K(x,y) = φ(x) T φ(y) For example, any two strings s 1, s 2 we can define edit distance e γedit(s 1,s 2 ) It s not a valid kernel [Cortes et al., 2003] Chih-Jen Lin (National Taiwan Univ.) MLSS 2006, Taipei 81 / 98

82 Mercer condition Parameter/kernel selection and practical issues What kind of K ij can be represented as φ(x i ) T φ(x j )? K(x,y) = φ(x) T φ(y) if and only if g s.t. g(x) 2 dx finite K(x,y)g(x)g(y)dxdy 0 A condition developed early last century However, still not easy to check Chih-Jen Lin (National Taiwan Univ.) MLSS 2006, Taipei 82 / 98

83 Outline Multi-class classification Basic concepts SVM primal/dual problems Training linear and nonlinear SVMs Parameter/kernel selection and practical issues Multi-class classification Discussion and conclusions Chih-Jen Lin (National Taiwan Univ.) MLSS 2006, Taipei 83 / 98

84 Multi-class Classification Multi-class classification k classes One-against-the rest: Train k binary SVMs: 1st class vs. (2 k)th class 2nd class vs.. (1, 3 k)th class k decision functions (w 1 ) T φ(x) + b 1. (w k ) T φ(x) + b k Chih-Jen Lin (National Taiwan Univ.) MLSS 2006, Taipei 84 / 98

85 Multi-class classification Prediction: arg max j (w j ) T φ(x) + b j Reason: If the 1st class, then we should have (w 1 ) T φ(x) + b 1 +1 (w 2 ) T φ(x) + b 2 1. (w k ) T φ(x) + b k 1 Chih-Jen Lin (National Taiwan Univ.) MLSS 2006, Taipei 85 / 98

86 Multi-class classification Multi-class Classification (Cont d) One-against-one: train k(k 1)/2 binary SVMs (1, 2), (1, 3),...,(1, k), (2, 3), (2, 4),...,(k 1, k) If 4 classes 6 binary SVMs y i = 1 y i = 1 Decision functions class 1 class 2 f 12 (x) = (w 12 ) T x + b 12 class 1 class 3 f 13 (x) = (w 13 ) T x + b 13 class 1 class 4 f 14 (x) = (w 14 ) T x + b 14 class 2 class 3 f 23 (x) = (w 23 ) T x + b 23 class 2 class 4 f 24 (x) = (w 24 ) T x + b 24 class 3 class 4 f 34 (x) = (w 34 ) T x + b 34 Chih-Jen Lin (National Taiwan Univ.) MLSS 2006, Taipei 86 / 98

87 Multi-class classification For a testing data, predicting all binary SVMs Classes winner Select the one with the largest vote class # votes May use decision values as well Chih-Jen Lin (National Taiwan Univ.) MLSS 2006, Taipei 87 / 98

88 More Complicated Forms Multi-class classification For example, [Vapnik, 1998, Weston and Watkins, 1999]: min w,b,ξ 1 2 k wmw T m + C m=1 l i=1 m y i ξ m i w T y i φ(x i ) + b yi w T mφ(x i ) + b m + 2 ξ m i, ξ m i 0, i = 1,...,l, m {1,...,k}\y i. y i : class of x i kl constraints Dual: kl variables; very large Chih-Jen Lin (National Taiwan Univ.) MLSS 2006, Taipei 88 / 98

89 Multi-class classification There are many other methods A comparison in [Hsu and Lin, 2002] Accuracy similar for many problems But 1-against-1 fastest for training Chih-Jen Lin (National Taiwan Univ.) MLSS 2006, Taipei 89 / 98

90 Multi-class classification Why 1vs1 Faster in Training 1 vs. 1 k(k 1)/2 problems, each 2l/k data on average 1 vs. all k problems, each l data If solving the optimization problem: polynomial of the size with degree d Their complexities (( ) d ) k(k 1) 2l O 2 k vs. ko(l d ) Chih-Jen Lin (National Taiwan Univ.) MLSS 2006, Taipei 90 / 98

91 Outline Discussion and conclusions Basic concepts SVM primal/dual problems Training linear and nonlinear SVMs Parameter/kernel selection and practical issues Multi-class classification Discussion and conclusions Chih-Jen Lin (National Taiwan Univ.) MLSS 2006, Taipei 91 / 98

92 Future Directions Discussion and conclusions I mentioned quite a few. Here are others. Better ways to handle unbalanced data i.e., some classes few data, some classes a lot Multi-label classification An instance associated with 2 labels e.g., a document in both politics, sports Structural data sets An instance may not be a vector e.g., a tree from a sentence Chih-Jen Lin (National Taiwan Univ.) MLSS 2006, Taipei 92 / 98

93 Conclusions Discussion and conclusions Dealing with data is interesting especially if you get good accuracy Some basic understandings are essential when applying classification methods SVM is a rather mature topic but still quite a few interesting research issues Chih-Jen Lin (National Taiwan Univ.) MLSS 2006, Taipei 93 / 98

94 References I Bakır, G. H., Bottou, L., and Weston, J. (2005). Breaking svm complexity with cross-training. In Saul, L. K., Weiss, Y., and Bottou, L., editors, Advances in Neural Information Processing Systems 17, pages MIT Press, Cambridge, MA. Boser, B., Guyon, I., and Vapnik, V. (1992). A training algorithm for optimal margin classifiers. In Proceedings of the Fifth Annual Workshop on Computational Learning Theory, pages ACM Press. Chu, W., Keerthi, S., and Ong, C. (2003). Bayesian trigonometric support vector classifier. Neural Computation, 15(9): Cortes, C., Haffner, P., and Mohri, M. (2003). Positive definite rational kernels. In Proceedings of the 16th Annual Conference on Learning Theory, pages Cortes, C. and Vapnik, V. (1995). Support-vector network. Machine Learning, 20: Chih-Jen Lin (National Taiwan Univ.) MLSS 2006, Taipei 94 / 98

95 References II Cristianini, N. and Shawe-Taylor, J. (2000). An Introduction to Support Vector Machines. Cambridge University Press, Cambridge, UK. Hsu, C.-W. and Lin, C.-J. (2002). A comparison of methods for multi-class support vector machines. IEEE Transactions on Neural Networks, 13(2): Joachims, T. (1998). Making large-scale SVM learning practical. In Schölkopf, B., Burges, C. J. C., and Smola, A. J., editors, Advances in Kernel Methods - Support Vector Learning, Cambridge, MA. MIT Press. Kao, W.-C., Chung, K.-M., Sun, C.-L., and Lin, C.-J. (2004). Decomposition methods for linear support vector machines. Neural Computation, 16(8): Keerthi, S. S., Chapelle, O., and DeCoste, D. (2006). Building support vector machines with reduced classifier complexity. Journal of Machine Learning Research, 7: Chih-Jen Lin (National Taiwan Univ.) MLSS 2006, Taipei 95 / 98

96 References III Keerthi, S. S. and Lin, C.-J. (2003). Asymptotic behaviors of support vector machines with Gaussian kernel. Neural Computation, 15(7): Lanckriet, G., Cristianini, N., Bartlett, P., El Ghaoui, L., and Jordan, M. (2004). Learning the Kernel Matrix with Semidefinite Programming. Journal of Machine Learning Research, 5: Lee, Y.-J. and Mangasarian, O. L. (2001). RSVM: Reduced support vector machines. In Proceedings of the First SIAM International Conference on Data Mining. Lin, C.-J. (2001). Formulations of support vector machines: a note from an optimization point of view. Neural Computation, 13(2): Lin, C.-J. (2002). A formal analysis of stopping criteria of decomposition methods for support vector machines. IEEE Transactions on Neural Networks, 13(5): Chih-Jen Lin (National Taiwan Univ.) MLSS 2006, Taipei 96 / 98

97 References IV Osuna, E., Freund, R., and Girosi, F. (1997). Training support vector machines: An application to face detection. In Proceedings of CVPR 97, pages , New York, NY. IEEE. Platt, J. C. (1998). Fast training of support vector machines using sequential minimal optimization. In Schölkopf, B., Burges, C. J. C., and Smola, A. J., editors, Advances in Kernel Methods - Support Vector Learning, Cambridge, MA. MIT Press. Schölkopf, B. and Smola, A. J. (2002). Learning with kernels. MIT Press. Syed, N. A., Liu, H., and Sung, K. K. (1999). Incremental learning with support vector machines. In Workshop on Support Vector Machines, IJCAI99. Vapnik, V. (1998). Statistical Learning Theory. Wiley, New York, NY. Chih-Jen Lin (National Taiwan Univ.) MLSS 2006, Taipei 97 / 98

98 References V Weston, J. and Watkins, C. (1999). Multi-class support vector machines. In Verleysen, M., editor, Proceedings of ESANN99, Brussels. D. Facto Press. Yu, H., Yang, J., and Han, J. (2003). Classifying large data sets using svms with hierarchical clusters. In KDD 03: Proceedings of the ninth ACM SIGKDD international conference on Knowledge discovery and data mining, pages , New York, NY, USA. ACM Press. Chih-Jen Lin (National Taiwan Univ.) MLSS 2006, Taipei 98 / 98