Support Vector Machines

1. Support Vector Machines Adapted from Lectures by Raymond Mooney (UT Austin) and Andrew Moore (CMU) L15SVM

2. Text classification • Earlier: Algorithms for text classification • K Nearest Neighbor classification • Simple, expensive at test time, low bias, high variance, non-linear • Vector space classification using centroids and hyperplanes that split them • Simple, linear classifier; perhaps too simple; high bias, low variance • Today • SVMs: Some empirical evaluation and comparison • Text-specific issues in classification L15SVM

3. Linear classifiers: Which Hyperplane? • Lots of possible solutions for a,b,c. • Some methods find a separating hyperplane, but not the optimal one [according to some criterion of expected goodness] • Support Vector Machine (SVM) finds an optimal solution. • Maximizes the distance between the hyperplane and the “difficult points” close to decision boundary • Intuition: Then there are fewer uncertain classification decisions This line represents the decision boundary: ax + by - c = 0 L15SVM 15.0

4. Another intuition • If we place a fat separator between classes, we have less choices, and so the capacity of the model has been decreased. Define the margin of a linear classifier as the width that the boundary could be increased by before hitting a datapoint.

5. L15SVM

6. Support vectors Maximize margin Support Vector Machine (SVM) • SVMs maximize the margin around the separating hyperplane. • A.k.a. large margin classifiers • The decision function is fully specified by a subset of training samples, the support vectors. • Solving SVMs is a Quadratic programming problem • Seen by many as most successful current text classification method L15SVM 15.1

7. Robustness of SVMs • If we make a small error in the location of the boundary, this approach minimizes misclassification. • Recall that support vectors are datapoints that the margin pushes up against. • The model is immune to removal of any non-support vector datapoints. • Rocchio centroids depend on all points of a class. • kNN boundaries depend on local neighborhoods. • Empirically SVMs work well. L15SVM

8. Maximum Margin: Formalization • w: decision hyperplane normal • xi: data point i • yi: class of data point i (+1 or -1) NB: Not 1/0 • Classifier is: f(xi) = sign(wTxi + b) • Functional margin of xi is: yi (wTxi + b) • But note that we can increase this margin simply by scaling w, b…. • Functional margin of dataset is minimum functional margin for any point • The factor of 2 comes from measuring the whole width of the margin L15SVM

9. The planar decision surface in data-space for the simple linear discriminant function: L15SVM

10. ρ Geometric Margin • Distance from example to the separator is • Examples closest to the hyperplane are support vectors. • Marginρof the separator is the width of separation between support vectors of classes. x r x′

11. Linear SVM Mathematically • Assume that all data is at least distance 1 from the hyperplane, then the following two constraints follow for a training set {(xi,yi)} • For support vectors, the inequality becomes an equality • Then, since each example’s distance from the hyperplane is • The geometric margin is: wTxi+ b≥ 1 if yi= 1 wTxi+ b ≤ -1 if yi= -1 L15SVM

12. Linear Support Vector Machine (SVM) • Hyperplane wT x + b = 0 • Extra scale constraint: mini=1,…,n |wTxi + b| = 1 • This implies: wT(xa–xb) = 2 ρ = ||xa–xb||2 = 2/||w||2 wTxa + b = 1 ρ wTxb + b = -1 wT x + b = 0 L15SVM

13. Linear SVMs Mathematically (cont.) • Then we can formulate the quadratic optimization problem: • A better formulation (min ||w|| = max 1/ ||w|| ): Find w and b such that is maximized; and for all {(xi, yi)} wTxi+ b≥ 1 if yi=1; wTxi+ b ≤ -1 if yi= -1 Find w and b such that Φ(w) =½ wTw is minimized; and for all {(xi,yi)}: yi (wTxi+ b)≥ 1

14. Recapitulation • We start with a training data set • We feed the data through a quadratic optimization • procedure to find the best separating hyperplane • Given a new point to classify, the classification • function computes the projection of the point onto • the hyperplane normal. • The sign of this function determines the class • If the point is within the margin of the classifier, the • classifier can return “don’t know”. • The value of may also be transformed into a • probability of classification 16

15. Multiclasssupportvectormachines • SVMs: inherentlytwo-classclassifiers. • Most common technique in practice: build |C| one-versus-rest classifiers (commonly referred to as “one-versus-all” or OVA classification), and choose the class which classifies the test datawithgreatestmargin • Another strategy: build a set of one-versus-one classifiers, and choose the class that is selected by the most classifiers. While this involves building |C|(|C| − 1)/2 classifiers, the time for training classifiers may actually decrease, since the training data set for each classifier is much smaller. 17

16. Walkthrough example: building an SVM over the data set shown in thefigure • Working geometrically: • The maximum margin weight vector will be parallel to the shortest line connecting points of the two classes, that is, the line between (1, 1) and • (2, 3), giving a weight vector of (1,2). • The optimal decision surface is orthogonal to that line and intersects it at the halfway point. Therefore, it passesthrough (1.5, 2). • So, the SVM decision boundary is: • y = x1 + 2x2 − 5.5 18

17. Walkthrough example: building an SVM over the data set shown in thefigure • Working algebraically: • Withtheconstraintsign • , we seek to • minimize • We know that the solution is • for some a. So: • a + 2a + b = −1, 2a + 6a + b = 1 • Hence, a = 2/5 and b = −11/5. • So the optimal hyperplane is given by and b = −11/5. • The marginρis 19

18. x 0 x 0 Non-linear SVMs • Datasets that are linearly separable (with some noise) work out great: • But what are we going to do if the dataset is just too hard? • How about … mapping data to a higher-dimensional space: x2 x 0 15.2.3

19. (0,1) + + - + -1 0 +1 + - (1,0) (0,0) Nonlinear SVMs: The Clever Bit! • Project the linearly inseparable data to high dimensional space where it is linearly separable and then we can use linear SVM L15SVM

20. Not linearly separable data. Linearly separable data. Angular degree (phase) polar coordinates 0 5 Distance from center (radius) Need to transform the coordinates: polar coordinates, kernel transformation into higher dimensional space (support vector machines). L15SVM

21. Non-linear SVMs: Feature spaces Φ: x→φ(x) L15SVM

22. f(.) f( ) f( ) f( ) f( ) f( ) f( ) f( ) f( ) f( ) f( ) f( ) f( ) f( ) f( ) f( ) f( ) f( ) f( ) Feature space Input space (cont’d) • Kernel functions and the kernel trick are used to transform data into a different linearly separable feature space L15SVM

23. Mathematical Details : SKIP L15SVM

24. Solving the Optimization Problem Find w and b such that Φ(w) =½ wTw is minimized; and for all {(xi,yi)}: yi (wTxi+ b)≥ 1 • This is now optimizing a quadratic function subject to linear constraints • Quadratic optimization problems are a well-known class of mathematical programming problems, and many (rather intricate) algorithms exist for solving them • The solution involves constructing a dual problem where a Lagrange multiplierαi is associated with every constraint in the primary problem: Find α1…αNsuch that Q(α) =Σαi- ½ΣΣαiαjyiyjxiTxjis maximized and (1)Σαiyi= 0 (2) αi≥ 0 for all αi

25. The Optimization Problem Solution • The solution has the form: • Each non-zero αi indicates that corresponding xi is a support vector. • Then the classifying function will have the form: • Notice that it relies on an inner product between the test point xand the support vectors xi. • Also keep in mind that solving the optimization problem involved computing the inner products xiTxj between all pairs of training points. w =Σαiyixi b= yk- wTxkfor any xksuch that αk 0 f(x) = ΣαiyixiTx + b L15SVM

26. Soft Margin Classification • If the training set is not linearly separable, slack variablesξican be added to allow misclassification of difficult or noisy examples. • Allow some errors • Let some points be moved to where they belong, at a cost • Still, try to minimize training set errors, and to place hyperplane “far” from each class (large margin) ξi ξj L15SVM 15.2.1

27. Soft Margin Classification Mathematically • The old formulation: • The new formulation incorporating slack variables: • Parameter C can be viewed as a way to control overfitting – a regularization term Find w and b such that Φ(w) =½ wTw is minimized and for all {(xi,yi)} yi (wTxi+ b)≥ 1 Find w and b such that Φ(w) =½ wTw + CΣξi is minimized and for all {(xi,yi)} yi(wTxi+ b)≥ 1- ξi and ξi≥ 0 for all i L15SVM

28. Soft Margin Classification – Solution • The dual problem for soft margin classification: • Neither slack variables ξinor their Lagrange multipliers appear in the dual problem! • Again, xi with non-zero αiwill be support vectors. • Solution to the dual problem is: Find α1…αNsuch that Q(α) =Σαi- ½ΣΣαiαjyiyjxiTxjis maximized and (1)Σαiyi= 0 (2) 0 ≤αi≤ C for all αi But w not needed explicitly for classification! w =Σαiyixi b= yk(1- ξk) - wTxkwhere k = argmax αk f(x) = ΣαiyixiTx + b k L15SVM

29. Classification with SVMs • Given a new point x = (x1,x2), score its projection onto the hyperplane normal: • In 2 dims: score = w1x1+w2x2+b. • I.e., compute score: wx + b= ΣαiyixiTx + b • Set confidence threshold t. Score > t: yes Score < -t: no Else: don’t know 7 5 3 L15SVM

30. Linear SVMs: Summary • The classifier is a separating hyperplane. • Most “important” training points are support vectors; they define the hyperplane. • Quadratic optimization algorithms can identify which training points xi are support vectors with non-zero Lagrangian multipliers αi. • Both in the dual formulation of the problem and in the solution training points appear only inside inner products: f(x) = ΣαiyixiTx + b Find α1…αNsuch that Q(α) =Σαi- ½ΣΣαiαjyiyjxiTxjis maximized and (1)Σαiyi= 0 (2) 0 ≤αi≤ C for all αi L15SVM

31. Non-linear SVMs: Feature spaces • General idea: the original feature space can always be mapped to some higher-dimensional feature space where the training set is separable: Φ: x→φ(x) L15SVM

32. The “Kernel Trick” • The linear classifier relies on an inner product between vectors K(xi,xj)=xiTxj • If every datapoint is mapped into high-dimensional space via some transformation Φ: x→φ(x), the inner product becomes: K(xi,xj)= φ(xi)Tφ(xj) • A kernel function is some function that corresponds to an inner product in some expanded feature space. • Example: 2-dimensional vectors x=[x1 x2]; let K(xi,xj)=(1 + xiTxj)2, Need to show that K(xi,xj)= φ(xi)Tφ(xj): K(xi,xj)=(1 + xiTxj)2,= 1+ xi12xj12 + 2 xi1xj1xi2xj2+ xi22xj22 + 2xi1xj1 + 2xi2xj2= = [1 xi12 √2 xi1xi2 xi22 √2xi1 √2xi2]T [1 xj12 √2 xj1xj2 xj22 √2xj1 √2xj2] = φ(xi)Tφ(xj) where φ(x) = [1 x12 √2 x1x2 x22 √2x1 √2x2]

33. Kernels • Why use kernels? • Make non-separable problem separable. • Map data into better representational space • Common kernels • Linear • Polynomial K(x,z) = (1+xTz)d • Radial basis function (infinite dimensional space) L15SVM

34. Evaluation: Classic Reuters Data Set • Most (over)used data set • 21578 documents • 9603 training, 3299 test articles (ModApte split) • 118 categories • An article can be in more than one category • Learn 118 binary category distinctions • Average document: about 90 types, 200 tokens • Average number of classes assigned • 1.24 for docs with at least one category • Only about 10 out of 118 categories are large • Earn (2877, 1087) • Acquisitions (1650, 179) • Money-fx (538, 179) • Grain (433, 149) • Crude (389, 189) • Trade (369,119) • Interest (347, 131) • Ship (197, 89) • Wheat (212, 71) • Corn (182, 56) Common categories (#train, #test)

35. Reuters Text Categorization data set (Reuters-21578) document <REUTERS TOPICS="YES" LEWISSPLIT="TRAIN" CGISPLIT="TRAINING-SET" OLDID="12981" NEWID="798"> <DATE> 2-MAR-1987 16:51:43.42</DATE> <TOPICS><D>livestock</D><D>hog</D></TOPICS> <TITLE>AMERICAN PORK CONGRESS KICKS OFF TOMORROW</TITLE> <DATELINE> CHICAGO, March 2 - </DATELINE><BODY>The American Pork Congress kicks off tomorrow, March 3, in Indianapolis with 160 of the nations pork producers from 44 member states determining industry positions on a number of issues, according to the National Pork Producers Council, NPPC. Delegates to the three day Congress will be considering 26 resolutions concerning various issues, including the future direction of farm policy and the tax law as it applies to the agriculture sector. The delegates will also debate whether to endorse concepts of a national PRV (pseudorabies virus) control and eradication program, the NPPC said. A large trade show, in conjunction with the congress, will feature the latest in technology in all areas of the industry, the NPPC added. Reuter &#3;</BODY></TEXT></REUTERS>

36. New Reuters: RCV1: 810,000 docs • Top topics in Reuters RCV1

37. Per class evaluation measures • Recall: Fraction of docs in class i classified correctly: • Precision: Fraction of docs assigned class i that are actually about class i: • “Correct rate”: (1- error rate) Fraction of docs classified correctly:

38. Dumais et al. 1998: Reuters - Accuracy Recall: % labeled in category among those stories that are really in category Precision: % really in category among those stories labeled in category Break Even: (Recall + Precision) / 2

39. Results for Kernels (Joachims 1998) L15SVM

40. Micro- vs. Macro-Averaging • If we have more than one class, how do we combine multiple performance measures into one quantity? • Macroaveraging: Compute performance for each class, then average. • Microaveraging: Collect decisions for all classes, compute contingency table, evaluate.

41. SKIP L15SVM

42. Micro- vs. Macro-Averaging: Example Class 1 Class 2 Micro.Av. Table • Macroaveraged precision: (0.5 + 0.9)/2 = 0.7 • Microaveraged precision: 100/120 = .83 • Why this difference?

44. Reuters ROC - Category Grain Recall LSVM Decision TreeNaïve Bayes Find Similar Precision Recall: % labeled in category among those stories that are really in category Precision: % really in category among those stories labeled in category

45. ROC for Category - Crude Recall LSVM Decision TreeNaïve Bayes Find Similar Precision

46. ROC for Category - Ship Recall LSVM Decision TreeNaïve Bayes Find Similar Precision

47. Yang&Liu: SVM vs. Other Methods

48. Good practice department:Confusion matrix • In a perfect classification, only the diagonal has non-zero entries This (i, j) entry means 53 of the docs actually in class i were put in class j by the classifier. Class assigned by classifier Actual Class 53

49. The Real World P. Jackson and I. Moulinier: Natural Language Processing for Online Applications • “There is no question concerning the commercial value of being able to classify documents automatically by content. There are myriad potential applications of such a capability for corporate Intranets, government departments, and Internet publishers” • “Understanding the data is one of the keys to successful categorization, yet this is an area in which most categorization tool vendors are extremely weak. Many of the ‘one size fits all’ tools on the market have not been tested on a wide range of content types.”

50. The Real World • Gee, I’m building a text classifier for real, now! • What should I do? • How much training data do you have? • None • Very little • Quite a lot • A huge amount and its growing