1. PrasadL15SVM1 Support Vector Machines Adapted from Lectures by Raymond Mooney (UT Austin) and Andrew Moore (CMU)

2. 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 PrasadL15SVM

3. 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 PrasadL15SVM 15.0

4. 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. Prasad Define the margin of a linear classifier as the width that the boundary could be increased by before hitting a datapoint.

5. PrasadL15SVM5 Decision hyperplane Recall that a decision hyperplane can be defined by:  the intercept term b  the normal vector w (weight vector), which is perpendicular to the hyperplane All points x on the hyperplane satisfy:

6. 6 Support Vector Machine (SVM) Support vectors Maximize margin 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 PrasadL15SVM 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. PrasadL15SVM7

8. 8 w: decision hyperplane normal x i : data point i y i : class of data point i (+1 or -1) NB: Not 1/0 Classifier is: f(x i ) = sign(w T x i + b) Functional margin of x i is: y i (w T x i + 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 Maximum Margin: Formalization PrasadL15SVM

9. 9 The planar decision surface in data-space for the simple linear discriminant function: PrasadL15SVM

10. 10 Functional Margin 10 Functional margin The functional margin of the i th example x i w.r.t. the hyperplane The functional margin of a data set w.r.t. a decision surface is twice the functional margin of any of the points in the data set with minimal functional margin  factor 2 comes from measuring across the whole width of the margin We are confident in the classification of a point if it is far away from the decision boundary. But we can increase functional margin by scaling w and b. So, we need to place some constraint on the size of the w vector.

11. 11 Geometric margin Geometric margin of the classifier: maximum width of the band that can be drawn separating the support vectors of the two classes. The geometric margin is clearly invariant to scaling of parameters: if we replace w by 5 w and b by 5b, then the geometric margin is the same, because it is inherently normalized by the length of w. 11

12. 12 Geometric Margin Distance from example to the separator is Examples closest to the hyperplane are support vectors. Margin ρ of the separator is the width of separation between support vectors of classes. r ρ x x′x′ Prasad

13. 13 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 {(x i,y i )} For support vectors, the inequality becomes an equality Then, since each example’s distance from the hyperplane is The geometric margin is: w T x i + b ≥ 1 if y i = 1 w T x i + b ≤ -1 if y i = -1 PrasadL15SVM

14. 14 Linear Support Vector Machine (SVM) Hyperplane w T x + b = 0 Extra scale constraint: min i=1,…,n |w T x i + b| = 1 This implies: w T (x a –x b ) = 2 ρ = ||x a –x b || 2 = 2/||w|| 2 w T x + b = 0 w T x a + b = 1 w T x b + b = -1 ρ PrasadL15SVM

15. 15 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 { ( x i, y i )} w T x i + b ≥ 1 if y i =1; w T x i + b ≤ -1 if y i = -1 Find w and b such that Φ(w) =½ w T w is minimized; and for all { ( x i,y i )} : y i (w T x i + b) ≥ 1 Prasad

16. 16 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

17. 17 Multiclass support vector machines SVMs: inherently two-class classifiers.  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 data with greatest margin  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.

18. 18 Walkthrough example: building an SVM over the data set shown in the figure 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 passes through (1.5, 2).  So, the SVM decision boundary is: y = x 1 + 2x 2 − 5.5 18

19. 19 Walkthrough example: building an SVM over the data set shown in the figure Working algebraically:  With the constraint sign, 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 margin ρ is

20. 20 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: 0 x2x2 x 0 x 0 x Prasad 15.2.3

21. 21 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 0+1 ++- (1,0) (0,0) (0,1) + + - PrasadL15SVM

22. 22 0 5 Not linearly separable data. Need to transform the coordinates: polar coordinates, kernel transformation into higher dimensional space (support vector machines). Distance from center (radius) Angular degree (phase) Linearly separable data. polar coordinates L15SVM

23. 23 Non-linear SVMs: Feature spaces Φ: x → φ(x) PrasadL15SVM

24. 24 (cont’d) Kernel functions and the kernel trick are used to transform data into a different linearly separable feature space  (.)  ( ) Feature space Input space PrasadL15SVM

25. 25 Mathematical Details : SKIP PrasadL15SVM

26. Solving the Optimization Problem 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 multiplier α i is associated with every constraint in the primary problem: Find w and b such that Φ(w) =½ w T w is minimized; and for all { ( x i,y i )} : y i (w T x i + b) ≥ 1 Find α 1 …α N such that Q( α ) = Σ α i - ½ ΣΣ α i α j y i y j x i T x j is maximized and (1) Σ α i y i = 0 (2) α i ≥ 0 for all α i Prasad

27. 27 The Optimization Problem Solution The solution has the form: Each non-zero α i indicates that corresponding x i is a support vector. Then the classifying function will have the form: Notice that it relies on an inner product between the test point x and the support vectors x i. Also keep in mind that solving the optimization problem involved computing the inner products x i T x j between all pairs of training points. w = Σ α i y i x i b= y k - w T x k for any x k such that α k  0 f(x) = Σ α i y i x i T x + b PrasadL15SVM

28. 28 Soft Margin Classification If the training set is not linearly separable, slack variables ξ i can 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) ξjξj ξiξi PrasadL15SVM 15.2.1

29. 29 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 w T w is minimized and for all { ( x i,y i )} y i (w T x i + b) ≥ 1 Find w and b such that Φ(w) =½ w T w + C Σ ξ i is minimized and for all { ( x i,y i )} y i (w T x i + b) ≥ 1- ξ i and ξ i ≥ 0 for all i PrasadL15SVM

30. 30 Soft Margin Classification – Solution The dual problem for soft margin classification: Neither slack variables ξ i nor their Lagrange multipliers appear in the dual problem! Again, x i with non-zero α i will be support vectors. Solution to the dual problem is: Find α 1 …α N such that Q( α ) = Σ α i - ½ ΣΣ α i α j y i y j x i T x j is maximized and (1) Σ α i y i = 0 (2) 0 ≤ α i ≤ C for all α i w = Σ α i y i x i b= y k (1- ξ k ) - w T x k where k = argmax α k k f(x) = Σ α i y i x i T x + b But w not needed explicitly for classification! PrasadL15SVM

31. 31 Classification with SVMs Given a new point x = (x 1,x 2 ), score its projection onto the hyperplane normal: In 2 dims: score = w 1 x 1 +w 2 x 2 +b. I.e., compute score: wx + b = Σα i y i x i T x + b Set confidence threshold t. 3 5 7 Score > t: yes Score < -t: no Else: don’t know PrasadL15SVM

32. 32 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 x i are support vectors with non-zero Lagrangian multipliers α i. Both in the dual formulation of the problem and in the solution training points appear only inside inner products: Find α 1 …α N such that Q( α ) = Σ α i - ½ ΣΣ α i α j y i y j x i T x j is maximized and (1) Σ α i y i = 0 (2) 0 ≤ α i ≤ C for all α i f(x) = Σ α i y i x i T x + b PrasadL15SVM

33. 33 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) PrasadL15SVM

34. The “Kernel Trick” The linear classifier relies on an inner product between vectors K(x i,x j )=x i T x j If every datapoint is mapped into high-dimensional space via some transformation Φ: x → φ(x), the inner product becomes: K(x i,x j )= φ(x i ) T φ(x j ) A kernel function is some function that corresponds to an inner product in some expanded feature space. Example: 2-dimensional vectors x=[x 1 x 2 ]; let K(x i,x j )=(1 + x i T x j ) 2, Need to show that K(x i,x j )= φ(x i ) T φ(x j ): K(x i,x j )=(1 + x i T x j ) 2, = 1+ x i1 2 x j1 2 + 2 x i1 x j1 x i2 x j2 + x i2 2 x j2 2 + 2x i1 x j1 + 2x i2 x j2 = = [1 x i1 2 √2 x i1 x i2 x i2 2 √2x i1 √2x i2 ] T [1 x j1 2 √2 x j1 x j2 x j2 2 √2x j1 √2x j2 ] = φ(x i ) T φ(x j ) where φ(x) = [1 x 1 2 √2 x 1 x 2 x 2 2 √2x 1 √2x 2 ]

35. 35 Kernels Why use kernels? Make non-separable problem separable. Map data into better representational space Common kernels Linear Polynomial K(x,z) = (1+x T z) d Radial basis function (infinite dimensional space) PrasadL15SVM

36. 36 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 Common categories (#train, #test) Evaluation: Classic Reuters Data Set 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) Prasad

37. 37 Reuters Text Categorization data set (Reuters-21578) document 2-MAR-1987 16:51:43.42 livestock hog AMERICAN PORK CONGRESS KICKS OFF TOMORROW CHICAGO, March 2 - 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

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

39. 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:

40. 40 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

41. 41 Results for Kernels (Joachims 1998) PrasadL15SVM

42. 42 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.

43. SKIP PrasadL15SVM43

44. 44 Micro- vs. Macro-Averaging: Example Truth: yes Truth: no Classifi er: yes 10 Classifi er: no 10970 Truth: yes Truth: no Classifi er: yes 9010 Classifi er: no 10890 Truth: yes Truth: no Classifie r: yes 10020 Classifie r: no 201860 Class 1Class 2Micro.Av. Table Macroaveraged precision: (0.5 + 0.9)/2 = 0.7 Microaveraged precision: 100/120 =.83 Why this difference?

45. 45

46. 46 Reuters ROC - Category Grain Precision Recall LSVM Decision Tree Naïve Bayes Find Similar Recall: % labeled in category among those stories that are really in category Precision: % really in category among those stories labeled in category

47. 47 ROC for Category - Crude LSVM Decision Tree Naïve Bayes Find Similar Precision Recall Prasad

48. 48 ROC for Category - Ship LSVM Decision Tree Naïve Bayes Find Similar Precision Recall Prasad

49. 49 Yang&Liu: SVM vs. Other Methods

50. 50 Good practice department: Confusion matrix In a perfect classification, only the diagonal has non-zero entries 53 Class assigned by classifier Actual Class This (i, j) entry means 53 of the docs actually in class i were put in class j by the classifier.

51. 51 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.”

52. 52 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

53. 53 Manually written rules No training data, adequate editorial staff? Never forget the hand-written rules solution! If (wheat or grain) and not (whole or bread) then Categorize as grain In practice, rules get a lot bigger than this Can also be phrased using tf or tf.idf weights With careful crafting (human tuning on development data) performance is high: Construe: 94% recall, 84% precision over 675 categories (Hayes and Weinstein 1990) Amount of work required is huge Estimate 2 days per class … plus maintenance

54. 54 Very little data? If you’re just doing supervised classification, you should stick to something high bias There are theoretical results that Naïve Bayes should do well in such circumstances (Ng and Jordan 2002 NIPS) The interesting theoretical answer is to explore semi-supervised training methods: Bootstrapping, EM over unlabeled documents, … The practical answer is to get more labeled data as soon as you can How can you insert yourself into a process where humans will be willing to label data for you??

55. 55 A reasonable amount of data? Perfect! We can use all our clever classifiers Roll out the SVM! But if you are using an SVM/NB etc., you should probably be prepared with the “hybrid” solution where there is a Boolean overlay Or else to use user-interpretable Boolean-like models like decision trees Users like to hack, and management likes to be able to implement quick fixes immediately

56. 56 A huge amount of data? This is great in theory for doing accurate classification… But it could easily mean that expensive methods like SVMs (train time) or kNN (test time) are quite impractical Naïve Bayes can come back into its own again! Or other advanced methods with linear training/test complexity like regularized logistic regression (though much more expensive to train)

57. 57 A huge amount of data? With enough data the choice of classifier may not matter much, and the best choice may be unclear Data: Brill and Banko on context-sensitive spelling correction But the fact that you have to keep doubling your data to improve performance is a little unpleasant

58. 58 How many categories? A few (well separated ones)? Easy! A zillion closely related ones? Think: Yahoo! Directory, Library of Congress classification, legal applications Quickly gets difficult! Classifier combination is always a useful technique Voting, bagging, or boosting multiple classifiers Much literature on hierarchical classification Mileage fairly unclear May need a hybrid automatic/manual solution

59. 59 How can one tweak performance? Aim to exploit any domain-specific useful features that give special meanings or that zone the data E.g., an author byline or mail headers Aim to collapse things that would be treated as different but shouldn’t be. E.g., part numbers, chemical formulas

60. 60 Does putting in “hacks” help? You bet! You can get a lot of value by differentially weighting contributions from different document zones: Upweighting title words helps (Cohen & Singer 1996) Doubling the weighting on the title words is a good rule of thumb Upweighting the first sentence of each paragraph helps (Murata, 1999) Upweighting sentences that contain title words helps (Ko et al, 2002)

61. 61 Two techniques for zones 1. Have a completely separate set of features/parameters for different zones like the title 2. Use the same features (pooling/tying their parameters) across zones, but upweight the contribution of different zones Commonly the second method is more successful: it costs you nothing in terms of sparsifying the data, but can give a very useful performance boost Which is best is a contingent fact about the data

62. 62 Text Summarization techniques in text classification Text Summarization: Process of extracting key pieces from text, normally by features on sentences reflecting position and content Much of this work can be used to suggest weightings for terms in text categorization See: Kolcz, Prabakarmurthi, and Kalita, CIKM 2001: Summarization as feature selection for text categorization Categorizing purely with title, Categorizing with first paragraph only Categorizing with paragraph with most keywords Categorizing with first and last paragraphs, etc.

63. 63 Does stemming/lowercasing/… help? As always it’s hard to tell, and empirical evaluation is normally the gold standard But note that the role of tools like stemming is rather different for TextCat vs. IR: For IR, you often want to collapse forms of the verb oxygenate and oxygenation, since all of those documents will be relevant to a query for oxygenation For TextCat, with sufficient training data, stemming does no good. It only helps in compensating for data sparseness (which can be severe in TextCat applications). Overly aggressive stemming can easily degrade performance.

64. 64 Measuring Classification Figures of Merit Not just accuracy; in the real world, there are economic measures: Your choices are: Do no classification That has a cost (hard to compute) Do it all manually Has an easy to compute cost if doing it like that now Do it all with an automatic classifier Mistakes have a cost Do it with a combination of automatic classification and manual review of uncertain/difficult/“new” cases Commonly the last method is most cost efficient and is adopted

65. 65 A common problem: Concept Drift Categories change over time Example: “president of the united states” 1999: clinton is great feature 2002: clinton is bad feature One measure of a text classification system is how well it protects against concept drift. Can favor simpler models like Naïve Bayes Feature selection: can be bad in protecting against concept drift

66. 66 Summary Support vector machines (SVM) Choose hyperplane based on support vectors Support vector = “critical” point close to decision boundary (Degree-1) SVMs are linear classifiers. Kernels: powerful and elegant way to define similarity metric Perhaps best performing text classifier But there are other methods that perform about as well as SVM, such as regularized logistic regression (Zhang & Oles 2001) Partly popular due to availability of SVMlight SVMlight is accurate and fast – and free (for research) Now lots of software: libsvm, TinySVM, …. Comparative evaluation of methods Real world: exploit domain specific structure!