Support Vector Machines

Transcription

1 Support Vector Machines Reading: Ben-Hur & Weston, A User s Guide to Support Vector Machines (linked from class web page)

2 Notation Assume a binary classification problem. Instances are represented by vector x R n. Training examples: x = (x 1, x 2,, x n ) S = {(x 1, t 1 ), (x 2, t 2 ),..., (x m, t m ) (x k, t k ) R n {+1, 1}} Hypothesis: A function h: R n {+1, 1}. h(x) = h(x 1, x 2,, x n ) {+1, 1}

3 Here, assume positive and negative instances are to be separated by the hyperplane where b is the bias. w x + b = 0 Equation of line: w x + b = w T x + b = w 1 x 1 + w 2 x 2 + b = 0 x 2 x 1

4 w x + b = 0 Intuition: the best hyperplane (for future generalization) will maximally separate the examples

5 Definition of Margin (with respect to a hyperplane): Distance from separating hyperplane to nearest positive (or negative) instance.

6 Definition of Margin (with respect to a hyperplane): Distance from separating hyperplane to nearest positive (or negative) instance. Vapnik (1979) showed that the hyperplane maximizing the margin of S will have minimal VC dimension in the set of all consistent hyperplanes, and will thus be optimal.

7 Definition of Margin (with respect to a hyperplane): Distance from separating hyperplane to nearest positive (or negative) instance. Vapnik (1979) showed that the hyperplane maximizing the margin of S will have minimal VC dimension in the set of all consistent hyperplanes, and will thus be optimal. This is an optimization problem!

8 Support vectors: Training examples that lie on the margin.

9 w x + b = 0 Support vectors: Training examples that lie on the margin.

10 w x + b = 0 w x + b = a Support vectors: Training examples that lie on the margin.

11 w x + b = a w x + b = 0 w x + b = a Support vectors: Training examples that lie on the margin.

12 w x + b = a w x + b = 0 w x + b = a Support vectors: Training examples that lie on the margin. Without changing the problem, we can rescale our data to set a = 1

13 w x + b = 1 w x + b = 0 w x + b =1 Support vectors: Training examples that lie on the margin. Without changing the problem, we can rescale our data to set a = 1

14 w x + b = 0 w x + b =1 Geometry of the margin w x + b = 1 Without changing the problem, we can rescale our data to set a = 1

15 w x + b = 1 w x + b = 0 w x + b =1 Geometry of the margin (w x 1 )+ b = +1 (w x 2 )+ b = 1 w (x 1 x 2 ) = 2 (x 1 x 2 ) = 2 w where w = w w = w w 2 2 Without changing the problem, we can rescale our data to set a = 1

16 The length of the margin is 1 w So to maximize the margin, we need to minimize. w

17 Minimizing w Find w and b by doing the following minimization: min w,b! 1 2 w 2 $ # & " % subject to: t k ( w x k + b) 1, k =1,..., m (t k { 1,+1}) This is a quadratic, constrained optimization problem. Use method of Lagrange multipliers to solve it.

18 Dual Representation It turns out that w can be expressed as a linear combination of the training examples: w = α k x k S x k where α k 0 only if x k is a support vector The results of the SVM training algorithm (involving solving a quadratic programming problem) are the α k and the bias b.

19 SVM Classification Classify new example x as follows: M h(x) = sgn α k x x k k=1 ( ) + b where S = M. (S is the training set)

20 SVM Classification Classify new example x as follows: M h(x) = sgn α k x x k k=1 ( ) + b where S = M. (S is the training set) Note that dot product is a kind of similarity measure

21 Example

22 Example Input to SVM optimzer: x 1 x 2 class Output from SVM optimizer: -2 Support vector α ( 1, 0).208 (1, 1).416 (0, 1).208 b =.376

23 Example Input to SVM optimzer: x 1 x 2 class Output from SVM optimizer: -2 Support vector α ( 1, 0).208 (1, 1).416 (0, 1).208 b =.376

24 Example Input to SVM optimzer: x 1 x 2 class Output from SVM optimizer: Weight vector: w = k {training examples} -2 α k x k =.208 ( 1, 0)+.416 (1,1).208 (0, 1) Support vector α ( 1, 0).208 (1, 1).416 (0, 1).208 b =.376 = (.624,.624)

25 Example Separation line: w 1 x 1 + w 2 x 2 + b = 0 Input to SVM optimzer: x x = 0 x 2 = x x 1 x 2 class Output from SVM optimizer: Weight vector: w = k {training examples} -2 α k x k =.208 ( 1, 0)+.416 (1,1).208 (0, 1) Support vector α ( 1, 0).208 (1, 1).416 (0, 1).208 b =.376 = (.624,.624)

26 Example Separation line: w 1 x 1 + w 2 x 2 + b = 0 Input to SVM optimzer: x x = 0 x 2 = x x 1 x 2 class Output from SVM optimizer: Weight vector: w = k {training examples} -2 α k x k =.208 ( 1, 0)+.416 (1,1).208 (0, 1) Support vector α ( 1, 0).208 (1, 1).416 (0, 1).208 b =.376 = (.624,.624)

27 Example Classifying a new point:

28 Example Classifying a new point: m h((2,2)) = sgn α k x k x k=1 ( ) = sgn.208 ( 1, 0) (2, 2) + b, where sgn(z) = 1 if z > 0 1 if z 0 ( [ ] [(1,1) (2, 2) ].208 [(0, 1) (2, 2) ].376) ( ) = +1 = sgn

29 Example Classifying a new point: m h((2,2)) = sgn α k x k x k=1 ( ) = sgn.208 ( 1, 0) (2, 2) + b, where sgn(z) = 1 if z > 0 1 if z 0 ( [ ] [(1,1) (2, 2) ].208 [(0, 1) (2, 2) ].376) ( ) = +1 = sgn

30 Example Classifying a new point: m h((2,2)) = sgn α k x k x k=1 ( ) = sgn.208 ( 1, 0) (2, 2) + b, where sgn(z) = 1 if z > 0 1 if z 0 ( [ ] [(1,1) (2, 2) ].208 [(0, 1) (2, 2) ].376) ( ) = +1 = sgn

31 In-Class Exercise 1

32 SVM summary Equation of line: w 1 x 1 + w 2 x 2 + b = 0 Define margin using: x k w + b +1 for positive instances (t k = +1) x k w + b 1 for negative instances (t k = 1) Margin distance: 1 w To maximize the margin, we minimize w subject to the constraint that positive examples fall on one side of the margin, and negative examples on the other side: t k ( w x k + b) 1, k =1,..., m where t k { 1,+1} We can relax this constraint using slack variables

33 SVM summary To do the optimization, we use the dual formulation: w = k {training examples} α k x k The results of the optimization black box are and b. {α k }

34 SVM review Once the optimization is done, we can classify a new example x as follows: h(x) = class(x) = sgn( w x + b) ## m & & = sgn% % α k x k ( x + b( $ $ k=1 ' ' ## m = sgn% % α k x k x $ $ k=1 ( ) & & (+ b( ' ' That is, classification is done entirely through a linear combination of dot products with training examples. This is a kernel method.

35 SVM_light demo

36 Non-linearly separable training examples What if the training examples are not linearly separable?

37 Non-linearly separable training examples What if the training examples are not linearly separable? Use old trick: Find a function that maps points to a higher dimensional space ( feature space ) in which they are linearly separable, and do the classification in that higherdimensional space.

38 x 2 x 2 x 1 x 1 x 3 Need to find a function Φ that will perform such a mapping: Φ: R n F Then can find hyperplane in higher dimensional feature space F, and do classification using that hyperplane in higher dimensional space.

39 Challenge Find a 3-dimensional feature space in which XOR is linearly separable. x 2 (0, 1) (1, 1) x 2 (1, 0) x 1 (0, 0) x 1 x 3

40 Problem: Recall that classification of instance x is expressed in terms of dot products of x and support vectors. $ Class(x) = sgn & % k {training examples} ' α k (x x k )+ b ) ( The quadratic programming problem of finding the support vectors and coefficients also depends only on dot products between training examples, rather than on the training examples outside of dot products.

41 So if each x k is replaced by Φ(x k ) in these procedures, we will have to calculate Φ(x k ) for each k as well as calculate a lot of dot products, Φ(x) Φ(x k ) But in general, if the feature space F is high dimensional, Φ(x i ) Φ(x j ) will be expensive to compute.

42 Second trick: Suppose that there were some magic function, K(x i, x j ) = Φ(x i ) Φ(x j ) such that K is cheap to compute even though Φ(x i ) Φ(x j ) is expensive to compute. Then we wouldn t need to compute the dot product directly; we d just need to compute K during both the training and testing phases. The good news is: such K functions exist! They are called kernel functions, and come from the theory of integral operators.

43 Example: Polynomial kernel: Suppose x = (x 1, x 2 ) and z = (z 1, z 2 ). k(x,z) = (x z) 2 Let Φ(x) = ( x 2 2 1, 2 x 1 x 2, x ) 2. Then : 2 x 1 2 z 1 Φ(x) Φ(z) = 2 x 1 x 2 2 x 2 z 1 z z 2 = x 1 x 2 z 1 z 2 2 = (x z) 2 = k(x,z)

44 In-Class Exercise 2

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48 More on Kernels So far we ve seen kernels that map instances in R n to instances in R z where z > n. One way to create a kernel: Figure out appropriate feature space Φ(x), and find kernel function k which defines inner product on that space. More practically, we usually don t know appropriate feature space Φ(x). What people do in practice is either: 1. Use one of the classic kernels (e.g., polynomial), or 2. Define their own function that is appropriate for their task, and show that it qualifies as a kernel.

49 How to define your own kernel Given training data (x 1, x 2,..., x n ) Algorithm for SVM learning uses kernel matrix (also called Gram matrix): K i, j = K(x i, x j ), for i, j =1,..., n We can choose some function K, and compute the kernel matrix K using the training data. We just have to guarantee that our kernel defines an inner product on some feature space. Not as hard as it sounds.

50 What counts as a kernel? Mercer s Theorem: If the kernel matrix K is positive definite, it defines a kernel, that is, it defines an inner product in some feature space. We don t even have to know what that feature space is! It can have a huge number of dimensions.

51 Structural Kernels In domains such as natural language processing and bioinformatics, the similarities we want to capture are often structural (e.g., parse trees, formal grammars). An important area of kernel methods is defining structural kernels that capture this similarity (e.g., sequence alignment kernels, tree kernels, graph kernels, etc.)

52 From Design criteria - we want kernels to be valid Satisfy Mercer condition of positive semidefiniteness good embody the true similarity between objects appropriate generalize well efficient the computation of K(x, x ) is feasible

53 Example: Watson used tree kernels and SVMs to classify question types for Jeopardy! questions From Moschitti et al., 2011

54 Summary of SVM algorithm Given training set S = {(x 1, t 1 ), (x 2, t 2 ),..., (x m, t m ) (x k, t k ) R n {+1, -1} 1. Choose a kernel function k(x,z). 2. Apply optimization procedure (using the kernel function K) to find support vectors x k, coefficients α k, and bias b. 3. Given a new instance, x, find the classification of x by computing class(x) = sgn α k k(x, x k ) + b k support vectors

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56 Given training data (x 1, x 2,..., x m ) How to define your own kernel Algorithm for SVM learning uses kernel matrix (also called Gram matrix): K i, j = K(x i, x j ), for i, j =1,..., m

57 How to define your own kernel We can choose some function K, and compute the kernel matrix K using the training data. We just have to guarantee that our kernel defines an inner product on some feature space Mercer s Theorem: : If K is positive semidefinite, it defines a kernel, that is, it defines an inner product in some feature space. We don t even have to know what that feature space is! K is symmetric if K = K T K is positive semidefinite if all the eigenvalues of K are positive.

58 Example of Simple Custom Kernel Similarity between DNA Sequences: E.g., s 1 = GAATGTCCTTTCTCTAAGTCCTAAG s 2 = GGAGACTTACAGGAAAGAGATTCG Define Hamming Distance Kernel : hamming(s 1, s 2 ) = number of sites where strings match

59 Kernel matrix for hamming kernel Suppose training set is s 1 = GAATGTCCTTTCTCTAAGTCCTAAG s 2 = GGAGACTTACAGGAAAGAGATTCG s 3 = GGAAACTTTCGGGAGAGAGTTTCG What is the Kernel matrix K? K s 1 s 2 s 3 s 1 s 2 s 3

60 In-Class Exercise 3

61 Homework 3

62 Data Standardization In general, we need to do data standardization for SVMs to avoid imbalance among feature scales: Let µ i denote the mean value of feature i in the training data, and σ i denote the corresponding standard deviation. For each training example x, replace each x i as follows:!! =!!!!!! Scale the test data in the same way, using the µ i and σ i values computed from the training data, not the test data.

63 Feature Selection Goal: Select a subset of d features (d < n) in order to maximize classification performance with fewer features.

64 Types of Feature Selection Methods Filter Wrapper Embedded

65 Filter Methods Independent of the classification algorithm. Training data Apply a filtering function to the features before applying classifier. Filter Examples of filtering functions: Information gain of individual features Selected features Statistical variance of individual features Statistical correlation among features Classifier

66 Wrapper Methods Training data Choose subset of features Accuracy on validation set Classifier (cross-validation) After cross-validation Classifier (Entire training set with best subset of features) Evaluate classifier on test set

67 Filter Methods Pros: Fast Cons: Chosen filter might not be relevant for a specific kind of classifier. Doesn t take into account interactions among features Often hard to know how many features to select.

68 Filter Methods Wrapper Methods Pros: Fast Cons: Chosen filter might not be relevant for a specific kind of classifier. Doesn t take into account interactions among features Pros: Features are evaluated in context of classification Wrapper method selects number of features to use Cons: Slow Often hard to know how many features to select.

69 Filter Methods Wrapper Methods Pros: Fast Cons: Chosen filter might not be relevant for a specific kind of classifier. Doesn t take into account interactions among features Often hard to know how many features to select. Intermediate method, often used with SVMs: Pros: Features are evaluated in context of classification Wrapper method selects number of features to use Cons: Slow Train SVM using all features Select features f i with highest w i Retrain SVM with selected features Test SVM on test set

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71 Embedded Methods Feature selection is intrinsic part of learning algorithm One example: L 1 SVM: Instead of minimizing w (the L2 norm ) we minimize the L 1 norm of the weight vector: w 1 = n w i i=1 Result is that most of the weights go to zero, leaving a small subset of the weights. Cf., Field of sparse coding

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73 Hard- vs. soft- margin SVMs

74 Hard-margin SVMs Find w and b by doing the following minimization: w x + b = 1 w x + b = 0 w x + b =1 min w,b! 1 2 w 2 $ # & " % subject to: t k ( w x k + b) 1, k =1,..., m (t k { 1,+1})

75 Extend to soft-margin SVMs w x + b =1 w x + b = 1 w x + b = 0

76 Extend to soft-margin SVMs Allow some instances to be misclassified, or fall within margins, but penalize them by distance to margin hyperplane. w x + b =1 w x + b = 1 w x + b = 0

77 Extend to soft-margin SVMs Allow some instances to be misclassified, or fall within margins, but penalize them by distance to margin hyperplane. w x + b =1 w x + b = 1 w x + b = 0 ξ 0

78 Extend to soft-margin SVMs Allow some instances to be misclassified, or fall within margins, but penalize them by distance to margin hyperplane. w x + b =1 w x + b = 0 w x + b = 1 ξ 1 ξ 0

79 Extend to soft-margin SVMs Allow some instances to be misclassified, or fall within margins, but penalize them by distance to margin hyperplane. w x + b =1 ξ k are called slack variables. w x + b = 1 w x + b = 0 ξ 1 ξ k > 0 only if x k is misclassified or inside margin ξ 0

80 Extend to soft-margin SVMs Allow some instances to be misclassified, or fall within margins, but penalize them by distance to margin hyperplane. Revised optimization problem: Find w and b by doing the following minimization: w x + b =1 w x + b = 0 w x + b = 1 ξ 1 min w,b 1 2 w 2 + C ξ k subject to: k ξ 0 t k ( w x k + b) 1 ξ k, k =1,..., m (t k { 1,+1})

81 Extend to soft-margin SVMs Allow some instances to be misclassified, or fall within margins, but penalize them by distance to margin hyperplane. Revised optimization problem: Find w and b by doing the following minimization: w x + b =1 w x + b = 0 w x + b = 1 ξ 1 min w,b 1 2 w 2 + C ξ k subject to: k ξ 0 t k ( w x k + b) 1 ξ k, k =1,..., m (t k { 1,+1}) Optimization tries to keep ξ k s to zero while maximizing margin. C is parameter that trades off margin width with misclassifications

82 Why use soft-margin SVMs? Always can be optimized (unlike hard-margin SVMs) More robust to outliers, noise However: Have to set C parameter

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