Support Vector Machine PNU Artificial Intelligence Lab. Kim, Minho

Overview  Support Vector Machines (SVM)  Linear SVMs  Non-Linear SVMs  Properties of SVM  Applications of SVMs  Gene Expression Data Classification  Text Categorization

Machine Learning

Type of Learning  Supervised (inductive) learning  Training data includes desired outputs  Unsupervised learning  Training data does not include desired outputs  Semi-supervised learning  Training data includes a few desired outputs  Reinforcement learning  Rewards from sequence of actions

f x  y est denotes +1 denotes -1 f(x,w,b) = sign(w x + b) How would you classify this data? w x + b=0 w x + b<0 w x + b>0 Linear Classifiers

f x  y est denotes +1 denotes -1 f(x,w,b) = sign(w x + b) How would you classify this data? Linear Classifiers

f x  y est denotes +1 denotes -1 f(x,w,b) = sign(w x + b) How would you classify this data? 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? Linear Classifiers

f x  y est denotes +1 denotes -1 f(x,w,b) = sign(w x + b) How would you classify this data? Misclassified to +1 class Linear Classifiers

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. 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. Classifier 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 Support Vectors are those datapoints that the margin pushes up against 1.Maximizing the margin is good according to intuition and PAC theory 2.Implies that only support vectors are important; other training examples are ignorable. 3.Empirically it works very very well. Maximum Margin

Hard Margin: So far we require all data points be classified correctly - No training error What if the training set is noisy? - Solution 1: use very powerful kernels denotes +1 denotes -1 OVERFITTING! Dataset With Noise

Slack variables ξi can be added to allow misclassification of difficult or noisy examples. wx+b=1 wx+b=0 wx+b=-1 77  11 22 Soft Margin Classification What should our quadratic optimization criterion be? Minimize

The old formulation: The new formulation incorporating slack variables: Parameter C can be viewed as a way to control overfitting. 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 Hard Margin versus Soft Margin

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. Linear SVMs: Overview

Non-Linear SVMs : Example

Non-Linear SVMs : After Modification

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 x 0 x 0 x x2x2 Non-Linear SVMs

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) Non-Linear SVMs: Feature Spaces

The linear classifier relies on dot product between vectors K(x i,x j )=x i T x j If every data point is mapped into high-dimensional space via some transformation Φ: x → φ(x), the dot 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 j x i1 x j1 x i2 x j2 + x i2 2 x j x 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 ] The “Kernel Trick”

Linear: K(x i,x j )= x i T x j Polynomial of power p: K(x i,x j )= (1+ x i T x j ) p Gaussian radial-basis function network): Examples of Kernel Functions

Example: Properties of Kernel Function Linear Kernel Polynomial KernelRadial Basis Function

Example: Kernel Selection

SVM locates a separating hyperplane in the feature space and classify points in that space It does not need to represent the space explicitly, simply by defining a kernel function The kernel function plays the role of the dot product in the feature space. Non-linear SVM - Overview

Properties of SVM  Flexibility in choosing a similarity function  Sparseness of solution when dealing with large data sets - only support vectors are used to specify the separating hyperplane  Ability to handle large feature spaces - complexity does not depend on the dimensionality of the feature space  Overfitting can be controlled by soft margin approach  Nice math property: a simple convex optimization problem which i s guaranteed to converge to a single global solution

Weakness of SVM  It is sensitive to noise - A relatively small number of mislabeled examples can dramatically de crease the performance  It only considers two classes - how to do multi-class classification with SVM? - Answer: 1) with output arity m, learn m SVM’s  SVM 1 learns “Output==1” vs “Output != 1”  SVM 2 learns “Output==2” vs “Output != 2”  :  SVM m learns “Output==m” vs “Output != m” 2)To predict the output for a new input, just predict with each SVM an d find out which one puts the prediction the furthest into the positive r egion.

SVM Applications  SVM has been used successfully in many real-world problems - text (and hypertext) categorization - image classification - bioinformatics (Protein classification, Cancer classification) - hand-written character recognition

Text Classification  Goal: to classify documents (news articles, s, Web pages, etc.) into predefined categories  Examples  To classify news articles into “business” and “sports”  To classify Web pages into personal home pages and others  To classify product reviews into positive reviews and negative reviews  Approach: supervised machine learning  For each pre-defined category, we need a set of training documents known to belong to the category.  From the training documents, we train a classifier.

Overview  Step 1—text pre-processing  to pre-process text and represent each document as a feature vector  Step 2—training  to train a classifier using a classification tool (e.g. LIBSVM, SVMlight)  Step 3—classification  to apply the classifier to new documents

Pre-processing: tokenization  Goal: to separate text into individual words  Example: “We’re attending a tutorial now.”  we ’re attending a tutorial now  Tool:  Word Splitter

Pre-processing: stop word removal (optional)  Goal: to remove common words that are usually not useful for text classification  Example: to remove words such as “a”, “the”, “I”, “he”, “she”, “is”, “are”, etc.  Stop word list:  /stop_words /stop_words

Pre-processing: stemming (optional)  Goal: to normalize words derived from the same root  Examples:  attending  attend  teacher  teach  Tool:  Porter stemmer

Pre-processing: feature extraction  Unigram features: to use each word as a feature  To use TF (term frequency) as feature value  To use TF*IDF (inverse document frequency) as feature value  IDF = log (total-number-of-documents / number-of- documents-containing-t)  Bigram features: to use two consecutive words as a feature

SVMlight  SVM-light: a command line C program that implements the SVM learning algorithm  Classification, regression, ranking  Download at  Documentation on the same page  Two programs  svm_learn for training  svm_classify for classification

SVMlight Examples  Input format 1 1:0.5 3:1 5: :0.9 3:0.1 4:2  To train a classifier from train.data  svm_learn train.data train.model  To classify new documents in test.data  svm_classify test.data train.model test.result  Output format  Positive score  positive class  Negative score  negative class  Absolute value of the score indicates confidence  Command line options  -c a tradeoff parameter (use cross validation to tune)

More on SVM-light  Kernel  Use the “-t” option  Polynomial kernel  User-defined kernel  Semi-supervised learning (transductive SVM)  Use “0” as the label for unlabeled examples  Very slow

Some Issues  Choice of kernel - Gaussian or polynomial kernel is default - if ineffective, more elaborate kernels are needed - domain experts can give assistance in formulating appropriate similari ty measures  Choice of kernel parameters - e.g. σ in Gaussian kernel - σ is the distance between closest points with different classifications - In the absence of reliable criteria, applications rely on the use of a valid ation set or cross-validation to set such parameters.  Optimization criterion – Hard margin v.s. Soft margin - a lengthy series of experiments in which various parameters are tested

Reference  Support Vector Machine Classification of Microarray Gene Expression Data, Michael P. S. Brown William N oble Grundy, David Lin, Nello Cristianini, Charles Sugne t, Manuel Ares, Jr., David Haussler   Text categorization with Support Vector Machines: learning with many relevant features T. Joachims, ECML - 98