Support Vector Machines

Advanced Computing Seminar Data Mining and Its Industrial Applications — Chapter 8 — Support Vector Machines Zhongzhi Shi, Markus Stumptner, Yalei Hao, Gerald Quirchmayr Knowledge and Software Engineering Lab Advanced Computing Research Centre School of Computer and Information Science University of South Australia

Outline Introduction Support Vector Machine Non-linear Classification SVM and PAC Applications Summary

History SVM is a classifier derived from statistical learning theory by Vapnik and Chervonenkis SVMs introduced by Boser, Guyon, Vapnik in COLT-92 Initially popularized in the NIPS community, now an important and active field of all Machine Learning research. Special issues of Machine Learning Journal, and Journal of Machine Learning Research.

What is SVM? SVMs are learning systems that use a hypothesis space of linear functions in a high dimensional feature space — Kernel function trained with a learning algorithm from optimization theory — Lagrange Implements a learning bias derived from statistical learning theory — Generalisation SVM is a classifier derived from statistical learning theory by Vapnik and Chervonenkis

Maximum 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 maximum margin. This is the simplest kind of SVM (Called an LSVM) Linear SVM Copyright © 2001, 2003, Andrew W. Moore

Model of Linear Classification Binary classification is frequently performed by using a real-valued hypothesis function: The input x is assigned to the positive class, if Otherwise to the negative class.

The concept of Hyperplane For a binary linear separable training set, we can find at least a hyperplane (w,b) which divides the space into two half spaces. The definition of hyperplane

Tuning the Hyperplane (w,b) The Perceptron Algorithm Proposed by Frank Rosenblatt in 1956 Preliminary definition The functional margin of an example (x i ,y i ) implies correct classification of (x i ,y i )

The Perceptron Algorithm The number of mistakes is at most

The Geometric margin -> The Euclidean distance of an example (x i ,y i ) from the decision boundary

The Geometric margin The margin of a training set S Maximal Margin Hyperplane A hyperplane realising the maximun geometric margin The optimal linear classifier If it can form the Maximal Margin Hyperplane.

How to Find the optimal solution ？ The drawback of the perceptron algorithm The algorithm may give a different solution depending on the order in which the examples are processed. The superiority of SVM The kind of learning machines tune the solution based on the optimization theory .

The Maximal Margin Classifier The simplest model of SVM Finds the maximal margin hyperplane in an chosen kernel-induced feature space. A convex optimization problem Minimizing a quadratic function under linear inequality constrains

Support Vector Classifiers Support vector machines Cortes and Vapnik (1995) well suited for high-dimensional data binary classification Training set D = {( x i ,y i ), i=1,…,n}, x i  R m and y i  {-1,1} Linear discriminant classifier Separating hyperplane { x : g( x ) = w T x + w 0 = 0 } model parameters: w  R m and w 0  R

Formalizi the geometric margin Assumes that The geometric margin In order to find the maximum ,we must find the minimum

Minimizing the norm -> Because We can re-formalize the optimization problem

Minimizing the norm -> Uses the Lagrangian function Obtained Resubstituting into the primal to obtain

Minimizing the norm Finds the minimum is equivalent to find the maximum The strategies for minimizing differentiable function Decomposition Sequential Minimal Optimization (SMO)

The Support Vector The condition of the optimization problem states that This implies that only for input xi for which the functional margin is one This implies that it lies closest to the hyperplane The corresponding

The optimal hypothesis (w,b) The two parameters can be obtained from The hypothesis is

Soft Margin Optimization The main problem with the maximal margin classifier is that it always products perfectly a consistent hypothesis a hypothesis with no training error Relax the boundary

Non-linear Classification The problem The maximal margin classifier is an important concept, but it cannot be used in many real-world problems There will in general be no linear separation in the feature space. The solution Maps the data into another space that can be separated linearly.

A learning machine A learning machine f takes an input x and transforms it, somehow using weights  , into a predicted output y est = +/- 1 f x  y est  is some vector of adjustable parameters

Some definitions Given some machine f And under the assumption that all training points (x k ,y k ) were drawn i.i.d from some distribution. And under the assumption that future test points will be drawn from the same distribution Define Official terminology

Some definitions Given some machine f And under the assumption that all training points (x k ,y k ) were drawn i.i.d from some distribution. And under the assumption that future test points will be drawn from the same distribution Define Official terminology R = #training set data points

Vapnik-Chervonenkis Dimension Given some machine f , let h be its VC dimension. h is a measure of f ’s power ( h does not depend on the choice of training set) Vapnik showed that with probability 1-  This gives us a way to estimate the error on future data based only on the training error and the VC-dimension of f

Structural Risk Minimization Let  (f) = the set of functions representable by f. Suppose Then We’re trying to decide which machine to use. We train each machine and make a table… f 4 4 f 5 5 f 6 6  f 3 3 f 2 2 f 1 1 Choice Probable upper bound on TESTERR VC-Conf TRAINERR f i i

Kernel-Induced Feature Space Mapping the data of space X into space F

Implicit Mapping into Feature Space For the non-linear separable data set, we can modify the hypothesis to map implicitly the data to another feature space

Kernel Function A Kernel is a function K , such that for all The benefits Solve the computational problem of working with many dimensions

The Polynomial Kernel The kind of kernel represents the inner product of two vector(point) in a feature space of dimension. For example

Text Categorization Inductive learning Inpute : Output : f(x) = confidence(class) In the case of text classification ,the attribute are words in the document ,and the classes are the categories.

PROPERTIES OF TEXT-CLASSIFICATION TASKS High-Dimensional Feature Space. Sparse Document Vectors. High Level of Redundancy.

Text representation and feature selection Binary feature term frequency Inverse document frequency n is the total number of documents DF(w) is the number of documents the word occurs in

Learning SVMS To learn the vector of feature weights Linear SVMS Polynomial classifiers Radial basis functions

Processing Text files are processed to produce a vector of words Select 300 words with highest mutual information with each category(remove stopwords) A separate classifier is learned for each category.

An example - Reuters ( trends & controversies) Category : interest Weight vector large positive weights : prime (.70), rate (.67), interest (.63), rates (.60), and discount (.46) large negative weights: group (–.24),year (–.25), sees (–.33) world (–.35), and dlrs (–.71)

Text Categorization Results Dumais et al. (1998)

Apply to the Linear Classifier Substitutes to the hypothesis Substitutes to the margin optimization

SVMs and PAC Learning Theorems connect PAC theory to the size of the margin Basically, the larger the margin, the better the expected accuracy See, for example, Chapter 4 of Support Vector Machines by Christianini and Shawe-Taylor, Cambridge University Press, 2002

PAC and the Number of Support Vectors The fewer the support vectors, the better the generalization will be Recall, non-support vectors are Correctly classified Don’t change the learned model if left out of the training set So

VC-dimension of an SVM Very loosely speaking there is some theory which under some different assumptions puts an upper bound on the VC dimension as where Diameter is the diameter of the smallest sphere that can enclose all the high-dimensional term-vectors derived from the training set. Margin is the smallest margin we’ll let the SVM use This can be used in SRM (Structural Risk Minimization) for choosing the polynomial degree, RBF  , etc. But most people just use Cross-Validation Copyright © 2001, 2003, Andrew W. Moore

Finding Non-Linear Separating Surfaces Map inputs into new space Example: features x 1 x 2 5 4 Example: features x 1 x 2 x 1 2 x 2 2 x 1 *x 2 5 4 25 16 20 Solve SVM program in this new space Computationally complex if many features But a clever trick exists

Summary Maximize the margin between positive and negative examples (connects to PAC theory) Non-linear Classification The support vectors contribute to the solution Kernels map examples into a new, usually non-linear space

References Vladimir Vapnik. The Nature of Statistical Learning Theory , Springer, 1995 Andrew W. Moore. cmsc726: SVMs. http:// www.cs.cmu.edu/~awm/tutorials C. Burges. A tutorial on support vector machines for pattern recognition. Data Mining and Knowledge Discovery, 2(2):955-974, 1998. http://citeseer.nj.nec.com/burges98tutorial.html Vladimir Vapnik. Statistical Learning Theory. Wiley-Interscience; 1998 Thorsten Joachims (joachims_01a): A Statistical Learning Model of Text Classification for Support Vector Machines

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Support Vector Machines

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