Support Vector Machines 主講人：虞台文

Content • Introduction • The VC Dimension & Structure Risk Minimization • Linear SVM  The Separable case • Linear SVM  The Non-Separable case • Lagrange Multipliers

Support Vector Machines Introduction

Learning Machines • A machine to learn the mapping • Defined as Learning by adjusting this parameter?

Generalization vs. Learning • How a machine learns? • Adjusting the parameters so as to partition the pattern (feature) space for classification. • How to adjust? Minimize the empirical risk (traditional approaches). • What the machine learned? • Memorize the patterns it sees? or • Memorize the rules it finds for different classes? • What does the machine actually learn if it minimizes empirical risk only?

Risks Expected Risk (test error) Empirical Risk (training error)

More on Empirical Risk • How can make the empirical risk arbitrarily small? • To let the machine have very large memorization capacity. • Does a machine with small empirical risk also get small expected risk? • How to avoid the machine to strain to memorize training patterns, instead of doing generalization, only? • How to deal with the straining-memorization capacity of a machine? • What the new criterion should be?

Structure Risk Minimization Learn both the right ‘structure’ and right `rules’ for classification. Goal: Right Structure: E.g., Right amount and right forms of components or parameters are to participate in a learning machine. Right Rules: The empirical risk will also be reduced if right rules are learned.

New Criterion Risk due to the structure of the learning machine Total Risk Empirical Risk = +

Support Vector Machines The VC Dimension & Structure Risk Minimization

VC: Vapnik Chervonenkis The VC Dimension • Consider a set of function f (x,) {1,1}. • A given set oflpoints can be labeled in 2lways. • If a member of the set {f ()} can be found which correctly assigns the labels for all labeling, then the set of points is shattered by that set of functions. • The VC dimension of {f ()} is the maximum number of training points that can be shattered by {f ()}.

VC dimension = 3 The VC Dimension for Oriented Lines in R2

More on VC Dimension • In general, the VC dimension of a set of oriented hyperplanes inRn is n+1. • VC dimension is a measure of memorization capability. • VC dimension is not directly related to number of parameters. Vapnik (1995) has an example with 1 parameter and infinite VC dimension.

Bound on Expected Risk Expected Risk Empirical Risk VC Confidence

Bound on Expected Risk Consider small  (e.g.,   0.5). VC Confidence

Bound on Expected Risk Consider small  (e.g.,   0.5). Structure risk minimization want to minimize the bound Traditional approaches minimize empirical risk only

VC Confidence Amongst machines with zero empirical risk, choose the one with smallest VC dimension How to evaluate VC dimension?  =0.05 and l=10,000

h3 h2 h1 h4 Structure Risk Minimization Nested subset of functions with different VC dimensions.

Support Vector Machines The Linear SVM  The Separable Case

The Linear Separability Linearly separable Not linearly separable

w wx + b = +1 wx + b = 1 The Linear Separability Linearly Separable wx + b = 0 Linearly separable

w d wx + b = +1 wx + b = 1 O Margin Width How about maximize the margin? wx + b = 0 What is the relation btw. the margin width and VC dimension?

O Supporters Maximum Margin Classifier How about maximize the margin? What is the relation btw. the margin width and VC dimension?

Building SVM Minimize Subject to This requires the knowledge about Lagrange Multiplier.

The Method of Lagrange Minimize Subject to The Lagrangian: Minimize it w.r.t w, while maximize it w.r.t. .

The Method of Lagrange How about if it is zero? Minimize Subject to What value of i should be if it is feasible and nonzero? The Lagrangian: Minimize it w.r.t w, while maximize it w.r.t. .

The Method of Lagrange Minimize Subject to The Lagrangian:

Duality Minimize Subject to Maximize Subject to

Duality Maximize Subject to

Maximize Duality

Maximize Duality

Duality Minimize The Primal Subject to Maximize The Dual Subject to

Quadratic Programming The Solution Find * by … Maximize The Dual Subject to

The Solution Call it a support vector is i> 0. Find * by … The Karush-Kuhn-Tucker Conditions The Lagrangian:

The Karush-Kuhn-Tucker Conditions

Classification

Classification Using Supporters The weight for the ith support vector. Bias The similarity measure btw. input and theithsupport vector.

Demonstration

Support Vector Machines The Linear SVM  The Non-Separable Case

wx + b = +1 wx + b = +1  i wx + b = 1 wx + b = 0 The Non-Separable Case We require that

For simplicity, we consider k = 1. Mathematic Formulation Minimize Subject to

The Lagrangian Minimize Subject to

Duality Minimize Subject to Maximize Subject to

Duality Maximize Subject to

Maximize this Duality

Maximize this Duality

Duality Minimize The Primal Subject to Maximize The Dual Subject to

The Karush-Kuhn-Tucker Conditions

Quadratic Programming The Solution Find * by … Maximize The Dual Subject to

The Solution Call it a support vector is 0