1 Machine Learning Support Vector Machines. 2 Perceptron Revisited: Linear Separators Binary classification can be viewed as the task of separating classes

1

Machine Learning

Support Vector Machines

2

Perceptron Revisited: Linear Separators

Binary classification can be viewed as the task of separating classes in feature space:

wTx + b = 0

wTx + b < 0wTx + b > 0

g(x) = sign(wTx + b)

3

Linear Discriminant Function

g(x) is a linear function:

( ) Tg b x w x

x1

x2

wT x + b = 0

wT x + b < 0

wT x + b > 0

A hyper-plane in the feature space

(Unit-length) normal vector of the hyper-plane:

w

nw

n

4

How would you classify these points using a linear discriminant function in order to minimize the error rate?

Linear Discriminant Functiondenotes +1

denotes -1

x1

x2

Infinite number of answers!

5



denotes -1

x1

x2


6



denotes -1

x1

x2


7

x1

x2 How would you classify these

points using a linear discriminant function in order to minimize the error rate?


denotes -1


Which one is the best?

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Large Margin Linear Classifier

“safe zone” The linear discriminant

function (classifier) with the maximum margin is the best

Margin is defined as the width that the boundary could be increased by before hitting a data point

Why it is the best? Robust to outliners and thus

strong generalization ability

Margin

x1

x2

denotes +1

denotes -1

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Classification Margin

Distance from example xi to the separator is Examples closest to the hyperplane are support vectors. Margin ρ of the separator is the distance between support vectors.

w

xw br i

T

r

ρ

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Maximum Margin Classification

Maximizing the margin is good according to intuition and PAC theory.

Implies that only support vectors matter; other training examples are ignorable.

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Large Margin Linear Classifier

We know that

The margin width is:

x1

x2

denotes +1

denotes -1

1

1

T

T

b

b

w x

w x

Margin

wT x + b = 0

wT x + b = -1w

T x + b = 1

x+

x+

x-

( )

2 ( )

M

x x n

wx x

w w

n

Support Vectors

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Linear SVMs Mathematically (cont.)

Then we can formulate the quadratic optimization problem:

Which can be reformulated as:

Find w and b such that

is maximized

and for all (xi, yi), i=1..n : yi(wTxi + b) ≥ 1

w

2


Φ(w) = ||w||2=wTw is minimized

and for all (xi, yi), i=1..n : yi (wTxi + b) ≥ 1

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Solving the Optimization Problem

Need to optimize a quadratic function subject to linear constraints. Quadratic optimization problems are a well-known class of

mathematical programming problems for which several (non-trivial) algorithms exist.

The solution involves constructing a dual problem where a Lagrange multiplier αi is associated with every inequality constraint in the primal (original) problem:

Find w and b such thatΦ(w) =wTw is minimized and for all (xi, yi), i=1..n : yi (wTxi + b) ≥ 1

Find α1…αn such that

Q(α) =Σαi - ½ΣΣαiαjyiyjxiTxj is maximized and

(1) Σαiyi = 0(2) αi ≥ 0 for all αi

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The Optimization Problem Solution

Given a solution α1…αn to the dual problem, solution to the primal is:

Each non-zero αi indicates that corresponding xi is a support vector. Then the classifying function is (note that we don’t need w explicitly):

Notice that it relies on an inner product between the test point x and the support vectors xi

Also keep in mind that solving the optimization problem involved computing the inner products xi

Txj between all training points.

w =Σαiyixi b = yk - Σαiyixi Txk for any αk > 0

f(x) = ΣαiyixiTx + b

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Soft Margin Classification

What if the training set is not linearly separable? Slack variables ξi can be added to allow misclassification of difficult

or noisy examples, resulting margin called soft.

ξi

ξi

R

kkεC

1

.2

1ww

What should our quadratic optimization criterion be?

Minimize

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Soft Margin Classification Mathematically

The old formulation:

Modified formulation incorporates slack variables:

Parameter C can be viewed as a way to control overfitting: it “trades off” the relative importance of maximizing the margin and fitting the training data.

Find w and b such thatΦ(w) =wTw is minimized and for all (xi ,yi), i=1..n : yi (wTxi + b) ≥ 1


Φ(w) =wTw + CΣξi is minimized

and for all (xi ,yi), i=1..n : yi (wTxi + b) ≥ 1 – ξi, , ξi ≥ 0

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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

0

0

x2

x

x

x

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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)

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The “Kernel Trick”

The linear classifier relies on inner product between vectors K(xi,xj)=xiTxj

If every datapoint is mapped into high-dimensional space via some transformation Φ: x → φ(x), the inner product becomes:

K(xi,xj)= φ(xi) Tφ(xj)

A kernel function is a function that is equivalent to an inner product in some feature space.

Example: 2-dimensional vectors x=[x1 x2]; let K(xi,xj)=(1 + xi

Txj)2,

Need to show that K(xi,xj)= φ(xi) Tφ(xj):

K(xi,xj)=(1 + xiTxj)2

,= 1+ xi12xj1

2 + 2 xi1xj1 xi2xj2+ xi2

2xj22 + 2xi1xj1 + 2xi2xj2=

= [1 xi12 √2 xi1xi2 xi2

2 √2xi1 √2xi2]T [1 xj12 √2 xj1xj2 xj2

2 √2xj1 √2xj2] =

= φ(xi) Tφ(xj), where φ(x) = [1 x1

2 √2 x1x2 x22 √2x1 √2x2]

Thus, a kernel function implicitly maps data to a high-dimensional space (without the need to compute each φ(x) explicitly).

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What Functions are Kernels?

For some functions K(xi,xj) checking that K(xi,xj)= φ(xi) Tφ(xj) can be

cumbersome. Mercer’s theorem:

Every semi-positive definite symmetric function is a kernel Semi-positive definite symmetric functions correspond to a semi-

positive definite symmetric Gram matrix:

K(x1,x1) K(x1,x2) K(x1,x3) … K(x1,xn)

K(x2,x1) K(x2,x2) K(x2,x3) K(x2,xn)

… … … … …

K(xn,x1) K(xn,x2) K(xn,x3) … K(xn,xn)

K=

For any non-zero vector x, xTKx>0

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Examples of Kernel Functions

Linear: K(xi,xj)= xi Txj

Polynomial of power p: K(xi,xj)= (1+ xi Txj)p

Gaussian (radial-basis function network):

Sigmoid: K(xi,xj)= tanh(β0xi Txj + β1)

)2

exp(),(2

2

ji

ji

xxxx

K

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Support Vector Machine: Algorithm

1. Choose a kernel function

2. Choose a value for C

3. Solve the quadratic programming problem (many software packages available)

4. Construct the discriminant function from the support vectors

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Some Issues Choice of kernel - Gaussian or polynomial kernel are the mostly used non-linear kernels - if ineffective, more elaborate kernels are needed - domain experts can give assistance in formulating appropriate similarity

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

validation 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

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April 18, 2023Data Mining: Concepts and

Techniques 24

Why Is SVM Effective on High Dimensional Data?

The complexity of trained classifier is characterized by the #

of support vectors rather than the dimensionality of the data

The support vectors are the essential or critical training

examples —they lie closest to the decision boundary

If all other training examples are removed and the training is

repeated, the same separating hyperplane would be found

The number of support vectors found can be used to

compute an (upper) bound on the expected error rate of the

SVM classifier, which is independent of the data

dimensionality

Thus, an SVM with a small number of support vectors can

have good generalization, even when the dimensionality of

the data is high

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SVM applications

SVMs were originally proposed by Boser, Guyon and Vapnik in 1992 and gained increasing popularity in late 1990s.

SVMs are currently among the best performers for a number of classification tasks ranging from text to genomic data.

SVMs can be applied to complex data types beyond feature vectors (e.g. graphs, sequences, relational data) by designing kernel functions for such data.

SVM techniques have been extended to a number of tasks such as regression [Vapnik et al. ’97], principal component analysis [Schölkopf et al. ’99], etc.

Most popular optimization algorithms for SVMs use decomposition to hill-climb over a subset of αi’s at a time, e.g. SMO [Platt ’99] and [Joachims ’99]

Tuning SVMs remains a black art: selecting a specific kernel and parameters is usually done in a try-and-see manner.

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SVM vs. Neural Network

SVM Relatively new concept Deterministic algorithm Nice Generalization

properties Hard to learn – learned in

batch mode using quadratic programming techniques, but faster with good optimization methods

Using kernels can learn very complex functions

Neural Network Relatively old Nondeterministic algorithm Generalizes well but doesn’t

have strong mathematical foundation

Can easily be learned in incremental fashion

To learn complex functions—use multilayer perceptron (not that trivial)

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Summary: Support Vector Machine

1. Large Margin Classifier Better generalization ability & less over-fitting

2. The Kernel Trick Map data points to higher dimensional space in order

to make them linearly separable. Since only dot product is used, we do not need to

represent the mapping explicitly.

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SVM resources

http://www.kernel-machines.org

http://www.csie.ntu.edu.tw/~cjlin/libsvm/

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Model Evaluation

Metrics for Performance Evaluation How to evaluate the performance of a model?

Methods for Performance Evaluation How to obtain reliable estimates?

Methods for Model Comparison How to compare the relative performance among

competing models?

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Metrics for Performance Evaluation

Focus on the predictive capability of a model Rather than how fast it takes to classify or build

models, scalability, etc. Confusion Matrix:

PREDICTED CLASS

ACTUALCLASS

Class=Yes Class=No

Class=Yes a b

Class=No c d

a: TP (true positive)

b: FN (false negative)

c: FP (false positive)

d: TN (true negative)

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Metrics for Performance Evaluation…

Most widely-used metric:

PREDICTED CLASS

ACTUALCLASS

Class=Yes Class=No

Class=Yes a(TP)

b(FN)

Class=No c(FP)

d(TN)

FNFPTNTPTNTP

dcbada

Accuracy

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Limitation of Accuracy

Consider a 2-class problem Number of Class 0 examples = 9990 Number of Class 1 examples = 10

If model predicts everything to be class 0, accuracy is 9990/10000 = 99.9 % Accuracy is misleading because model does not detect

any class 1 example

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Cost Matrix

PREDICTED CLASS

ACTUALCLASS

C(i|j) Class=Yes Class=No

Class=Yes C(Yes|Yes) C(No|Yes)

Class=No C(Yes|No) C(No|No)

C(i|j): Cost of misclassifying class j example as class i

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Cost-Sensitive Measures

cbaa

prrp

baa

caa

222

(F) measure-F

(r) Recall

(p)Precision

Precision is biased towards C(Yes|Yes) & C(Yes|No) Recall is biased towards C(Yes|Yes) & C(No|Yes) F-measure is biased towards all except C(No|No)

dwcwbwawdwaw

4321

41Accuracy Weighted

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Model Evaluation




competing models?

36

Methods for Performance Evaluation

How to obtain a reliable estimate of performance?

Performance of a model may depend on other factors besides the learning algorithm: Class distribution Cost of misclassification Size of training and test sets

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Learning Curve

Learning curve shows how accuracy changes with varying sample size

Requires a sampling schedule for creating learning curve:

Arithmetic sampling(Langley, et al)

Geometric sampling(Provost et al)

Effect of small sample size:- Bias in the estimate- Variance of estimate

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Methods of Estimation

Holdout Reserve 2/3 for training and 1/3 for testing

Random subsampling Repeated holdout

Cross validation Partition data into k disjoint subsets k-fold: train on k-1 partitions, test on the remaining one Leave-one-out: k=n

Stratified sampling oversampling vs undersampling

Bootstrap Sampling with replacement

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Model Evaluation




competing models?

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ROC (Receiver Operating Characteristic)

Characterize the trade-off between positive hits and false alarms

ROC curve plots TP (on the y-axis) against FP (on the x-axis)

Performance of each classifier represented as a point on the ROC curve changing the threshold of algorithm, sample

distribution or cost matrix changes the location of the point

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ROC Curve

At threshold t:

TP=0.5, FN=0.5, FP=0.12, FN=0.88

- 1-dimensional data set containing 2 classes (positive and negative)

- any points located at x > t is classified as positive

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ROC Curve

(TP,FP): (0,0): declare everything

to be negative class (1,1): declare everything

to be positive class (1,0): ideal

Diagonal line: Random guessing Below diagonal line:

prediction is opposite of the true class

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Using ROC for Model Comparison

No model consistently outperform the other M1 is better for

small FPR M2 is better for

large FPR

Area Under the ROC curve

Ideal: Area = 1

Random guess: Area = 0.5

Documents

1 Machine Learning Support Vector Machines. 2 Perceptron Revisited: Linear Separators Binary classification can be viewed as the task of separating classes