Linear classifiers CE-717: Machine Learning Sharif University of - - PowerPoint PPT Presentation

Linear classifiers

CE-717: Machine Learning

Sharif University of Technology

• M. Soleymani

Fall 2018

Topics

} Linear classifiers

} Perceptron } Fisher

} Multi-class classification

2

SVM will be covered in the later lectures

Classification problem

3

} Given:Training set

} labeled set of 𝑂 input-output pairs 𝐸 =

𝒚 % , 𝑧 %

%() *

} 𝑧 ∈ {1, … , 𝐿}

} Goal: Given an input 𝒚, assign it to one of 𝐿 classes } Examples:

} Spam filter } Handwritten digit recognition } …

Linear classifiers

4

} Decision boundaries are linear in 𝒚, or linear in some

given set of functions of 𝒚

} Linearly separable data: data points that can be exactly

classified by a linear decision surface.

} Why linear classifier?

} Even when they are not optimal, we can use their simplicity

} are relatively easy to compute } In the absence of information suggesting otherwise, linear classifiers are an

attractive candidates for initial, trial classifiers.

Two Category

5

} 𝑔 𝒚; 𝒙 = 𝒙4𝒚 + 𝑥7 = 𝑥7 + 𝑥)𝑦) + . . . 𝑥;𝑦;

} 𝒚 = 𝑦) 𝑦< … 𝑦; } 𝒙 = [𝑥) 𝑥< … 𝑥;] } 𝑥7: bias

} if 𝒙4𝒚 + 𝑥7 ≥ 0 then 𝒟) } else 𝒟<

Decision surface (boundary): 𝒙4𝒚 + 𝑥7 = 0

𝒙 is orthogonal to every vector lying within the decision surface

Example

6

𝑦1 𝑦2

1 2 3 1 2 3 4

3 − 3 4 𝑦) − 𝑦< = 0 if 𝒙4𝒚 + 𝑥7 ≥ 0 then 𝒟) else 𝒟<

Linear classifier: Two Category

7

} Decision boundary is a (𝑒 − 1)-dimensional hyperplane 𝐼 in

the 𝑒-dimensional feature space

} The orientation of 𝐼 is determined by the normal vector 𝑥), … , 𝑥; } 𝑥7 determine the location of the surface.

} The normal distance from the origin to the decision surface is HI

𝒙

𝒚 = 𝒚J + 𝑠 𝒙 𝒙 𝒙4𝒚 + 𝑥7 = 𝑠 𝒙 ⇒ 𝑠 = 𝒙4𝒚 + 𝑥7 𝒙 gives a signed measure of the perpendicular distance 𝑠 of the point 𝒚 from the decision surface

𝑔 𝒚 = 0

𝒚J

Linear boundary: geometry

8

𝒙4𝒚 + 𝑥7 = 0 𝒙4𝒚 + 𝑥7 > 0 𝒙4𝒚 + 𝑥7 < 0 𝒙4𝒚 + 𝑥7 𝒙

Non-linear decision boundary

9

} Choose non-linear features } Classifier still linear in parameters 𝒙 𝑦1 𝑦2

1 1

−1 + 𝑦)

< + 𝑦< < = 0

if 𝒙4𝝔(𝒚) ≥ 0 then 𝑧 = 1 else 𝑧 = −1 𝒙 = 𝑥7, 𝑥), … , 𝑥R = [−1, 0, 0,1,1,0]

1

• 1

𝝔 𝒚 = [1, 𝒚), 𝒚< , 𝒚)

<, 𝒚< <, 𝒚)𝒚<]

𝒚 = [𝒚), 𝒚<]

Cost Function for linear classification

10

} Finding linear classifiers can be formulated as an optimization

problem:

} Select how to measure the prediction loss

} Based on the training set 𝐸 =

𝒚 % , 𝑧 %

%() S

, a cost function 𝐾 𝒙 is defined

} Solve the resulting optimization problem to find parameters:

} Find optimal 𝑔

U 𝒚 = 𝑔 𝒚; 𝒙 V where 𝒙 V = argmin

𝒙

𝐾 𝒙

} Criterion or cost functions for classification:

} We will investigate several cost functions for the classification problem

SSE cost function for classification

11

SSE cost function is not suitable for classification:

} Least square loss penalizes ‘too correct’ predictions (that they lie a long

way on the correct side of the decision)

} Least square loss also lack robustness to noise

𝐾 𝒙 = ] 𝒙4𝒚 % − 𝑧 %

< * %() 𝐿 = 2

SSE cost function for classification

12

𝒙4𝒚 𝑧 = 1 𝒙4𝒚 − 𝑧 < 1 𝒙4𝒚 𝑧 = −1 𝒙4𝒚 − 𝑧 < −1 Correct predictions that are penalized by SSE [Bishop] 𝐿 = 2

SSE cost function for classification

13

𝐾(𝒙)

} Is it more suitable if we set 𝑔 𝒚; 𝒙 = 𝑕 𝒙4𝒚 ?

𝐾 𝒙 = ] sign 𝒙4𝒚 % − 𝑧 %

< * %()

sign 𝑨 = a− 1, 𝑨 < 0 1, 𝑨 ≥ 0

} 𝐾 𝒙

is a piecewise constant function shows the number

• f misclassifications

𝐿 = 2 𝒙4𝒚 𝑧 = 1 sign 𝒙4𝒚 − 𝑧 <

Training error incurred in classifying training samples

SSE cost function

14

} Is it more suitable if we set 𝑔 𝒚; 𝒙 = 𝜏 𝒙4𝒚 ?

𝐾 𝒙 = ] 𝜏 𝒙4𝒚 % − 𝑧 %

< * %()

𝜏 𝑨 = 1 − 𝑓de 1 + 𝑓de

} We see later in this lecture than the cost function of the

logistic regression method is more suitable than this cost function for the classification problem

𝐿 = 2

Perceptron algorithm

15

} Linear classifier } Two-class: 𝑧 ∈ {−1,1}

} 𝑧 = −1 for 𝐷<,

𝑧 = 1 for 𝐷)

} Goal: ∀𝑗, 𝒚(%) ∈ 𝐷) ⇒ 𝒙4𝒚(%) > 0 }

∀𝑗, 𝒚 % ∈ 𝐷< ⇒ 𝒙4𝒚 % < 0

} 𝑔 𝒚; 𝒙 = sign(𝒙4𝒚)

Perceptron criterion

16

𝐾i 𝒙 = − ] 𝒙4𝒚 % 𝑧 %

• %∈ℳ

ℳ: subset of training data that are misclassified Many solutions? Which solution among them?

Cost function

17

[Duda, Hart, and Stork, 2002] 𝐾(𝒙) 𝐾i(𝒙) 𝑥7 𝑥) 𝑥7 𝑥) # of misclassifications as a cost function Perceptron’s cost function There may be many solutions in these cost functions

Batch Perceptron

18

“Gradient Descent” to solve the optimization problem: 𝒙lm) = 𝒙l − 𝜃𝛼

𝒙𝐾i(𝒙l)

𝛼

𝒙𝐾i 𝒙 = − ] 𝒚 % 𝑧 %

• %∈ℳ

Batch Perceptron converges in finite number of steps for linearly separable data:

Initialize 𝒙 Repeat 𝒙 = 𝒙 + 𝜃 ∑ 𝒚 % 𝑧 %

• %∈ℳ

Until 𝜃 ∑ 𝒚 % 𝑧 %

• %∈ℳ

< 𝜄

Stochastic gradient descent for Perceptron

19

} Single-sample perceptron:

} If 𝒚(%) is misclassified:

𝒙lm) = 𝒙l + 𝜃𝒚(%)𝑧(%)

} Perceptron convergence theorem: for linearly separable data

} If training data are linearly separable, the single-sample perceptron is

also guaranteed to find a solution in a finite number of steps

Initialize 𝒙, 𝑢 ← 0 repeat 𝑢 ← 𝑢 + 1 𝑗 ← 𝑢 mod 𝑂 if 𝒚(%) is misclassified then 𝒙 = 𝒙 + 𝒚(%)𝑧(%) Until all patterns properly classified

Fixed-Increment single sample Perceptron 𝜃 can be set to 1 and proof still works

Perceptron Convergence

20

} It can be shown:

𝑁 = max

(𝒚,x)∈y 𝒚 <

𝜈 = 2 min

(𝒚,x)∈y 𝑧𝒙

V 0 4𝒚 𝑏 = min

(𝒚,x)∈y 𝑧𝒙

V ∗4𝒚

Example

21

Perceptron: Example

22

Change 𝒙 in a direction that corrects the error [Bishop]

Convergence of Perceptron

23

} For data sets that are not linearly separable, the single-sample

perceptron learning algorithm will never converge

[Duda, Hart & Stork, 2002]

Pocket algorithm

24

} For the data that are not linearly separable due to noise:

} Keeps in its pocket the best 𝒙 encountered up to now.

Initialize 𝒙 for 𝑢 = 1, … , 𝑈 𝑗 ← 𝑢 mod 𝑂 if 𝒚(%) is misclassified then 𝒙S~H = 𝒙 + 𝒚(%)𝑧(%) if 𝐹l€•%S 𝒙S~H < 𝐹l€•%S 𝒙 then 𝒙 = 𝒙S~H end

𝐹l€•%S 𝒙 = 1 𝑂 ] 𝑡𝑗𝑕𝑜(𝒙4𝒚(S)) ≠ 𝑧(S)

* S()

Linear Discriminant Analysis (LDA)

25

} Fisher’s Linear Discriminant Analysis :

} Dimensionality reduction

} Finds linear combinations of features with large ratios of between-

groups scatters to within-groups scatters (as discriminant new variables)

} Classification

} Predicts the class of an observation 𝒚 by first projecting it to the

space of discriminant variables and then classifying it in this space

Good Projection for Classification

} What is a good criterion?

} Separating different classes in the projected space

26

Good Projection for Classification

} What is a good criterion?

} Separating different classes in the projected space

27

Good Projection for Classification

} What is a good criterion?

} Separating different classes in the projected space

28

𝒙

LDA Problem

} Problem definition:

} 𝐷 = 2 classes }

𝒚(%), 𝑧(%)

%() *

training samples with 𝑂) samples from the first class (𝒟)) and 𝑂< samples from the second class (𝒟<)

} Goal: finding the best direction 𝒙 that we hope to enable accurate

classification

} The projection of sample 𝒚 onto a line in direction 𝒙 is 𝒙4𝒚 } What is the measure of the separation between the projected

points of different classes?

29

Measure of Separation in the Projected Direction

30

[Bishop] } Is the direction of the line jointing the class means a good

candidate for 𝒙?

Measure of Separation in the Projected Direction

31

} The direction of the line jointing the class means is the

solution of the following problem:

} Maximizes the separation of the projected class means

max

𝒙 𝐾 𝒙 = 𝜈) … − 𝜈< … <

• s. t. 𝒙

= 1

} What is the problem with the criteria considering only

𝜈)

… − 𝜈< … ?

} It does not consider the variances of the classes in the projected direction

𝜈)

… = 𝒙4 𝝂)

𝝂) =

∑ 𝒚(ˆ)

• 𝒚(ˆ)∈𝒟‰

*‰

𝜈<

… = 𝒙4 𝝂<

𝝂< =

∑ 𝒚(ˆ)

• 𝒚(ˆ)∈𝒟Š

*Š

LDA Criteria

32

} Fisher idea: maximize a function that will give

} large separation between the projected class means } while also achieving a small variance within each class, thereby

minimizing the class overlap.

𝐾 𝒙 = 𝜈)

… − 𝜈< … <

𝑡)

…< + 𝑡< …<

LDA Criteria

} The scatters of projected data are:

𝑡)

…< =

] 𝒙4𝒚 % − 𝒙4𝝂)

<

• 𝒚(ˆ)∈𝒟‰

𝑡<

…< =

] 𝒙4𝒚 % − 𝒙4𝝂)

<

• 𝒚(ˆ)∈𝒟Š

33

LDA Criteria

34

𝐾 𝒙 = 𝜈)

… − 𝜈< … <

𝑡)

…< + 𝑡< …<

𝜈)

… − 𝜈< … < = 𝒙4𝝂) − 𝒙4𝝂< <

= 𝒙4 𝝂) − 𝝂< 𝝂) − 𝝂< 4𝒙 𝑡)

…< =

] 𝒙4𝒚 % − 𝒙4𝝂)

<

• 𝒚(ˆ)∈𝒟‰

= 𝒙4 ] 𝒚 % − 𝝂) 𝒚 % − 𝝂)

4

• 𝒚(ˆ)∈𝒟‰

𝒙

LDA Criteria

35

𝐾 𝒙 =

𝒙4𝑻Œ𝒙 𝒙4𝑻•𝒙

𝑻Œ = 𝝂) − 𝝂< 𝝂) − 𝝂< 4 𝑻• = 𝑻) + 𝑻< 𝑻) = ] 𝒚 % − 𝝂) 𝒚 % − 𝝂)

4

• 𝒚(ˆ)∈𝒟‰

𝑻< = ] 𝒚 % − 𝝂< 𝒚 % − 𝝂<

4

• 𝒚(ˆ)∈𝒟Š

scatter matrix=N×covariance matrix Between-class scatter matrix Within-class scatter matrix

LDA Derivation

( )

T B T W

J = w S w w w S w

( )

( ) ( )

( )

2 2

2 2 ( )

T T T T W B T T W B B W W B T T W W

J ¶ ¶ ´

• ´
• ¶

¶ ¶ = = ¶ w S w w S w w S w w S w S w w S w S w w S w w w w w w S w w S w

( )

B W

J l ¶ = Þ = ¶ w S w S w w

36

LDA Derivation

} 𝑻𝐶𝒙 (for any vector 𝒙) points in the same direction as

𝝂) − 𝝂<:

} Thus, we can solve the eigenvalue problem immediately

rank

• is full

𝑻𝑋 If

37

𝑻𝐶𝒙 = 𝜇𝑻𝑋𝒙 𝑻•

d)𝑻𝐶𝒙 = 𝜇𝒙

𝑻𝐶𝒙 = 𝝂) − 𝝂< 𝝂) − 𝝂< 4𝒙 ∝ 𝝂) − 𝝂< 𝒙 ∝ 𝑻•

d) 𝝂) − 𝝂<

LDA Algorithm

38

} Find 𝝂) and 𝝂< as the mean of class 1 and 2 respectively } Find 𝑻) and 𝑻< as scatter matrix of class 1 and 2 respectively } 𝑻• = 𝑻) + 𝑻< } 𝑻Œ = 𝝂) − 𝝂<

𝝂) − 𝝂< 4

} Feature Extraction

} 𝒙 = 𝑻H

d) 𝝂) − 𝝂<

as the eigenvector corresponding to the largest eigenvalue of 𝑻H

d)𝑻“

} Classification

} 𝒙 = 𝑻H

d) 𝝂) − 𝝂<

} Using a threshold on 𝒙4𝒚, we can classify 𝒚

𝝂< 𝝂)

Converting multi-class problem to a set of two-class problems

39

} “one versus rest” or “one against all”

} For each class 𝐷%, a linear discriminant function that separates

samples of 𝐷% from all the other samples is found.

} Totally linearly separable

} “one versus one”

} 𝑑(𝑑 − 1)/2 linear discriminant functions are used, one to

separate samples of a pair of classes.

} Pairwise linearly separable

Multi-class classification

40

} One-vs-all (one-vs-rest)

Class 1: Class 2: Class 3:

𝑦2 𝑦2 𝑦1 𝑦2 𝑦1 𝑦1 𝑦2 𝑦1

Multi-class classification

41

} One-vs-one

Class 1: Class 2: Class 3:

𝑦2 𝑦1

𝑦2 𝑦1 𝑦2 𝑦1 𝑦2 𝑦1

Multi-class classification: ambiguity

42

• ne versus rest
• ne versus one

[Duda, Hart & Stork, 2002] } Converting the multi-class problem to a set of two-class

problems can lead to regions in which the classification is undefined

Discriminant functions

43

} Discriminant function can directly assign each vector 𝒚 to a

specific class 𝑙

} A popular way of representing a classifier

} Many classification methods are based on discriminant functions

} Assumption: the classes are taken to be disjoint

} The input space is thereby divided into decision regions

} boundaries are called decision boundaries or decision surfaces.

Discriminant Functions

44

} Discriminant functions: A discriminant function 𝑔

% 𝒚

for each class 𝒟% (𝑗 = 1, … , 𝐿):

} 𝒚 is assigned to class 𝒟% if:

𝑔𝑗(𝒚) > 𝑔𝑘(𝒚) "𝑘 ¹ 𝑗

} Thus, we can easily divide the feature space into 𝐿 decision

regions

∀𝒚, 𝑔𝑗(𝒚) > 𝑔𝑘(𝒚) "𝑘 ¹ 𝑗 ⇒ 𝒚 ∈ ℛ%

} Decision surfaces (or boundaries) can also be found using

discriminant functions

} Boundary of the ℛ% and ℛ™ separating samples of these two categories:

∀𝒚, 𝑔

% 𝒚 = 𝑔𝑘(𝒚)

ℛ%: Region of the 𝑗-th class

Discriminant Functions: Two-Category

45

} Decision surface: 𝑔 𝒚 = 0 } For two-category problem, we can only find a function 𝑔 ∶

ℝœ → ℝ

} 𝑔

) 𝒚 = 𝑔(𝒚)

} 𝑔

< 𝒚 = −𝑔(𝒚)

} First, we explain two-category classification problem and then

discuss the multi-category problems.

} Binary classification: a target variable 𝑧 ∈ 0,1 or 𝑧 ∈ −1,1

Multi-class classification

46

} Solutions to multi-category problems:

} Extend the learning algorithm to support multi-class:

} A function 𝑔

%(𝒚) for each class 𝑗 is found

¨ 𝑧

ž = argmax

%(),…,Ÿ

𝑔

%(𝒚)

} Converting the problem to a set of two-class problems: 𝑦1

𝑦2

𝑔𝑗(𝒚) > 𝑔

™(𝒚) "𝑘 ¹ 𝑗

if 𝐷% is assigned to class 𝒚

Multi-class classification: linear machine

47

} A discriminant function 𝑔

% 𝒚 = 𝒙% 4𝒚 + 𝑥%7 for each class

𝒟% (𝑗 = 1, … , 𝐿):

} 𝒚 is assigned to class 𝒟% if:

𝑔𝑗(𝒚) > 𝑔𝑘(𝒚) "𝑘 ¹ 𝑗

} Decision surfaces (boundaries) can also be found using

discriminant functions

} Boundary of the contiguous ℛ% and ℛ™: ∀𝒚, 𝑔

% 𝒚 = 𝑔𝑘(𝒚)

}

𝒙% − 𝒙™

4𝒚 + 𝑥%7 − 𝑥 ™7 = 0

Multi-class classification: linear machine

48

[Duda, Hart & Stork, 2002]

Perceptron: multi-class

49

𝑧 ž = argmax

%(),…,Ÿ

𝒙%

4𝒚

𝐾i 𝑿 = − ] 𝒙x ˆ − 𝒙x

ž ˆ 4

𝒚 %

• %∈ℳ

ℳ: subset of training data that are misclassified ℳ = 𝑗|𝑧 ž % ≠ 𝑧(%)

Initialize 𝑿 = 𝒙), … , 𝒙Ÿ , 𝑙 ← 0 repeat 𝑙 ← 𝑙 + 1 mod 𝑂 if 𝒚(%) is misclassified then 𝒙x

ž ˆ = 𝒙x ž ˆ − 𝒚(%)

𝒙x ˆ = 𝒙x ˆ + 𝒚(%) Until all patterns properly classified

Resources

50

} C. Bishop, “Pattern Recognition and Machine Learning”,

Chapter 4.1.