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 SVM will be covered in the later lectures } Fisher } Multi-class classification 2

Classification problem } Given:Training set * 𝒚 % , 𝑧 % } labeled set of 𝑂 input-output pairs 𝐸 = %() } 𝑧 ∈ {1, … , 𝐿} } Goal: Given an input 𝒚 , assign it to one of 𝐿 classes } Examples: } Spam filter } Handwritten digit recognition } … 3

Linear classifiers } 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. 4

Two Category } 𝑔 𝒚; 𝒙 = 𝒙 4 𝒚 + 𝑥 7 = 𝑥 7 + 𝑥 ) 𝑦 ) + . . . 𝑥 ; 𝑦 ; } 𝒚 = 𝑦 ) 𝑦 < … 𝑦 ; } 𝒙 = [𝑥 ) 𝑥 < … 𝑥 ; ] } 𝑥 7 : bias } if 𝒙 4 𝒚 + 𝑥 7 ≥ 0 then 𝒟 ) } else 𝒟 < Decision surface (boundary): 𝒙 4 𝒚 + 𝑥 7 = 0 𝒙 is orthogonal to every vector lying within the decision surface 5

Example 3 − 3 4 𝑦 ) − 𝑦 < = 0 𝑦 2 3 if 𝒙 4 𝒚 + 𝑥 7 ≥ 0 then 𝒟 ) 2 else 𝒟 < 1 𝑦 1 4 1 2 3 6

Linear classifier: Two Category } 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 ; } 𝑥 7 determine the location of the surface. } The normal distance from the origin to the decision surface is H I 𝒙 𝒚 = 𝒚 J + 𝑠 𝒙 𝒙 ⇒ 𝑠 = 𝒙 4 𝒚 + 𝑥 7 𝒚 J 𝒙 4 𝒚 + 𝑥 7 = 𝑠 𝒙 𝒙 𝑔 𝒚 = 0 gives a signed measure of the perpendicular distance 𝑠 of the point 𝒚 from the decision surface 7

Linear boundary: geometry 𝒙 4 𝒚 + 𝑥 7 > 0 𝒙 4 𝒚 + 𝑥 7 = 0 𝒙 4 𝒚 + 𝑥 7 < 0 𝒙 4 𝒚 + 𝑥 7 𝒙 8

Non-linear decision boundary } Choose non-linear features } Classifier still linear in parameters 𝒙 < + 𝑦 < < = 0 𝑦 2 −1 + 𝑦 ) < , 𝒚 < < , 𝒚 ) 𝒚 < ] 𝝔 𝒚 = [1, 𝒚 ) , 𝒚 < , 𝒚 ) 1 𝒙 = 𝑥 7 , 𝑥 ) , … , 𝑥 R = [−1, 0, 0,1,1,0] 𝑦 1 1 1 - 1 if 𝒙 4 𝝔(𝒚) ≥ 0 then 𝑧 = 1 else 𝑧 = −1 𝒚 = [𝒚 ) , 𝒚 < ] 9

Cost Function for linear classification } Finding linear classifiers can be formulated as an optimization problem : } Select how to measure the prediction loss S 𝒚 % , 𝑧 % } Based on the training set 𝐸 = , a cost function 𝐾 𝒙 is defined %() } Solve the resulting optimization problem to find parameters: U 𝒚 = 𝑔 𝒚; 𝒙 } Find optimal 𝑔 V where 𝒙 V = argmin 𝐾 𝒙 𝒙 } Criterion or cost functions for classification: } We will investigate several cost functions for the classification problem 10

SSE cost function for classification 𝐿 = 2 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 = ] 𝒙 4 𝒚 % − 𝑧 % %() 11

SSE cost function for classification 𝐿 = 2 𝒙 4 𝒚 − 𝑧 < 𝑧 = 1 1 𝒙 4 𝒚 Correct predictions that 𝒙 4 𝒚 − 𝑧 < are penalized by SSE 𝑧 = −1 [Bishop] −1 𝒙 4 𝒚 12

SSE cost function for classification 𝐿 = 2 } Is it more suitable if we set 𝑔 𝒚; 𝒙 = 𝑕 𝒙 4 𝒚 ? * sign 𝒙 4 𝒚 − 𝑧 < < 𝐾 𝒙 = ] sign 𝒙 4 𝒚 % − 𝑧 % 𝑧 = 1 %() sign 𝑨 = a− 1, 𝑨 < 0 1, 𝑨 ≥ 0 𝒙 4 𝒚 } 𝐾 𝒙 is a piecewise constant function shows the number of misclassifications 𝐾(𝒙) Training error incurred in classifying training samples 13

SSE cost function 𝐿 = 2 } Is it more suitable if we set 𝑔 𝒚; 𝒙 = 𝜏 𝒙 4 𝒚 ? * < 𝐾 𝒙 = ] 𝜏 𝒙 4 𝒚 % − 𝑧 % %() 𝜏 𝑨 = 1 − 𝑓 de 1 + 𝑓 de } 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 14

Perceptron algorithm } Linear classifier } Two-class: 𝑧 ∈ {−1,1} } 𝑧 = −1 for 𝐷 < , 𝑧 = 1 for 𝐷 ) } Goal: ∀𝑗, 𝒚 (%) ∈ 𝐷 ) ⇒ 𝒙 4 𝒚 (%) > 0 ∀𝑗, 𝒚 % ∈ 𝐷 < ⇒ 𝒙 4 𝒚 % < 0 } } 𝑔 𝒚; 𝒙 = sign(𝒙 4 𝒚) 15

� Perceptron criterion 𝐾 i 𝒙 = − ] 𝒙 4 𝒚 % 𝑧 % %∈ℳ ℳ : subset of training data that are misclassified Many solutions? Which solution among them? 16

Cost function 𝐾(𝒙) 𝐾 i (𝒙) 𝑥 7 𝑥 7 𝑥 ) 𝑥 ) # of misclassifications Perceptron’s as a cost function cost function There may be many solutions in these cost functions 17 [Duda, Hart, and Stork, 2002]

� � � Batch Perceptron “Gradient Descent” to solve the optimization problem: 𝒙 lm) = 𝒙 l − 𝜃𝛼 𝒙 𝐾 i (𝒙 l ) 𝒙 𝐾 i 𝒙 = − ] 𝒚 % 𝑧 % 𝛼 %∈ℳ Batch Perceptron converges in finite number of steps for linearly separable data: Initialize 𝒙 Repeat 𝒚 % 𝑧 % 𝒙 = 𝒙 + 𝜃 ∑ %∈ℳ 𝒚 % 𝑧 % Until 𝜃 ∑ < 𝜄 %∈ℳ 18

Stochastic gradient descent for Perceptron } Single-sample perceptron: } If 𝒚 (%) is misclassified: 𝒙 lm) = 𝒙 l + 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 Fixed-Increment single sample Perceptron Initialize 𝒙, 𝑢 ← 0 repeat 𝜃 can be set to 1 and 𝑢 ← 𝑢 + 1 proof still works 𝑗 ← 𝑢 mod 𝑂 if 𝒚 (%) is misclassified then 𝒙 = 𝒙 + 𝒚 (%) 𝑧 (%) Until all patterns properly classified 19

Perceptron Convergence } It can be shown: (𝒚,x)∈y 𝒚 < 𝑁 = max V 0 4 𝒚 𝜈 = 2 min (𝒚,x)∈y 𝑧𝒙 V ∗4 𝒚 𝑏 = min (𝒚,x)∈y 𝑧𝒙 20

Example 21

Perceptron: Example Change 𝒙 in a direction that corrects the error 22 [Bishop]

Convergence of Perceptron [Duda, Hart & Stork, 2002] } For data sets that are not linearly separable, the single-sample perceptron learning algorithm will never converge 23

Pocket algorithm } 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 𝒙 S~H = 𝒙 + 𝒚 (%) 𝑧 (%) if 𝐹 l€•%S 𝒙 S~H < 𝐹 l€•%S 𝒙 then 𝒙 = 𝒙 S~H end * 𝐹 l€•%S 𝒙 = 1 𝑂 ] 𝑡𝑗𝑕𝑜(𝒙 4 𝒚 (S) ) ≠ 𝑧 (S) S() 24

Linear Discriminant Analysis (LDA) } 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 25

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 𝒙 28

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 𝒙 4 𝒚 } What is the measure of the separation between the projected points of different classes? 29

Measure of Separation in the Projected Direction } Is the direction of the line jointing the class means a good candidate for 𝒙 ? [Bishop] 30

� � Measure of Separation in the Projected Direction } The direction of the line jointing the class means is the solution of the following problem: } Maximizes the separation of the projected class means … − 𝜈 < … < max 𝒙 𝐾 𝒙 = 𝜈 ) s. t. 𝒙 = 1 𝒚 (ˆ) ∑ … = 𝒙 4 𝝂 ) 𝒚(ˆ)∈𝒟‰ 𝜈 ) 𝝂 ) = * ‰ 𝒚 (ˆ) ∑ … = 𝒙 4 𝝂 < 𝒚(ˆ)∈𝒟Š 𝜈 < 𝝂 < = * Š } What is the problem with the criteria considering only … − 𝜈 < … ? 𝜈 ) } It does not consider the variances of the classes in the projected direction 31

LDA Criteria } 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. … − 𝜈 < … < 𝐾 𝒙 = 𝜈 ) …< + 𝑡 < …< 𝑡 ) 32