Jeff Howbert Introduction to Machine Learning Winter 2014 1

Classification / Regression

Support Vector Machines

Jeff Howbert Introduction to Machine Learning Winter 2014 2

Topics– SVM classifiers for linearly separable classes– SVM classifiers for non-linearly separable

classes– SVM classifiers for nonlinear decision

boundaries kernel functions

– Other applications of SVMs– Software

Support vector machines

Jeff Howbert Introduction to Machine Learning Winter 2014 3

Support vector machines

Goal: find a linear decision boundary (hyperplane)that separates the classes

Linearlyseparableclasses

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Support vector machines

One possible solution

B1

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Support vector machines

B2

Another possible solution

Jeff Howbert Introduction to Machine Learning Winter 2014 6

Support vector machines

B2

Other possible solutions

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Support vector machines

Which one is better? B1 or B2? How do you define better?

B1

B2

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Support vector machines

Hyperplane that maximizes the margin will have better generalization=> B1 is better than B2

B1

B2

b11

b12

b21b22

margin

Jeff Howbert Introduction to Machine Learning Winter 2014 9

Support vector machines

Hyperplane that maximizes the margin will have better generalization=> B1 is better than B2

B1

B2

b11

b12

b21

b22

margin

test sample

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Support vector machines

Hyperplane that maximizes the margin will have better generalization=> B1 is better than B2

B1

B2

b11

b12

b21

b22

margin

test sample

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0 bxw

Support vector machines

B1

b11

b12

W

1 bxw

1 if1

1 if1)(

bb

fyi xwxw

x ||||2margin w

1 bxw

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We want to maximize:

Which is equivalent to minimizing:

But subject to the following constraints:

– This is a constrained convex optimization problem– Solve with numerical approaches, e.g. quadratic

programming

Support vector machines

||||2margin w

2||||)(

2ww L

1 if1

1 if1)(

bb

fyi xwxw

x

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Solving for w that gives maximum margin:1. Combine objective function and constraints into new

objective function, using Lagrange multipliers i

2. To minimize this Lagrangian, we take derivatives of w and b and set them to 0:

Support vector machines

N

iiiiprimal byL

1

2 1)(21 xww

N

iii

p

N

iiii

p

ybL

yL

1

1

00

0

xww

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Solving for w that gives maximum margin:3. Substituting and rearranging gives the dual of the

Lagrangian:

which we try to maximize (not minimize).4. Once we have the i, we can substitute into previous

equations to get w and b.5. This defines w and b as linear combinations of the

training data.

Support vector machines

ji

jijiji

N

iidual yyL

,1 21 xx

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Optimizing the dual is easier.– Function of i only, not i and w.

Convex optimization guaranteed to find global optimum.

Most of the i go to zero.– The xi for which i 0 are called the support vectors.

These “support” (lie on) the margin boundaries.– The xi for which i = 0 lie away from the margin

boundaries are not required for defining the maximum margin hyperplane.

Support vector machines

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Support vector machines

Example of solving for maximum margin hyperplane

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Support vector machines

What if the classes are not linearly separable?

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Support vector machines

Now which one is better? B1 or B2? How do you define better?

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i

ii b

bfy

1 if11 if1

)(xwxw

x

What if the problem is not linearly separable? Solution: introduce slack variables

– Need to minimize:

– Subject to:

– C is an important hyperparameter, whose value is usually optimized by cross-validation.

N

i

kiCL

1

2

2||||)( ww

Support vector machines

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Slack variables for nonseparable data

Support vector machines

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Support vector machines

What if decision boundary is not linear?

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Support vector machinesSolution: nonlinear transform of attributes

])(,[],[: 421121 xxxxx

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Support vector machinesSolution: nonlinear transform of attributes

)](),[(],[: 22

212

121 xxxxxx

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Issues with finding useful nonlinear transforms– Not feasible to do manually as number of attributes

grows (i.e. any real world problem)– Usually involves transformation to higher dimensional

space increases computational burden of SVM optimization curse of dimensionality

With SVMs, can circumvent all the above via the kernel trick

Support vector machines

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Support vector machines

Kernel trick– Don’t need to specify the attribute transform ( x )– Only need to know how to calculate the dot product of

any two transformed samples:k( x1, x2 ) = ( x1 ) ( x2 )

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Support vector machines

Kernel trick (cont’d.)– The kernel function k( , ) is substituted into the dual

of the Lagrangian, allowing determination of a maximum margin hyperplane in the (implicitly) transformed space ( x ):

– All subsequent calculations, including predictions on test samples, are done using the kernel in place of( x1 ) ( x2 )

jijijiji

N

ii

jijijiji

N

iidual

yy

yyL

,1

,1

),k(21

)()(21

xx

xx

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Common kernel functions for SVM

– linear

– polynomial

– Gaussian or radial basis

– sigmoid

Support vector machines

) tanh(),(

exp),(

) (),(

),(

2121

22121

2121

2121

ck

k

ck

k

d

xxxx

xxxx

xxxx

xxxx

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For some kernels (e.g. Gaussian) the implicit transform ( x ) is infinite-dimensional!– But calculations with kernel are done in original space,

so computational burden and curse of dimensionality aren’t a problem.

Support vector machines

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Support vector machines

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Support vector machines

Applications of SVMs to machine learning– Classification

binary multiclass one-class

– Regression– Transduction (semi-supervised learning)– Ranking– Clustering– Structured labels

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Support vector machines

Software– SVMlight

http://svmlight.joachims.org/

– libSVM http://www.csie.ntu.edu.tw/~cjlin/libsvm/ includes MATLAB / Octave interface

– MATLAB svmtrain / svmclassify only supports binary classification

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Online demos

– http://cs.stanford.edu/people/karpathy/svmjs/demo/

Support vector machines