Jeff Howbert Introduction to Machine Learning Winter 2012 1
Classification / Regression
Support Vector Machines
Jeff Howbert Introduction to Machine Learning Winter 2012 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 2012 3
Support vector machines
Goal: find a linear decision boundary (hyperplane)that separates the classes
Linearlyseparableclasses
Jeff Howbert Introduction to Machine Learning Winter 2012 4
Support vector machines
One possible solution
B1
Jeff Howbert Introduction to Machine Learning Winter 2012 5
Support vector machines
B2
Another possible solution
Jeff Howbert Introduction to Machine Learning Winter 2012 6
Support vector machines
B2
Other possible solutions
Jeff Howbert Introduction to Machine Learning Winter 2012 7
Support vector machines
Which one is better? B1 or B2? How do you define better?
B1
B2
Jeff Howbert Introduction to Machine Learning Winter 2012 8
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 2012 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
Jeff Howbert Introduction to Machine Learning Winter 2012 10
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
Jeff Howbert Introduction to Machine Learning Winter 2012 11
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
Jeff Howbert Introduction to Machine Learning Winter 2012 13
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
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iiiiprimal byL
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2 1)(21 xww
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p
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Jeff Howbert Introduction to Machine Learning Winter 2012 14
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
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iidual yyL
,1 21 xx
Jeff Howbert Introduction to Machine Learning Winter 2012 15
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. They 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?
Jeff Howbert Introduction to Machine Learning Winter 2012 19
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
Jeff Howbert Introduction to Machine Learning Winter 2012 20
Slack variables for nonseparable data
Support vector machines
Jeff Howbert Introduction to Machine Learning Winter 2012 21
Support vector machines
What if decision boundary is not linear?
Jeff Howbert Introduction to Machine Learning Winter 2012 22
Support vector machinesSolution: nonlinear transform of attributes
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Jeff Howbert Introduction to Machine Learning Winter 2012 23
Support vector machinesSolution: nonlinear transform of attributes
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212
121 xxxxxx
Jeff Howbert Introduction to Machine Learning Winter 2012 24
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
Jeff Howbert Introduction to Machine Learning Winter 2012 25
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 )
– 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 )
Jeff Howbert Introduction to Machine Learning Winter 2012 26
Common kernel functions for SVM
– linear
– polynomial
– Gaussian or radial basis
– sigmoid
Support vector machines
) tanh(),(
exp),(
) (),(
),(
2121
22121
2121
2121
ck
k
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Jeff Howbert Introduction to Machine Learning Winter 2012 27
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
Jeff Howbert Introduction to Machine Learning Winter 2012 28
Support vector machines
Jeff Howbert Introduction to Machine Learning Winter 2012 29
Support vector machines
Applications of SVMs to machine learning– Classification
binary multiclass one-class
– Regression– Transduction (semi-supervised learning)– Ranking– Clustering– Structured labels
Jeff Howbert Introduction to Machine Learning Winter 2012 30
Support vector machines
Software
– SVMlight
http://svmlight.joachims.org/
– libSVM http://www.csie.ntu.edu.tw/~cjlin/libsvm/ includes MATLAB / Octave interface