classification / regression support vector machines
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Classification / Regression Support Vector Machines. Support vector machines. 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. - PowerPoint PPT PresentationTRANSCRIPT
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
Jeff Howbert Introduction to Machine Learning Winter 2014 7
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
Jeff Howbert Introduction to Machine Learning Winter 2014 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 2014 11
0 bxw
Support vector machines
B1
b11
b12
W
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1 if1
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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
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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
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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
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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.
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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
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Jeff Howbert Introduction to Machine Learning Winter 2014 23
Support vector machinesSolution: nonlinear transform of attributes
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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
Jeff Howbert Introduction to Machine Learning Winter 2014 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 )
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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 )
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Common kernel functions for SVM
– linear
– polynomial
– Gaussian or radial basis
– sigmoid
Support vector machines
) tanh(),(
exp),(
) (),(
),(
2121
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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