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Support Vector Machines (and Kernel Methods in general)

Machine Learning

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Last Time

• Multilayer Perceptron/Logistic Regression Networks– Neural Networks– Error Backpropagation

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Today

• Support Vector Machines

• Note: we’ll rely on some math from Optimality Theory that we won’t derive.

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Maximum Margin

• Perceptron (and other linear classifiers) can lead to many equally valid choices for the decision boundary

Are these really “equally valid”?

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Max Margin

• How can we pick which is best?

• Maximize the size of the margin.

Are these really “equally valid”?

Small Margin

Large Margin

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Support Vectors

• Support Vectors are those input points (vectors) closest to the decision boundary

• 1. They are vectors• 2. They “support”

the decision hyperplane

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Support Vectors

• Define this as a decision problem

• The decision hyperplane:

• No fancy math, just the equation of a hyperplane.

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Support Vectors

• Aside: Why do some cassifiers use or – Simplicity of the math and

interpretation.– For probability density

function estimation 0,1 has a clear correlate.

– For classification, a decision boundary of 0 is more easily interpretable than .5.

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Support Vectors

• Define this as a decision problem

• The decision hyperplane:

• Decision Function:

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Support Vectors

• Define this as a decision problem

• The decision hyperplane:

• Margin hyperplanes:

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Support Vectors

• The decision hyperplane:

• Scale invariance

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Support Vectors

• The decision hyperplane:

• Scale invariance

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Support Vectors

• The decision hyperplane:

• Scale invariance

This scaling does not change the decision hyperplane, or the supportvector hyperplanes. But we willeliminate a variable from the optimization

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What are we optimizing?

• We will represent the size of the margin in terms of w.

• This will allow us to simultaneously– Identify a decision

boundary– Maximize the margin

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How do we represent the size of the margin in terms of w?

1. There must at least one point that lies on each support hyperplanes

Proof outline: If not, we could define a larger margin support hyperplane that does touch the nearest point(s).

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How do we represent the size of the margin in terms of w?

1. There must at least one point that lies on each support hyperplanes

Proof outline: If not, we could define a larger margin support hyperplane that does touch the nearest point(s).

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How do we represent the size of the margin in terms of w?

1. There must at least one point that lies on each support hyperplanes

2. Thus:

3. And:

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How do we represent the size of the margin in terms of w?

1. There must at least one point that lies on each support hyperplanes

2. Thus:

3. And:

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• The vector w is perpendicular to the decision hyperplane– If the dot product of two

vectors equals zero, the two vectors are perpendicular.

How do we represent the size of the margin in terms of w?

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• The margin is the projection of x1 – x2 onto w, the normal of the hyperplane.

How do we represent the size of the margin in terms of w?

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Aside: Vector Projection

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• The margin is the projection of x1 – x2 onto w, the normal of the hyperplane.

How do we represent the size of the margin in terms of w?

Size of the Margin:

Projection:

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Maximizing the margin

• Goal: maximize the margin

Linear Separability of the data by the decision boundary

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Max Margin Loss Function

• If constraint optimization then Lagrange Multipliers

• Optimize the “Primal”

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Max Margin Loss Function

• Optimize the “Primal”

Partial wrt b

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Max Margin Loss Function

• Optimize the “Primal”

Partial wrt w

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Max Margin Loss Function

• Optimize the “Primal”

Partial wrt w

Now have to find αi.Substitute back to the Loss function

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Max Margin Loss Function

• Construct the “dual”

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Dual formulation of the error

• Optimize this quadratic program to identify the lagrange multipliers and thus the weights

There exist (rather) fast approaches to quadratic optimization in both C, C++, Python, Java and R

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Quadratic Programming

•If Q is positive semi definite, then f(x) is convex.

•If f(x) is convex, then there is a single maximum.

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Support Vector Expansion

• When αi is non-zero then xi is a support vector

• When αi is zero xi is not a support vector

New decision FunctionIndependent of the

Dimension of x!

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Kuhn-Tucker Conditions

• In constraint optimization: At the optimal solution– Constraint * Lagrange Multiplier = 0

Only points on the decision boundary contribute to the solution!

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Visualization of Support Vectors

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Interpretability of SVM parameters

• What else can we tell from alphas?– If alpha is large, then the associated data point is

quite important.– It’s either an outlier, or incredibly important.

• But this only gives us the best solution for linearly separable data sets…

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Basis of Kernel Methods

• The decision process doesn’t depend on the dimensionality of the data.• We can map to a higher dimensionality of the data space.

• Note: data points only appear within a dot product.• The error is based on the dot product of data points – not the data

points themselves.

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Basis of Kernel Methods

• Since data points only appear within a dot product.• Thus we can map to another space through a replacement

• The error is based on the dot product of data points – not the data points themselves.

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Learning Theory bases of SVMs

• Theoretical bounds on testing error.– The upper bound doesn’t depend on the

dimensionality of the space– The lower bound is maximized by maximizing the

margin, γ, associated with the decision boundary.

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Why we like SVMs

• They work– Good generalization

• Easily interpreted.– Decision boundary is based on the data in the

form of the support vectors.• Not so in multilayer perceptron networks

• Principled bounds on testing error from Learning Theory (VC dimension)

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SVM vs. MLP

• SVMs have many fewer parameters– SVM: Maybe just a kernel parameter– MLP: Number and arrangement of nodes and eta

learning rate • SVM: Convex optimization task– MLP: likelihood is non-convex -- local minima

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Soft margin classification• There can be outliers on the other side of the decision

boundary, or leading to a small margin.• Solution: Introduce a penalty term to the constraint function

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Soft Max Dual

Still Quadratic Programming!

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• Points are allowed within the margin, but cost is introduced.

Soft margin example

Hinge Loss

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Probabilities from SVMs

• Support Vector Machines are discriminant functions

– Discriminant functions: f(x)=c– Discriminative models: f(x) = argmaxc p(c|x)

– Generative Models: f(x) = argmaxc p(x|c)p(c)/p(x)

• No (principled) probabilities from SVMs• SVMs are not based on probability distribution

functions of class instances.

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Efficiency of SVMs

• Not especially fast.• Training – n^3– Quadratic Programming efficiency

• Evaluation – n– Need to evaluate against each support vector

(potentially n)

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Good Bye

• Next time: – The Kernel “Trick” -> Kernel Methods– or– How can we use SVMs that are not linearly

separable?