Machine LearningQueens College

Lecture 13: SVM Again

Today

• Completion of Support Vector Machines

• Project Description and Topics

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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

• The decision hyperplane:

• Scale invariance

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Support Vectors

• The decision hyperplane:

• Scale invariance

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

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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Max Margin Loss Function

• If constraint optimization then Lagrange Multipliers

• Optimize the “Primal”

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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

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Still Quadratic Programming!

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

Soft margin example

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Hinge Loss

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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Research Projects• Run a machine learning experiment

– Identify a problem/task.– Find appropriate data– Implement one or more ML algorithm– Evaluate the performance.

• Write a report of the experiment– 4 pages including references– Abstract

• One paragraph describing the experiment– Introduction

• Describe the problem/task– Data

• Describe the data set, features extracted, cleaning processes– Method

• Describe the algorithm/approach– Results

• Present and Discuss results– Conclusion

• Summarize the experiment and results• Teams of two people are acceptable.

– Requires a report from each participant (written independently) describing who was responsible for the components of the work. 21

Sample Problems/Tasks• Vision/Graphics

– Object Classification– Facial Recognition– Fingerprint Identification– Fingerprint ID– Handwriting recognition

• Non English languages?• Language

– Topic classification– Sentiment analysis– Speech recognition– Speaker identification– Punctuation restoration– Semantic Segmentation– Recognition of Emotion, Sarcasm, etc.– SMS Text normalization– Chat participant Id– Twitter classification– Twitter threading

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Sample Problems/Tasks• Games

– Chess– Checkers– Poker– Blackjack– Go

• Recommenders (Collaborative Filtering)– Netflix– Courses– Jokes– Books– Facebook

• Video Classification– Motion classification– Segmentation

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ML Topics to explore in the project

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Data

• UCI Machine Learning Repository– http://archive.ics.uci.edu/ml/

• Ask Me

• Collect some of your own

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Next Time

• Kernel Methods

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