Fun with Hyperplanes: Perceptrons, SVMs, and Friends

1. Fun with Hyperplanes:Perceptrons, SVMs, and Friends Adapted from slides by Ryan Gabbard CIS 391 – Introduction to Artificial Intelligence

2. Naïve Bayes Classifiersare one example Universal Machine Learning Diagram

3. Generative v. Discriminative Models • Generative question: “How can we model the joint distribution of the classes and the features?” Why waste energy on stuff we don’t care about? Let’s optimize the job we’re trying to do directly! • Discriminative question: “What features distinguish the classes from one another?”

4. Example Modeling what sort of bizarre distribution produced these training points is hard, but distinguishing the classes is a piece of cake! chart from MIT tech report #507, Tony Jebara

5. Linear Classification

6. Representing Lines • How do we represent a line? In general a hyperplane is defined by Why bother with this weird representation?

7. Projections alternate intuition: recall the dot product of two vectors is simply the product of their lengths and the cosine of the angle between them

8. Now classification is easy! But... how do we learn this mysterious model vector?

9. Perceptron Learning Algorithm

10. Perceptron Update Example I

11. Perceptron Update Example II

12. Properties of the Simple Perceptron • You can prove that • If it’s possible to separate the data with a hyperplane(i.e. if it’s linearly separable), • Then the algorithm will converge to that hyperplane. • But what if it isn’t? Then perceptron is very unstable and bounces all over the place

13. Voted Perceptron • Works just like a regular perceptron, except you keep track of all the intermediate models you created • When you want to classify something, you let each of the (many, many) models vote on the answer and take the majority

14. Properties of Voted Perceptron • Simple! • Much better generalization performance than regular perceptron • For later: (almost as good as SVMs) • For later: Can use the ‘kernel trick’ • Training as fast as regular perceptron • But run-time is slower

15. Averaged Perceptron • Extremely simple! • Return as your final model the average of all your intermediate models • Approximation to voted perceptron • Nearly as fast to train and exactly as fast to run as regular perceptron

16. What’s wrong with these hyperplanes?

17. They’re unjustifiably biased!

18. A less biased choice

19. Margin • The margin is the distance to closest point in the training data • We tend to get better generalization to unseen data if we choose the separating hyperplane which maximizes the margin

20. Support Vector Machines • Another learning method which explicitly calculates the maximum margin hyperplane by solving a gigantic quadratic programming problem. • Generally considered the highest-performing current machine learning technique. • But it’s relatively slow and very complicated.

21. Margin-Infused Relaxed Algorithm (MIRA) • Multiclass; each class has a prototype vector • Classify an instance by choosing the class whose prototype vector has the greatest dot product with the instance • During training, when updating make the ‘smallest’ (in a sense) change to the prototype vectors which guarantees correct classification by a minimum margin • Pays attention to the margin directly

22. What if it isn’t separable?

23. Project it to someplace where it is!

24. Kernel Trick • If our data isn’t linearly separable, we can define a projection to map it into a much higher dimensional feature space where it is. • For some algorithms where everything can be expressed as the dot products of instances (SVM, voted perceptron, MIRA) this can be done efficiently using something called the `kernel trick’