fun with hyperplanes 2007

Fun with Hyperplanes:Perceptrons, SVMs, and Friends: 

Fun with Hyperplanes: Perceptrons, SVMs, and Friends Ryan Gabbard CIS 391 – Introduction to Artificial Intelligence

Universal Machine Learning Diagram: 

Universal Machine Learning Diagram

Generative v. Discriminative Models: 

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?”

Example: 

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

Linear Classification: 

Linear Classification

Representing Lines: 

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

Projections: 

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

Now classification is easy!: 

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

Perceptron Learning Algorithm : 

Perceptron Learning Algorithm

Perceptron Update Example I: 

Perceptron Update Example I

Perceptron Update Example II: 

Perceptron Update Example II

Properties of the Simple Perceptron: 

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), the algorithm will converge to that hyperplane. But what if it isn’t? Then perceptron is very unstable and bounces all over the place.

Voted Perceptron: 

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.

Properties of Voted Perceptron: 

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

Averaged Perceptron: 

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.

What’s wrong with these hyperplanes?: 

What’s wrong with these hyperplanes?

They’re unjustifiably biased!: 

They’re unjustifiably biased!

A less biased choice: 

A less biased choice

Margin: 

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.

Support Vector Machines: 

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.

Margin-Infused Relaxed Algorithm (MIRA): 

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

What if it isn’t separable?: 

What if it isn’t separable?

Project it to someplace where it is!: 

Project it to someplace where it is!

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’ Kernel Trick