Lecture 17: Supervised Learning Recap Machine Learning April 6, 2010

Last Time Support Vector Machines Kernel Methods

Today Short recap of Kernel Methods Review of Supervised Learning Unsupervised Learning – (Soft) K-means clustering – Expectation Maximization – Spectral Clustering – Principle Components Analysis – Latent Semantic Analysis

Kernel Methods Feature extraction to higher dimensional spaces. Kernels describe the relationship between vectors (points) rather than the new feature space directly.

When can we use kernels? Any time training and evaluation are both based on the dot product between two points. SVMs Perceptron k-nearest neighbors k-means etc.

Kernels in SVMs Optimize α i ’s and bias w.r.t. kernel Decision function:

Kernels in Perceptrons Training Decision function

Good and Valid Kernels Good: Computing K(x i,x j ) is cheaper than ϕ (x i ) Valid: – Symmetric: K(x i,x j ) =K(x j,x i ) – Decomposable into ϕ (x i ) T ϕ (x j ) Positive Semi Definite Gram Matrix Popular Kernels – Linear, Polynomial – Radial Basis Function – String (technically infinite dimensions) – Graph

Supervised Learning Linear Regression Logistic Regression Graphical Models – Hidden Markov Models Neural Networks Support Vector Machines – Kernel Methods

Major concepts Gaussian, Multinomial, Bernoulli Distributions Joint vs. Conditional Distributions Marginalization Maximum Likelihood Risk Minimization Gradient Descent Feature Extraction, Kernel Methods

Some favorite distributions Bernoulli Multinomial Gaussian

Maximum Likelihood Identify the parameter values that yield the maximum likelihood of generating the observed data. Take the partial derivative of the likelihood function Set to zero Solve NB: maximum likelihood parameters are the same as maximum log likelihood parameters

Maximum Log Likelihood Why do we like the log function? It turns products (difficult to differentiate) and turns them into sums (easy to differentiate) log(xy) = log(x) + log(y) log(x c ) = c log(x)

Risk Minimization Pick a loss function – Squared loss – Linear loss – Perceptron (classification) loss Identify the parameters that minimize the loss function. – Take the partial derivative of the loss function – Set to zero – Solve

Frequentists v. Bayesians Point estimates vs. Posteriors Risk Minimization vs. Maximum Likelihood L2-Regularization – Frequentists: Add a constraint on the size of the weight vector – Bayesians: Introduce a zero-mean prior on the weight vector – Result is the same!

L2-Regularization Frequentists: – Introduce a cost on the size of the weights Bayesians: – Introduce a prior on the weights

Types of Classifiers Generative Models – Highest resource requirements. – Need to approximate the joint probability Discriminative Models – Moderate resource requirements. – Typically fewer parameters to approximate than generative models Discriminant Functions – Can be trained probabilistically, but the output does not include confidence information

Linear Regression Fit a line to a set of points

Linear Regression Extension to higher dimensions – Polynomial fitting – Arbitrary function fitting Wavelets Radial basis functions Classifier output

Logistic Regression Fit gaussians to data for each class The decision boundary is where the PDFs cross No “closed form” solution to the gradient. Gradient Descent

Graphical Models General way to describe the dependence relationships between variables. Junction Tree Algorithm allows us to efficiently calculate marginals over any variable.

Junction Tree Algorithm Moralization – “Marry the parents” – Make undirected Triangulation – Remove cycles >4 Junction Tree Construction – Identify separators such that the running intersection property holds Introduction of Evidence – Pass slices around the junction tree to generate marginals

Hidden Markov Models Sequential Modeling – Generative Model Relationship between observations and state (class) sequences

Perceptron Step function used for squashing. Classifier as Neuron metaphor.

Perceptron Loss Classification Error vs. Sigmoid Error – Loss is only calculated on Mistakes Perceptrons use strictly classification error

Neural Networks Interconnected Layers of Perceptrons or Logistic Regression “neurons”

Neural Networks There are many possible configurations of neural networks – Vary the number of layers – Size of layers

Support Vector Machines Maximum Margin Classification Small Margin Large Margin

Support Vector Machines Optimization Function Decision Function

Visualization of Support Vectors 30

Questions? Now would be a good time to ask questions about Supervised Techniques.

Clustering Identify discrete groups of similar data points Data points are unlabeled

Recall K-Means Algorithm – Select K – the desired number of clusters – Initialize K cluster centroids – For each point in the data set, assign it to the cluster with the closest centroid – Update the centroid based on the points assigned to each cluster – If any data point has changed clusters, repeat

k-means output

Soft K-means In k-means, we force every data point to exist in exactly one cluster. This constraint can be relaxed. Minimizes the entropy of cluster assignment

Soft k-means example

Soft k-means We still define a cluster by a centroid, but we calculate the centroid as the weighted mean of all the data points Convergence is based on a stopping threshold rather than changed assignments

Gaussian Mixture Models Rather than identifying clusters by “nearest” centroids Fit a Set of k Gaussians to the data.

GMM example

Gaussian Mixture Models Formally a Mixture Model is the weighted sum of a number of pdfs where the weights are determined by a distribution,

Graphical Models with unobserved variables What if you have variables in a Graphical model that are never observed? – Latent Variables Training latent variable models is an unsupervised learning application laughing amused sweating uncomfortable

Latent Variable HMMs We can cluster sequences using an HMM with unobserved state variables We will train the latent variable models using Expectation Maximization

Expectation Maximization Both the training of GMMs and Gaussian Models with latent variables are accomplished using Expectation Maximization – Step 1: Expectation (E-step) Evaluate the “responsibilities” of each cluster with the current parameters – Step 2: Maximization (M-step) Re-estimate parameters using the existing “responsibilities” Related to k-means

Questions One more time for questions on supervised learning…

Next Time Gaussian Mixture Models (GMMs) Expectation Maximization