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Supervised Learning Recap Machine Learning Last Time • Support Vector Machines • Kernel Methods Today • Review of Supervised Learning • Unsupervised Learning – (Soft) K-means clustering – Expectation Maximization – Spectral Clustering – Principle Components Analysis – Latent Semantic Analysis 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(xc) = 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 25 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 uncomfortable sweating amused laughing 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