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

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Last Time Support Vector Machines Kernel Methods

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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

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

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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.

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

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Kernels in Perceptrons Training Decision function

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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

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

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

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Some favorite distributions Bernoulli Multinomial Gaussian

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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

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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)

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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

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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!

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

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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

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Linear Regression Fit a line to a set of points

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

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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

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

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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

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

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

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

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

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

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Support Vector Machines Maximum Margin Classification Small Margin Large Margin

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Support Vector Machines Optimization Function Decision Function

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Visualization of Support Vectors 30

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

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

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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

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k-means output

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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

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Soft k-means example

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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

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

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GMM example

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

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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

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

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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

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Questions One more time for questions on supervised learning…

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Next Time Gaussian Mixture Models (GMMs) Expectation Maximization

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