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Lirong Xia Kernel (continued), Clustering Friday, April 18, 2014

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Project 4 due tonight Project 5 (last project!) out later –Q1: Naïve Bayes with auto tuning (for smoothing) –Q2: freebie –Q3: Perceptron –Q4: another freebie –Q5: MIRA with auto tuning (for C) –Q6: feature design for Naïve Bayes 1 Projects

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Last time 2 Fixing the Perceptron: MIRA Support Vector Machines k-nearest neighbor (KNN)

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MIRA 3 MIRA*: choose an update size that fixes the current mistake …but, minimizes the change to w Solution *Margin Infused Relaxed Algorithm

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MIRA with Maximum Step Size 4 In practice, its also bad to make updates that are too large –Solution: cap the maximum possible value of τ with some constant C –tune C on the held-out data

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Support Vector Machines 5 Maximizing the margin: good according to intuition, theory, practice SVM

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Parametric / Non-parametric 6 Parametric models: –Fixed set of parameters –More data means better settings Non-parametric models: –Complexity of the classifier increases with data –Better in the limit, often worse in the non-limit (K)NN is non-parametric

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Basic Similarity 7 Many similarities based on feature dot products: If features are just the pixels: Note: not all similarities are of this form

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Perceptron Weights 8 What is the final value of a weight w y of a perceptron? –Can it be any real vector? –No! its build by adding up inputs Can reconstruct weight vectors (the primal representation) from update counts (the dual representation) primal dual

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Dual Perceptron 9 Start with zero counts (alpha) Pick up training instances one by one Try to classify x n, If correct, no change! If wrong: lower count of wrong class (for this instance), raise score of right class (for this instance)

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The kernel trick Neuron network Clustering (k-means) 10 Outline

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Kernelized Perceptron 11 If we had a black box (kernel) which told us the dot product of two examples x and y: –Could work entirely with the dual representation –No need to ever take dot products (kernel trick) Like nearest neighbor – work with black-box similarities Downside: slow if many examples get nonzero alpha

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Kernelized Perceptron Structure 12

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Kernels: Who Cares? 13 So far: a very strange way of doing a very simple calculation Kernel trick: we can substitute many similarity function in place of the dot product Lets us learn new kinds of hypothesis

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Non-Linear Separators 14 Data that is linearly separable (with some noise) works out great: But what are we going to do if the dataset is just too hard? How about… mapping data to a higher-dimensional space: This and next few slides adapted from Ray Mooney, UT

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Non-Linear Separators 15 General idea: the original feature space can always be mapped to some higher-dimensional feature space where the training set is separable:

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Some Kernels 16 Kernels implicitly map original vectors to higher dimensional spaces, take the dot product there, and hand the result back Linear kernel: Quadratic kernel: RBF: infinite dimensional representation Mercers theorem: if the matrix A with A i,j =K(x i,x j ) is positive definite, then K can be represented as the inner product of a higher dimensional space

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Why Kernels? 17 Cant you just add these features on your own? –Yes, in principle, just compute them –No need to modify any algorithms –But, number of features can get large (or infinite) Kernels let us compute with these features implicitly –Example: implicit dot product in quadratic kernel takes much less space and time per dot product –Of course, theres the cost for using the pure dual algorithms: you need to compute the similarity to every training datum

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The kernel trick Neuron network Clustering (k-means) 18 Outline

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Neuron = Perceptron with a non-linear activation function a i =g(Σ j W j,i a j ) 19 Neuron

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(a) step function or threshold function –perceptron (b) sigmoid function 1/(1+e -x ) –logistic regression Does the range of g(in i ) matter? 20 Activation function

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AND, OR, and NOT can be implemented XOR cannot be implemented 21 Implementation by perceptron

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Feed-forward networks –single-layer (perceptrons) –multi-layer (perceptrons) can represent any boolean function –Learning Minimize squared error of prediction Gradient descent: backpropagation algorithm Recurrent networks –Hopfield networks have symmetric weights W i,j =W j,i –Boltzmann machines use stochastic activation functions 22 Neural networks

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a 5 = g 5 ( W 3,5 × a 3 + W 4,5 × a 4 ) = g 5 ( W 3,5 × g 3 ( W 1,3 × a 1 + W 2,3 × a 2 )+ W 4,5 × g 4 ( W 1,4 × a 1 + W 2,4 × a 2 )) 23 Feed-forward example

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The kernel trick Neuron network Clustering (k-means) 24 Outline

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Recap: Classification 25 Classification systems: –Supervised learning –Make a prediction given evidence –Weve seen several methods for this –Useful when you have labeled data

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Clustering 26 Clustering systems: –Unsupervised learning –Detect patterns in unlabeled data E.g. group s or search results E.g. find categories of customers E.g. detect anomalous program executions –Useful when dont know what youre looking for –Requires data, but no labels –Often get gibberish

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Clustering 27 Basic idea: group together similar instances Example: 2D point patterns What could similar mean? –One option: small (squared) Euclidean distance

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K-Means 28 An iterative clustering algorithm –Pick K random points as cluster centers (means) –Alternate: Assign data instances to closest mean Assign each mean to the average of its assigned points –Stop when no points assignments change

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K-Means Example 29

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Example: K-Means 30 [web demo] –http://www.cs.washington.edu/research/imagedatabase/ demo/kmcluster/http://www.cs.washington.edu/research/imagedatabase/ demo/kmcluster/

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K-Means as Optimization 31 Consider the total distance to the means: Each iteration reduces phi Two states each iteration: –Update assignments: fix means c, change assignments a –Update means: fix assignments a, change means c points assignments means

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Phase I: Update Assignments 32 For each point, re-assign to closest mean: Can only decrease total distance phi!

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Phase II: Update Means 33 Move each mean to the average of its assigned points Also can only decrease total distance…(Why?) Fun fact: the point y with minimum squared Euclidean distance to a set of points {x} is their mean

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Initialization 34 K-means is non-deterministic –Requires initial means –It does matter what you pick! –What can go wrong? –Various schemes for preventing this kind of thing: variance- based split / merge, initialization heuristics

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K-Means Getting Stuck 35 A local optimum:

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K-Means Questions 36 Will K-means converge? –To a global optimum? Will it always find the true patterns in the data? –If the patterns are very very clear? Will it find something interesting? Do people ever use it? How many clusters to pick?

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Agglomerative Clustering 37 Agglomerative clustering: –First merge very similar instances –Incrementally build larger clusters out of smaller clusters Algorithm: –Maintain a set of clusters –Initially, each instance in its own cluster –Repeat: Pick the two closest clusters Merge them into a new cluster Stop when there is only one cluster left Produces not one clustering, but a family of clusterings represented by a dendrogram

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Agglomerative Clustering 38 How should we define closest for clusters with multiple elements? Many options –Closest pair (single-link clustering) –Farthest pair (complete-link clustering) –Average of all pairs –Wards method (min variance, like k-means) Different choices create different clustering behaviors

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Clustering Application 39

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