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Lirong Xia MIRA, SVM, k-NN Tue, April 15, 2014

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Linear Classifiers (perceptrons) 1 Inputs are feature values Each feature has a weight Sum is the activation If the activation is: –Positive: output +1 –Negative, output -1

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Classification: Weights 2 Binary case: compare features to a weight vector Learning: figure out the weight vector from examples

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Binary Decision Rule 3 In the space of feature vectors –Examples are points –Any weight vector is a hyperplane –One side corresponds to Y = +1 –Other corresponds to Y = -1

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Learning: Binary Perceptron 4 Start with weights = 0 For each training instance: –Classify with current weights –If correct (i.e. y=y*), no change! –If wrong: adjust the weight vector by adding or subtracting the feature vector. Subtract if y* is -1.

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Multiclass Decision Rule 5 If we have multiple classes: –A weight vector for each class: –Score (activation) of a class y: –Prediction highest score wins Binary = multiclass where the negative class has weight zero

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Learning: Multiclass Perceptron 6 Start with all weights = 0 Pick up training examples one by one Predict with current weights If correct, no change! If wrong: lower score of wrong answer, raise score of right answer

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Examples: Perceptron 7 Separable Case

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Today 8 Fixing the Perceptron: MIRA Support Vector Machines k-nearest neighbor (KNN)

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Properties of Perceptrons 9 Separability: some parameters get the training set perfectly correct Convergence: if the training is separable, perceptron will eventually converge (binary case) Mistake Bound: the maximum number of mistakes (binary case) related to the margin or degree of separability

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Examples: Perceptron 10 Non-Separable Case

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Problems with the Perceptron 11 Noise: if the data isn’t separable, weights might thrash –Averaging weight vectors over time can help (averaged perceptron) Mediocre generalization: finds a “barely” separating solution Overtraining: test / held-out accuracy usually rises, then falls –Overtraining is a kind of overfitting

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Fixing the Perceptron 12 Idea: adjust the weight update to mitigate these effects MIRA*: choose an update size that fixes the current mistake …but, minimizes the change to w The +1 helps to generalize *Margin Infused Relaxed Algorithm

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Minimum Correcting Update 13 min not τ =0, or would not have made an error, so min will be where equality holds

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Maximum Step Size 14 In practice, it’s also bad to make updates that are too large –Example may be labeled incorrectly –You may not have enough features –Solution: cap the maximum possible value of τ with some constant C –Corresponds to an optimization that assumes non-separable data –Usually converges faster than perceptron –Usually better, especially on noisy data

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Outline 15 Fixing the Perceptron: MIRA Support Vector Machines k-nearest neighbor (KNN)

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Linear Separators 16 Which of these linear separators is optimal?

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Support Vector Machines 17 Maximizing the margin: good according to intuition, theory, practice Only support vectors matter; other training examples are ignorable Support vector machines (SVMs) find the separator with max margin Basically, SVMs are MIRA where you optimize over all examples at once MIRA SVM

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Classification: Comparison 18 Naive Bayes –Builds a model training data –Gives prediction probabilities –Strong assumptions about feature independence –One pass through data (counting) Perceptrons / MIRA: –Makes less assumptions about data –Mistake-driven learning –Multiple passes through data (prediction) –Often more accurate

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Extension: Web Search 19 Information retrieval: –Given information needs, produce information –Includes, e.g. web search, question answering, and classic IR Web search: not exactly classification, but rather ranking x = “Apple Computers”

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Feature-Based Ranking 20 x = “Apple Computers”

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Perceptron for Ranking 21 Input x Candidates y Many feature vectors: f(x,y) One weight vector: w –Prediction: –Update (if wrong):

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Pacman Apprenticeship! 22 Examples are states s Candidates are pairs (s,a) “correct” actions: those taken by expert Features defined over (s,a) pairs: f(s,a) Score of a q-state (s,a) given by: w×f(s,a) How is this VERY different from reinforcement learning?

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Outline 23 Fixing the Perceptron: MIRA Support Vector Machines k-nearest neighbor (KNN)

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Case-Based Reasoning 24 Similarity for classification –Case-based reasoning –Predict an instance’s label using similar instances Nearest-neighbor classification –1-NN: copy the label of the most similar data point –K-NN: let the k nearest neighbors vote (have to devise a weighting scheme) –Key issue: how to define similarity –Trade-off: Small k gives relevant neighbors Large k gives smoother functions Generated data 1-NN......

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Parametric / Non-parametric 25 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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Nearest-Neighbor Classification 26 Nearest neighbor for digits: –Take new image –Compare to all training images –Assign based on closest example Encoding: image is vector of intensities: What’s the similarity function? –Dot product of two images vectors? –Usually normalize vectors so ||x||=1 –min = 0 (when?), max = 1(when?)

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Basic Similarity 27 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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Invariant Metrics 28 Better distances use knowledge about vision Invariant metrics: –Similarities are invariant under certain transformations –Rotation, scaling, translation, stroke-thickness… –E.g.: 16*16=256 pixels; a point in 256-dim space Small similarity in R 256 (why?) –How to incorporate invariance into similarities? This and next few slides adapted from Xiao Hu, UIUC

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Invariant Metrics 29 Each example is now a curve in R 256 Rotation invariant similarity: s’=max s(r( ),r( )) E.g. highest similarity between images’ rotation lines

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Invariant Metrics 30 Problems with s’: –Hard to compute –Allows large transformations(6→9) Tangent distance: –1 st order approximation at original points. Easy to compute Models small rotations

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Template Deformation 31 Deformable templates: –An “ideal” version of each category –Best-fit to image using min variance –Cost for high distortion of template –Cost for image points being far from distorted template Used in many commercial digit recognizers Examples from [Hastie 94]

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A Tale of Two Approaches… 32 Nearest neighbor-like approaches –Can use fancy similarity functions –Don’t actually get to do explicit learning Perceptron-like approaches –Explicit training to reduce empirical error –Can’t use fancy similarity, only linear –Or can they? Let’s find out!

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Perceptron Weights 33 What is the final value of a weight w y of a perceptron? –Can it be any real vector? –No! it’s 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 34 How to classify a new example x? If someone tells us the value of K for each pair of examples, never need to build the weight vectors!

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Dual Perceptron 35 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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Kernelized Perceptron 36 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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