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Lirong Xia MIRA, SVM, k-NN Tue, April 15, 2014 Slide 2 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 Slide 3 Classification: Weights 2 Binary case: compare features to a weight vector Learning: figure out the weight vector from examples Slide 4 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 Slide 5 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. Slide 6 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 Slide 7 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 Slide 8 Examples: Perceptron 7 Separable Case Slide 9 Today 8 Fixing the Perceptron: MIRA Support Vector Machines k-nearest neighbor (KNN) Slide 10 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 Slide 11 Examples: Perceptron 10 Non-Separable Case Slide 12 Problems with the Perceptron 11 Noise: if the data isnt 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 Slide 13 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 Slide 14 Minimum Correcting Update 13 min not =0, or would not have made an error, so min will be where equality holds Slide 15 Maximum Step Size 14 In practice, its 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 with some constant C Corresponds to an optimization that assumes non-separable data Usually converges faster than perceptron Usually better, especially on noisy data Slide 16 Outline 15 Fixing the Perceptron: MIRA Support Vector Machines k-nearest neighbor (KNN) Slide 17 Linear Separators 16 Which of these linear separators is optimal? Slide 18 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 Slide 19 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 Slide 20 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 Slide 21 Feature-Based Ranking 20 x = Apple Computers Slide 22 Perceptron for Ranking 21 Input x Candidates y Many feature vectors: f(x,y) One weight vector: w Prediction: Update (if wrong): Slide 23 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: wf(s,a) How is this VERY different from reinforcement learning? Slide 24 Outline 23 Fixing the Perceptron: MIRA Support Vector Machines k-nearest neighbor (KNN) Slide 25 Case-Based Reasoning 24 Similarity for classification Case-based reasoning Predict an instances 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...... Slide 26 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 Slide 27 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: Whats the similarity function? Dot product of two images vectors? Usually normalize vectors so ||x||=1 min = 0 (when?), max = 1(when?) Slide 28 Basic Similarity 27 Many similarities based on feature dot products: If features are just the pixels: Note: not all similarities are of this form Slide 29 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 Slide 30 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 Slide 31 Invariant Metrics 30 Problems with s: Hard to compute Allows large transformations(69) Tangent distance: 1 st order approximation at original points. Easy to compute Models small rotations Slide 32 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] Slide 33 A Tale of Two Approaches 32 Nearest neighbor-like approaches Can use fancy similarity functions Dont actually get to do explicit learning Perceptron-like approaches Explicit training to reduce empirical error Cant use fancy similarity, only linear Or can they? Lets find out! Slide 34 Perceptron Weights 33 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 Slide 35 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! Slide 36 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) Slide 37 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