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CS 541: Artificial Intelligence

CS 541: Artificial Intelligence

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CS 541: Artificial Intelligence

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  1. CS 541: Artificial Intelligence Lecture IX: Supervised Machine Learning • Slides Credit:Peter Norvig and Sebastian Thrun

  2. Schedule

  3. Re-cap of Lecture VIII • Temporal models use state and sensor variables replicated over time • Markov assumptions and stationarity assumption, so we need • Transition model P(Xt|Xt-1) • Sensor model P(Et|Xt) • Tasks are filtering, prediction, smoothing, most likely sequence; • All done recursively with constant cost per time step • Hidden Markov models have a single discrete state variable; • Widely used for speech recognition • Kalman filters allow n state variables, linear Gaussian, O(n3) update • Dynamic Bayesian nets subsume HMMs, Kalman filters; exact update intractable • Particle filtering is a good approximate filtering algorithm for DBNs

  4. Outline • Machine Learning • Classification (Naïve Bayes) • Classification (Decision Tree) • Regression (Linear, Smoothing) • Linear Separation (Perceptron, SVMs) • Non-parametric classification (KNN)

  5. Machine Learning

  6. Machine Learning • Up until now: how to reason in a give model • Machine learning: how to acquire a model on the basis of data / experience • Learning parameters (e.g. probabilities) • Learning structure (e.g. BN graphs) • Learning hidden concepts (e.g. clustering)

  7. Machine Learning Lingo

  8. Supervised Machine Learning Given a training set: (x1, y1), (x2, y2), (x3, y3), …(xn, yn) Where each yiwas generated by an unknown y = f (x), Discover a function h that approximates the true function f.

  9. Classification (Naïve Bayes)

  10. Classification Example: Spam Filter • Input: x = email • Output: y = “spam” or “ham” • Setup: • Get a large collection of example emails, each labeled “spam” or “ham” • Note: someone has to hand label all this data! • Want to learn to predict labels of new, future emails • Features: The attributes used to make the ham / spam decision • Words: FREE! • Text Patterns: $dd, CAPS • Non-text: SenderInContacts • … Dear Sir. First, I must solicit your confidence in this transaction, this is by virture of its nature as being utterly confidencial and top secret. … TO BE REMOVED FROM FUTURE MAILINGS, SIMPLY REPLY TO THIS MESSAGE AND PUT "REMOVE" IN THE SUBJECT. 99 MILLION EMAIL ADDRESSES FOR ONLY $99 Ok, Iknow this is blatantly OT but I'm beginning to go insane. Had an old Dell Dimension XPS sitting in the corner and decided to put it to use, I know it was working pre being stuck in the corner, but when I plugged it in, hit the power nothing happened.

  11. A Spam Filter • Naïve Bayes spam filter • Data: • Collection of emails, labeled spam or ham • Note: someone has to hand label all this data! • Split into training, held-out, test sets • Classifiers • Learn on the training set • (Tune it on a held-out set) • Test it on new emails Dear Sir. First, I must solicit your confidence in this transaction, this is by virture of its nature as being utterly confidencial and top secret. … TO BE REMOVED FROM FUTURE MAILINGS, SIMPLY REPLY TO THIS MESSAGE AND PUT "REMOVE" IN THE SUBJECT. 99 MILLION EMAIL ADDRESSES FOR ONLY $99 Ok, Iknow this is blatantly OT but I'm beginning to go insane. Had an old Dell Dimension XPS sitting in the corner and decided to put it to use, I know it was working pre being stuck in the corner, but when I plugged it in, hit the power nothing happened.

  12. Naïve Bayes for Text • Bag-of-Words Naïve Bayes: • Predict unknown class label (spam vs. ham) • Assume evidence features (e.g. the words) are independent • Generative model • Tied distributions and bag-of-words • Usually, each variable gets its own conditional probability distribution P(F|Y) • In a bag-of-words model • Each position is identically distributed • All positions share the same conditional probs P(W|C) • Why make this assumption? Word at position i, not ith word in the dictionary!

  13. General Naïve Bayes • General probabilistic model: • General naive Bayes model: • We only specify how each feature depends on the class • Total number of parameters is linear in n |Y| x |F|nparameterss Y F1 F2 Fn n x |F| x |Y| parameters |Y| parameters

  14. Example: Spam Filtering • Model: • What are the parameters? • Where do these tables come from? ham : 0.66 spam: 0.33 the : 0.0156 to : 0.0153 and : 0.0115 of : 0.0095 you : 0.0093 a : 0.0086 with: 0.0080 from: 0.0075 ... the : 0.0210 to : 0.0133 of : 0.0119 2002: 0.0110 with: 0.0108 from: 0.0107 and : 0.0105 a : 0.0100 ... Counts from examples!

  15. Spam Example P(spam | words) = e-76 / (e-76 + e-80.5) = 98.9%

  16. Example: Overfitting • Posteriors determined by relative probabilities (odds ratios): south-west : inf nation : inf morally : inf nicely : inf extent : inf seriously : inf ... screens : inf minute : inf guaranteed : inf $205.00 : inf delivery : inf signature : inf ... What went wrong here?

  17. Generalization and Overfitting • Raw counts will overfit the training data! • Unlikely that every occurrence of “minute” is 100% spam • Unlikely that every occurrence of “seriously” is 100% ham • What about all the words that don’t occur in the training set at all? 0/0? • In general, we can’t go around giving unseen events zero probability • At the extreme, imagine using the entire email as the only feature • Would get the training data perfect (if deterministic labeling) • Wouldn’t generalize at all • Just making the bag-of-words assumption gives us some generalization, but isn’t enough • To generalize better: we need to smooth or regularize the estimates

  18. Estimation: Smoothing • Maximum likelihood estimates: • Problems with maximum likelihood estimates: • If I flip a coin once, and it’s heads, what’s the estimate for P(heads)? • What if I flip 10 times with 8 heads? • What if I flip 10M times with 8M heads? • Basic idea: • We have some prior expectation about parameters (here, the probability of heads) • Given little evidence, we should skew towards our prior • Given a lot of evidence, we should listen to the data r g g

  19. Estimation: Laplace Smoothing • Laplace’s estimate (extended): • Pretend you saw every outcome k extra times • What’s Laplace with k = 0? • k is the strength of the prior • Laplace for conditionals: • Smooth each condition independently: H H T

  20. Estimation: Linear Interpolation • In practice, Laplace often performs poorly for P(X|Y): • When |X| is very large • When |Y| is very large • Another option: linear interpolation • Also get P(X) from the data • Make sure the estimate of P(X|Y) isn’t too different from P(X) • What if is 0? 1?

  21. Real NB: Smoothing • For real classification problems, smoothing is critical • New odds ratios: helvetica : 11.4 seems : 10.8 group : 10.2 ago : 8.4 areas : 8.3 ... verdana : 28.8 Credit : 28.4 ORDER : 27.2 <FONT> : 26.9 money : 26.5 ... Do these make more sense?

  22. Tuning on Held-Out Data • Now we’ve got two kinds of unknowns • Parameters: the probabilities P(Y|X), P(Y) • Hyperparameters, like the amount of smoothing to do: k • How to learn? • Learn parameters from training data • Must tune hyperparameters on different data • Why? • For each value of the hyperparameters, train and test on the held-out (validation)data • Choose the best value and do a final test on the test data

  23. How to Learn • Data: labeled instances, e.g. emails marked spam/ham • Training set • Held out (validation) set • Test set • Features: attribute-value pairs which characterize each x • Experimentation cycle • Learn parameters (e.g. model probabilities) on training set • Tune hyperparameters on held-out set • Compute accuracy on test set • Very important: never “peek” at the test set! • Evaluation • Accuracy: fraction of instances predicted correctly • Overfitting and generalization • Want a classifier which does well on test data • Overfitting: fitting the training data very closely, but not generalizing well to test data Training Data Held-Out Data Test Data

  24. What to Do About Errors? • Need more features– words aren’t enough! • Have you emailed the sender before? • Have 1K other people just gotten the same email? • Is the sending information consistent? • Is the email in ALL CAPS? • Do inline URLs point where they say they point? • Does the email address you by (your) name? • Can add these information sources as new variables in the Naïve Bayes model

  25. A Digit Recognizer • Input: x = pixel grids • Output: y = a digit 0-9

  26. Example: Digit Recognition • Input: x = images (pixel grids) • Output: y = a digit 0-9 • Setup: • Get a large collection of example images, each labeled with a digit • Note: someone has to hand label all this data! • Want to learn to predict labels of new, future digit images • Features: The attributes used to make the digit decision • Pixels: (6,8)=ON • Shape Patterns: NumComponents, AspectRatio, NumLoops • … 0 1 2 1 ??

  27. Naïve Bayes for Digits • Simple version: • One feature Fij for each grid position <i,j> • Boolean features • Each input maps to a feature vector, e.g. • Here: lots of features, each is binary valued • Naïve Bayes model:

  28. Learning Model Parameters

  29. Problem: Overfitting 2 wins!!

  30. Classification (Decision Tree) Slides credit: GuopingQiu

  31. Trees • Node • Root • Leaf • Branch • Path • Depth

  32. Decision Trees • A hierarchical data structure that represents data by implementing a divide and conquer strategy • Can be used as a non-parametric classification method. • Given a collection of examples, learn a decision tree that represents it. • Use this representation to classify new examples

  33. Decision Trees • Each node is associated with a feature (one of the elements of a feature vector that represent an object); • Each node test the value of its associated feature; • There is one branch for each value of the feature • Leaves specify the categories (classes) • Can categorize instances into multiple disjoint categories – multi-class Outlook Sunny Overcast Rain Humidity Wind Yes High Normal Strong Weak No Yes No Yes

  34. Decision Trees • Play Tennis Example • Feature Vector = (Outlook, Temperature, Humidity, Wind) Outlook Sunny Overcast Rain Humidity Wind Yes High Normal Strong Weak No Yes No Yes

  35. Decision Trees Node associated with a feature Node associated with a feature Outlook Sunny Overcast Rain Humidity Yes Wind High Normal Strong Weak No Yes No Yes Node associated with a feature

  36. Decision Trees • Play Tennis Example • Feature values: • Outlook = (sunny, overcast, rain) • Temperature =(hot, mild, cool) • Humidity = (high, normal) • Wind =(strong, weak)

  37. Decision Trees • Outlook = (sunny, overcast, rain) One branch for each value One branch for each value Outlook Sunny Overcast Rain Humidity Yes Wind One branch for each value High Normal Strong Weak No Yes No Yes

  38. Decision Trees • Class = (Yes, No) Outlook Sunny Overcast Rain Humidity Yes Wind High Normal Strong Weak No Yes No Yes Leaf nodes specify classes Leaf nodes specify classes

  39. Decision Trees • Design Decision Tree Classifier • Picking the root node • Recursively branching

  40. Decision Trees • Picking the root node • Consider data with two Boolean attributes (A,B) and two classes + and – { (A=0,B=0), - }: 50 examples { (A=0,B=1), - }: 50 examples { (A=1,B=0), - }: 3 examples { (A=1,B=1), + }: 100 examples

  41. B A 1 1 0 0 - - A B 1 1 0 0 - - + + Decision Trees • Picking the root node • Trees looks structurally similar; which attribute should we choose?

  42. Decision Trees • Picking the root node • The goal is to have the resulting decision tree as small as possible (Occam’s Razor) • The main decision in the algorithm is the selection of the next attribute to condition on (start from the root node). • We want attributes that split the examples to sets that are relatively pure in one label; this way we are closer to a leaf node. • The most popular heuristics is based on information gain, originated with the ID3 system of Quinlan.

  43. Entropy • S is a sample of training examples • p+ is the proportion of positive examples in S • p- is the proportion of negative examples in S • Entropy measures the impurity of S p+

  44. - - + + + - + - + + - + - + + + - - + - + - - + - + - - + - + - + - - - + - - - - + - - + - - - + + + + + + + + - - + - + - + - + + + - - - - - - - - - - - - - - - - - - - - - - - - + + + + + + + + + + + + + + - - - - - + + + + + + + - - + + + - - + - + - + + - - + + + - - + - + - - + - - + - + - - + - + - + + - - + + - - - + - + - + + - - + + + - - + - + - + + - - + - + Highly Disorganized High Entropy Much Information Required Highly Organized Low Entropy Little Information Required

  45. Information Gain • Gain (S, A) = expected reduction in entropy due to sorting on A • Values (A) is the set of all possible values for attribute A, Sv is the subset of S which attribute A has value v, |S| and | Sv | represent the number of samples in set S and set Sv respectively • Gain(S,A) is the expected reduction in entropy caused by knowing the value of attribute A.

  46. Information Gain Example: Choose A or B ? Split on A Split on B B A 1 0 1 0 100 + 50- 53- 100 + 3 - 100 -

  47. Example Play Tennis Example

  48. Example Humidity High Normal 3+,4- 6+,1- E=.985 E=.592 Gain(S, Humidity) = .94 - 7/14 * 0.985 - 7/14 *.592 = 0.151

  49. Example Wind Weak Strong 6+2- 3+,3- E=.811 E=1.0 Gain(S, Wind) = .94 - 8/14 * 0.811 - 6/14 * 1.0 = 0.048

  50. Example Outlook Sunny Overcast Rain 1,2,8,9,11 3,7,12,13 4,5,6,10,14 2+,3- 4+,0- 3+,2- 0.0 0.970 0.970 Gain(S, Outlook) = 0.246