1 / 66

Topics on Final

Topics on Final. Perceptrons SVMs Precision/Recall/ROC Decision Trees Naive Bayes Bayesian networks Adaboost Genetic algorithms Q learning Not on the final: MLPs , PCA. Rules for Final. Open book, notes, computer, calculator No discussion with others

galvin-hoffman
Download Presentation

Topics on Final

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Topics on Final • Perceptrons • SVMs • Precision/Recall/ROC • Decision Trees • Naive Bayes • Bayesian networks • Adaboost • Genetic algorithms • Q learning • Not on the final: MLPs, PCA

  2. Rules for Final • Open book, notes, computer, calculator • No discussion with others • You can ask me or Dona general questions about a topic • Read each question carefully • Hand in your own work only • Turn in to box at CS front desk or to me (hardcopy or e-mail) by 5pm Wednesday, March 21. • No extensions

  3. Short recap of important topics

  4. Perceptrons

  5. Training a perceptron • Start with random weights, w= (w1, w2, ... , wn). • Select training example (xk, tk). • Run the perceptron with input xkand weights w to obtain o. • Let  be the learning rate (a user-set parameter). Now, • Go to 2.

  6. Support Vector Machines

  7. Here, assume positive and negative instances are to be separated by the hyperplane Equation of line: x2 x1

  8. Intuition: the best hyperplane (for future generalization) will “maximally” separate the examples

  9. Definition of Margin

  10. Minimizing ||w|| Find w and b by doing the following minimization: This is a quadratic optimization problem. Use “standard optimization tools” to solve it.

  11. Dual formulation: It turns out that w can be expressed as a linear combination of a small subset of the training examples xi: those that lie exactly on margin (minimum distance to hyperplane): such that xi lie exactly on the margin. • These training examples are called “support vectors”. They carry all relevant information about the classification problem.

  12. The results of the SVM training algorithm (involving solving a quadratic programming problem) are the i and the bias b. • The support vectors are all xi such that i > 0. • Clarification:In the slides below we use ito denote |i|yi, where yi {−1, 1}.

  13. For a new example x, We can now classify x using the support vectors: • This is the resulting SVM classifier.

  14. SVM review • Equation of line: w1x1 + w2x2 + b = 0 • Define margin using: • Margin distance: • To maximize the margin, we minimize ||w|| • subject to the constraint that positive examples • fall on one side of the margin, and negative • examples on the other side: • We can relax this constraint using “slack variables”

  15. SVM review • To do the optimization, we use the dual formulation: • The results of the optimization “black box” are • and b . • The support vectors are all xisuch that i!= 0.

  16. SVM review • Once the optimization is done, we can classify a new • example x as follows: • That is, classification is done entirely through a linear • combination of dot products with training examples. • This is a “kernel” method.

  17. Example 2 1 1 2 -2 -1 -1 -2

  18. Example Input to SVM optimzer: x1 x2 class 1 1 1 1 2 1 2 1 1 -1 0 -1 0 -1 -1 -1 -1 -1 2 1 1 2 -2 -1 -1 -2

  19. Example Input to SVM optimzer: x1 x2 class 1 1 1 1 2 1 2 1 1 -1 0 -1 0 -1 -1 -1 -1 -1 Output from SVM optimzer: Support vectorα (-1, 0) -.208 (1, 1) .416 (0, -1) -.208 b = -.376 2 1 1 2 -2 -1 -1 -2

  20. Example Input to SVM optimzer: x1 x2 class 1 1 1 1 2 1 2 1 1 -1 0 -1 0 -1 -1 -1 -1 -1 Output from SVM optimzer: Support vectorα (-1, 0) -.208 (1, 1) .416 (0, -1) -.208 b = -.376 2 1 1 2 -2 -1 Weight vector: -1 -2

  21. Example Input to SVM optimzer: x1 x2 class 1 1 1 1 2 1 2 1 1 -1 0 -1 0 -1 -1 -1 -1 -1 Output from SVM optimzer: Support vectorα (-1, 0) -.208 (1, 1) .416 (0, -1) -.208 b = -.376 2 1 1 2 -2 -1 Weight vector: -1 -2 Separation line:

  22. Example 2 1 1 2 -2 -1 -1 Classifying a new point: -2

  23. Precision/Recall/ROC

  24. Creating a Precision/Recall Curve Results of classifier

  25. Creating a ROC Curve Results of classifier

  26. Precision/Recall versus ROC curves http://blog.crowdflower.com/2008/06/aggregate-turker-judgments-threshold-calibration/

  27. Decision Trees

  28. Naive Bayes

  29. Naive Bayes classifier: Assume Given this assumption, here’s how to classify an instance x= <a1, a2, ...,an>: We can estimate the values of these various probabilities over the training set.

  30. In-class example Training set: a1a2a3class 0 1 0 + 1 0 1 + 0 0 1 − 1 1 0 − 1 0 0 − What class would be assigned by a NB classifier to 1 1 1 ?

  31. Laplace smoothing (also called “add-one” smoothing) For each class cj and attribute aiwith valuez,add one “virtual” instance. That is, recalculate: where k is the number of possible values of attribute a. a1a2a3 class Smoothed P(a1=1 | +) = 0 1 0 + Smoothed P(a1=0 | +) = 0 0 1 + Smoothed P(a1=1 |−) = 1 1 1 − Smoothed P(a1=0 |−) = 1 1 0 − 1 0 1 −

  32. Bayesian Networks

  33. Methods used in computing probabilities • Definition of conditional probability: P(A | B) = P (A,B) / P(B) • Bayes theorem: P(A | B) = P(B | A) P(A) / P(B) • Semantics of Bayesian networks: P(A ^ B ^ C ^ D) = P(A | Parents(A))  P(B | Parents(B)) P(C | Parents(C))P(D |Parents(D)) • Caculating marginal probabilities

  34. What is P(Cloudy| Sprinkler)?

  35. What is P(Cloudy| Wet Grass)?

  36. Markov Chain Monte Carlo Algorithm • Markov blanket of a variable Xi: • parents, children, children’s other parents • MCMC algorithm: For a given set of evidence variables {Xj=xk} Repeat for NumSamples: • Start with random sample from variables, with evidence variables fixed: (x1, ..., xn). This is the current “state” of the algorithm. • Next state: Randomly sample value for one non-evidence variable Xi , conditioned on current values in “Markov Blanket” of Xi. Finally, return the estimated distribution of each non-evidence variable Xi

  37. Example • Query: What is P(Sprinkler =true | WetGrass = true)? • MCMC: • Random sample, with evidence variables fixed: [Cloudy, Sprinkler, Rain, WetGrass] = [true, true, false, true] • Repeat: • Sample Cloudy, given current values of its Markov blanket: Sprinkler = true, Rain = false. Suppose result is false. New state: [false, true, false, true] Note that current values of Markov blanket remain fixed. • Sample Sprinkler, given current values of its Markov blanket: Cloudy = false, Rain= false, Wet = true. Suppose result is true. New state: [false, true, false, true].

  38. Each sample contributes to estimate for query P(Sprinkler = true| WetGrass = true) • Suppose we perform 50 such samples, 20 with Sprinkler = true and 30 with Sprinkler= false. • Then answer to the query is Normalize (20,30) = .4,.6

  39. Adaboost

  40. Sketch of algorithm Given data S and learning algorithm L: • Repeatedly runLon training sets StS to produce h1, h2, ... , hT. • At each step, derive St from S by choosing examples probabilistically according to probability distributionwt. Use St to learn ht. • At each step, derivewt+1 by giving more probability to examples that were misclassified at step t. • The final ensemble classifier H is a weighted sum of the ht’s, with each weight being a function of the corresponding ht’s error on its training set.

  41. Adaboost algorithm • Given S= {(x1, y1), ..., (xN, yN)} where xX,yi {+1, -1} • Initializew1(i) = 1/N. (Uniform distribution over data)

  42. For t = 1, ..., T: • Select new training set St from S with replacement, according to wt • Train L on St to obtain hypothesis ht • Compute the training error tof ht on S : • If t0.5, break from loop. • Compute coefficient

  43. Compute new weights on data: where Zt is a normalization factor chosen so that wt+1 will be a probability distribution:

  44. At the end of T iterations of this algorithm, we have h1, h2, . . . , hT We also have 1, 2, . . . ,T, where • Ensemble classifier: • Note that hypotheses with higher accuracy on their training sets are weighted more strongly.

  45. A Simple Example t =1 • S = Spam8.train: x1, x2, x3, x4 (class +1) x5, x6, x7, x8 (class -1) • w1 = {1/8, 1/8, 1/8, 1/8, 1/8, 1/8, 1/8, 1/8} • S1 = {x1, x2, x2, x5,x5,x6, x7, x8} • Run svm_light on S1 to get h1 • Run h1 on S. Classifications: {1, -1, -1, -1, -1, -1, -1, -1} • Calculate error:

  46. Calculate ’s: • Calculate new w’s:

  47. t =2 • w2 = {0.102, 0.163, 0.163, 0.163, 0.102, 0.102, 0.102, 0.102} • S2 = {x1, x2, x2, x3,x4,x4,x7, x8} • Run svm_light on S2 to get h2 • Run h2 on S. Classifications: {1, 1, 1, 1, 1, 1, 1, 1} • Calculate error:

  48. Calculate ’s: • Calculate w’s:

More Related