Logistic Regression

1. Logistic Regression Rong Jin

2. Logistic Regression Model • In Gaussian generative model: • Generalize the ratio to a linear model • Parameters: w and c

3. Logistic Regression Model • In Gaussian generative model: • Generalize the ratio to a linear model • Parameters: w and c

4. Logistic Regression Model • The log-ratio of positive class to negative class • Results

5. Logistic Regression Model • The log-ratio of positive class to negative class • Results

6. Logistic Regression Model • Assume the inputs and outputs are related in the log linear function • Estimate weights: MLE approach

7. Example 1: Heart Disease 1: 25-29 2: 30-34 3: 35-39 4: 40-44 5: 45-49 6: 50-54 7: 55-59 8: 60-64 • Input feature x: age group id • output y: having heart disease or not • +1: having heart disease • -1: no heart disease

8. Example 1: Heart Disease • Logistic regression model • Learning w and c: MLE approach • Numerical optimization: w = 0.58, c = -3.34

9. Example 1: Heart Disease • W = 0.58 • An old person is more likely to have heart disease • C = -3.34 • xw+c < 0  p(+|x) < p(-|x) • xw+c > 0  p(+|x) > p(-|x) • xw+c = 0  decision boundary • x* = 5.78  53 year old

10. Naïve Bayes Solution • Inaccurate fitting • Non Gaussian distribution • i* = 5.59 • Close to the estimation by logistic regression • Even though naïve Bayes does not fit input patterns well, it still works fine for the decision boundary

11. Problems with Using Histogram Data?

12. Uneven Sampling for Different Ages

13. Solution w = 0.63, c = -3.56  i* = 5.65

14. Solution w = 0.63, c = -3.56  i* = 5.65 < 5.78

15. Solution w = 0.63, c = -3.56  i* = 5.65 < 5.78

16. Example: Text Classification • Learn to classify text into predefined categories • Input x: a document • Represented by a vector of words • Example: {(president, 10), (bush, 2), (election, 5), …} • Output y: if the document is politics or not • +1 for political document, -1 for not political document • Training data:

17. Example 2: Text Classification • Logistic regression model • Every term ti is assigned with a weight wi • Learning parameters: MLE approach • Need numerical solutions

18. Example 2: Text Classification • Logistic regression model • Every term ti is assigned with a weight wi • Learning parameters: MLE approach • Need numerical solutions

19. Example 2: Text Classification • Weight wi • wi > 0: term ti is a positive evidence • wi < 0: term ti is a negative evidence • wi = 0: term ti is irrelevant to the category of documents • The larger the | wi |, the more important ti term is determining whether the document is interesting. • Threshold c

20. Example 2: Text Classification • Weight wi • wi > 0: term ti is a positive evidence • wi < 0: term ti is a negative evidence • wi = 0: term ti is irrelevant to the category of documents • The larger the | wi |, the more important ti term is determining whether the document is interesting. • Threshold c

21. Example 2: Text Classification • Dataset: Reuter-21578 • Classification accuracy • Naïve Bayes: 77% • Logistic regression: 88%

22. Why Logistic Regression Works better for Text Classification? • Optimal linear decision boundary • Generative model • Weight ~ logp(w|+) - logp(w|-) • Sub-optimal weights • Independence assumption • Naive Bayes assumes that each word is generated independently • Logistic regression is able to take into account of the correlation of words

23. Discriminative Model • Logistic regression model is a discriminative model • Models the conditional probability p(y|x), i.e., the decision boundary • Gaussian generative model • Models p(x|y), i.e., input patterns of different classes

24. Comparison • Discriminative Model • Model P(y|x) directly • Model the decision boundary • Usually good performance • But • Slow convergence • Expensive computation • Sensitive to noise data • Generative Model • Model P(x|y) • Model the input patterns • Usually fast converge • Cheap computation • Robust to noise data • But • Usually performs worse

25. Comparison • Discriminative Model • Model P(y|x) directly • Model the decision boundary • Usually good performance • But • Slow convergence • Expensive computation • Sensitive to noise data • Generative Model • Model P(x|y) • Model the input patterns • Usually fast converge • Cheap computation • Robust to noise data • But • Usually performs worse

26. A Few Words about Optimization • Convex objective function • Solution could be non-unique

27. Problems with Logistic Regression? How about words that only appears in one class?

28. Overfitting Problem with Logistic Regression • Consider word t that only appears in one document d, and d is a positive document. Let w be its associated weight • Consider the derivative of l(Dtrain) with respect to w • w will be infinite !

29. Overfitting Problem with Logistic Regression • Consider word t that only appears in one document d, and d is a positive document. Let w be its associated weight • Consider the derivative of l(Dtrain) with respect to w • w will be infinite !

30. Example of Overfitting for LogRes Decrease in the classification accuracy of test data Iteration

31. Solution: Regularization • Regularized log-likelihood • s||w||2 is called the regularizer • Favors small weights • Prevents weights from becoming too large

32. The Rare Word Problem • Consider word t that only appears in one document d, and d is a positive document. Let w be its associated weight

33. The Rare Word Problem • Consider the derivative of l(Dtrain) with respect to w • When s is small, the derivative is still positive • But, it becomes negative when w is large

34. The Rare Word Problem • Consider the derivative of l(Dtrain) with respect to w • When w is small, the derivative is still positive • But, it becomes negative when w is large

35. Using regularization Without regularization Iteration Regularized Logistic Regression

36. Interpretation of Regularizer • Many interpretation of regularizer • Bayesian stat.: model prior • Statistical learning: minimize the generalized error • Robust optimization: min-max solution

37. Regularizer: Robust Optimization • assume each data point is unknown-but-bounded in a sphere of radius s and center xi • find the classifier w that is able to classify the unknown-but-bounded data point with high classification confidence

38. Sparse Solution • What does the solution of regularized logistic regression look like ? • A sparse solution • Most weights are small and close to zero

39. Sparse Solution • What does the solution of regularized logistic regression look like ? • A sparse solution • Most weights are small and close to zero

40. Why do We Need Sparse Solution? • Two types of solutions • Many non-zero weights but many of them are small • Only a small number of non-zero weights, and many of them are large • Occam’s Razor: the simpler the better • A simpler model that fits data unlikely to be coincidence • A complicated model that fit data might be coincidence • Smaller number of non-zero weights  less amount of evidence to consider  simpler model  case 2 is preferred

41. Occam’s Razer

42. Occam’s Razer: Power = 1

43. Occam’s Razer: Power = 3

44. Occam’s Razor: Power = 10

45. Finding Optimal Solutions • Concave objective function • No local maximum • Many standard optimization algorithms work

46. Predication Errors Preventing weights from being too large Gradient Ascent • Maximize the log-likelihood by iteratively adjusting the parameters in small increments • In each iteration, we adjust w in the direction that increases the log-likelihood (toward the gradient)

47. Graphical Illustration No regularization case

48. Gradient Ascent • Maximize the log-likelihood by iteratively adjusting the parameters in small increments • In each iteration, we adjust w in the direction that increases the log-likelihood (toward the gradient)

49. Using regularization Without regularization Iteration

50. When should Stop? • The gradient ascent learning method converges when there is no incentive to move the parameters in any particular direction: