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Winnow vs perceptron

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Winnow vs perceptron

  1. 1. SIMS 290-2: Applied Natural Language Processing Barbara Rosario October 4, 2004 1
  2. 2. Today Algorithms for Classification Binary classification Perceptron Winnow Support Vector Machines (SVM) Kernel Methods Multi-Class classification Decision Trees Naïve Bayes K nearest neighbor 2
  3. 3. Binary Classification: examples Spam filtering (spam, not spam) Customer service message classification (urgent vs. not urgent) Information retrieval (relevant, not relevant) Sentiment classification (positive, negative) Sometime it can be convenient to treat a multi-way problem like a binary one: one class versus all the others, for all classes 3
  4. 4. Binary Classification Given: some data items that belong to a positive (+1 ) or a negative (-1 ) class Task: Train the classifier and predict the class for a new data item Geometrically: find a separator 4
  5. 5. Linear versus Non Linear algorithms Linearly separable data: if all the data points can be correctly classified by a linear (hyperplanar) decision boundary 5
  6. 6. Linearly separable data Linear Decision boundary Class1 Class2 6
  7. 7. Non linearly separable data Class1 Class2 7
  8. 8. Non linearly separable data Non Linear Classifier Class1 Class2 8
  9. 9. Linear versus Non Linear algorithms Linear or Non linear separable data? We can find out only empirically Linear algorithms (algorithms that find a linear decision boundary) When we think the data is linearly separable Advantages – Simpler, less parameters Disadvantages – High dimensional data (like for NLT) is usually not linearly separable Examples: Perceptron, Winnow, SVM Note: we can use linear algorithms also for non linear problems (see Kernel methods) 9
  10. 10. Linear versus Non Linear algorithms Non Linear When the data is non linearly separable Advantages – More accurate Disadvantages – More complicated, more parameters Example: Kernel methods Note: the distinction between linear and non linear applies also for multi-class classification (we’ll see this later) 10
  11. 11. Simple linear algorithms Perceptron and Winnow algorithm Linear Binary classification Online (process data sequentially, one data point at the time) Mistake driven Simple single layer Neural Networks 11
  12. 12. Linear binary classification Data: {(xi,yi)}i=1...n x in Rd (x is a vector in d-dimensional space)  feature vector y in {-1,+1}  label (class, category) Question: Design a linear decision boundary: wx + b (equation of hyperplane) such that the classification rule associated with it has minimal probability of error classification rule : – y = sign(w x + b) which means: – if wx + b > 0 then y = +1 – if wx + b < 0 then y = -1 From Gert Lanckriet, Statistical Learning Theory Tutorial 12
  13. 13. Linear binary classification Find a good hyperplane (w,b) in R d+1 that correctly classifies data points as much as possible In online fashion: one data point at the time, update weights as necessary wx + b = 0 Classification Rule: y = sign(wx + b) From Gert Lanckriet, Statistical Learning Theory Tutorial 13
  14. 14. Perceptron algorithm Initialize: w1 = 0 Updating rule For each data point x If class(x) != decision(x,w) wk then wk+1  wk + yixi k k+1 else wk+1  wk 0 +1 wk+1 -1 wk x + b = 0 Function decision(x, w) If wx + b > 0 return +1 Else return -1 From Gert Lanckriet, Statistical Learning Theory Tutorial Wk+1 x + b = 0 14
  15. 15. Perceptron algorithm Online: can adjust to changing target, over time Advantages Simple and computationally efficient Guaranteed to learn a linearly separable problem (convergence, global optimum) Limitations Only linear separations Only converges for linearly separable data Not really “efficient with many features” From Gert Lanckriet, Statistical Learning Theory Tutorial 15
  16. 16. Winnow algorithm Another online algorithm for learning perceptron weights: f(x) = sign(wx + b) Linear, binary classification Update-rule: again error-driven, but multiplicative (instead of additive) From Gert Lanckriet, Statistical Learning Theory Tutorial 16
  17. 17. Winnow algorithm Initialize: w1 = 0 Updating rule For each data point x wk If class(x) != decision(x,w) then wk+1  wk + yixi  Perceptron w k+1  w k *exp(y i x i )  Winnow k k+1 else wk+1  wk 0 +1 wk+1 -1 wk x + b= 0 Function decision(x, w) If wx + b > 0 return +1 Else return -1 From Gert Lanckriet, Statistical Learning Theory Tutorial Wk+1 x + b = 0 17
  18. 18. Perceptron vs. Winnow Assume N available features only K relevant items, with K<<N Perceptron: number of mistakes: O( K N) Winnow: number of mistakes: O(K log N) Winnow is more robust to high-dimensional feature spaces From Gert Lanckriet, Statistical Learning Theory Tutorial 18
  19. 19. Perceptron vs. Winnow Perceptron Online: can adjust to changing target, over time Advantages Simple and computationally efficient Guaranteed to learn a linearly separable problem Limitations only linear separations only converges for linearly separable data not really “efficient with many features” Winnow Online: can adjust to changing target, over time Advantages Simple and computationally efficient Guaranteed to learn a linearly separable problem Suitable for problems with many irrelevant attributes Limitations only linear separations only converges for linearly separable data not really “efficient with many features” Used in NLP From Gert Lanckriet, Statistical Learning Theory Tutorial 19
  20. 20. Weka Winnow in Weka 20
  21. 21. Large margin classifier Another family of linear algorithms Intuition (Vapnik, 1965) If the classes are linearly separable: Separate the data Place hyper-plane “far” from the data: large margin Statistical results guarantee good generalization BAD From Gert Lanckriet, Statistical Learning Theory Tutorial 21
  22. 22. Large margin classifier Intuition (Vapnik, 1965) if linearly separable: Separate the data Place hyperplane “far” from the data: large margin Statistical results guarantee good generalization GOOD  Maximal Margin Classifier From Gert Lanckriet, Statistical Learning Theory Tutorial 22
  23. 23. Large margin classifier If not linearly separable Allow some errors Still, try to place hyperplane “far” from each class From Gert Lanckriet, Statistical Learning Theory Tutorial 23
  24. 24. Large Margin Classifiers Advantages Theoretically better (better error bounds) Limitations Computationally more expensive, large quadratic programming 24
  25. 25. Support Vector Machine (SVM) M Large Margin Classifier wTxa + b = 1 wTxb + b = -1 Linearly separable case Goal: find the hyperplane that maximizes the margin wT x + b = 0 Support vectors From Gert Lanckriet, Statistical Learning Theory Tutorial 25
  26. 26. Support Vector Machine (SVM) Text classification Hand-writing recognition Computational biology (e.g., micro-array data) Face detection Face expression recognition Time series prediction From Gert Lanckriet, Statistical Learning Theory Tutorial 26
  27. 27. Non Linear problem 27
  28. 28. Non Linear problem 28
  29. 29. Non Linear problem Kernel methods A family of non-linear algorithms Transform the non linear problem in a linear one (in a different feature space) Use linear algorithms to solve the linear problem in the new space From Gert Lanckriet, Statistical Learning Theory Tutorial 29
  30. 30. Main intuition of Kernel methods (Copy here from black board) 30
  31. 31. Basic principle kernel methods Φ : Rd  RD (D >> d) wTΦ(x)+b=0 Φ(X)=[x2 z2 xz] X=[x z] f(x) = sign(w1x2+w2z2+w3xz +b) From Gert Lanckriet, Statistical Learning Theory Tutorial 31
  32. 32. Basic principle kernel methods Linear separability : more likely in high dimensions Mapping: Φ maps input into high-dimensional feature space Classifier: construct linear classifier in highdimensional feature space Motivation: appropriate choice of Φ leads to linear separability We can do this efficiently! From Gert Lanckriet, Statistical Learning Theory Tutorial 32
  33. 33. Basic principle kernel methods We can use the linear algorithms seen before (Perceptron, SVM) for classification in the higher dimensional space 33
  34. 34. Multi-class classification Given: some data items that belong to one of M possible classes Task: Train the classifier and predict the class for a new data item Geometrically: harder problem, no more simple geometry 34
  35. 35. Multi-class classification 35
  36. 36. Multi-class classification: Examples Author identification Language identification Text categorization (topics) 36
  37. 37. (Some) Algorithms for Multi-class classification Linear Parallel class separators: Decision Trees Non parallel class separators: Naïve Bayes Non Linear K-nearest neighbors 37
  38. 38. Linear, parallel class separators (ex: Decision Trees) 38
  39. 39. Linear, NON parallel class separators (ex: Naïve Bayes) 39
  40. 40. Non Linear (ex: k Nearest Neighbor) 40
  41. 41. Decision Trees Decision tree is a classifier in the form of a tree structure, where each node is either: Leaf node - indicates the value of the target attribute (class) of examples, or Decision node - specifies some test to be carried out on a single attribute-value, with one branch and sub-tree for each possible outcome of the test. A decision tree can be used to classify an example by starting at the root of the tree and moving through it until a leaf node, which provides the classification of the instance. http://dms.irb.hr/tutorial/tut_dtrees.php 41
  42. 42. Training Examples Goal: learn when we can play Tennis and when we cannot Day Outlook Temp. Humidity Wind Play Tennis D1 Sunny Hot High Weak No D2 Sunny Hot High Strong No D3 Overcast Hot High Weak Yes D4 Rain Mild High Weak Yes D5 Rain Cool Normal Weak Yes D6 Rain Cool Normal Strong No D7 Overcast Cool Normal Weak Yes D8 Sunny Mild High Weak No D9 Sunny Cold Normal Weak Yes D10 Rain Mild Normal Strong Yes D11 Sunny Mild Normal Strong Yes D12 Overcast Mild High Strong Yes D13 Overcast Hot Normal Weak Yes D14 Rain Mild High Strong No 42
  43. 43. Decision Tree for PlayTennis Outlook Sunny Humidity High No Overcast Rain Yes Normal Yes www.math.tau.ac.il/~nin/ Courses/ML04/DecisionTreesCLS.pp Wind Strong No Weak Yes 43
  44. 44. Decision Tree for PlayTennis Outlook Sunny Humidity High No Overcast Rain Each internal node tests an attribute Normal Yes www.math.tau.ac.il/~nin/ Courses/ML04/DecisionTreesCLS.pp Each branch corresponds to an attribute value node Each leaf node assigns a classification 44
  45. 45. Decision Tree for PlayTennis Outlook Temperature Humidity Wind Sunny Hot High Weak PlayTennis ? No Outlook Sunny Humidity High No Overcast Rain Yes Normal Yes www.math.tau.ac.il/~nin/ Courses/ML04/DecisionTreesCLS.pp Wind Strong No Weak Yes 45
  46. 46. Decision Tree for Reuter classification Foundations of Statistical Natural Language Processing, Manning and Schuetze 46
  47. 47. Decision Tree for Reuter classification Foundations of Statistical Natural Language Processing, Manning and Schuetze 47
  48. 48. Building Decision Trees Given training data, how do we construct them? The central focus of the decision tree growing algorithm is selecting which attribute to test at each node in the tree. The goal is to select the attribute that is most useful for classifying examples. Top-down, greedy search through the space of possible decision trees. That is, it picks the best attribute and never looks back to reconsider earlier choices. 48
  49. 49. Building Decision Trees Splitting criterion Finding the features and the values to split on – for example, why test first “cts” and not “vs”? – Why test on “cts < 2” and not “cts < 5” ? Split that gives us the maximum information gain (or the maximum reduction of uncertainty) Stopping criterion When all the elements at one node have the same class, no need to split further In practice, one first builds a large tree and then one prunes it back (to avoid overfitting) See Foundations of Statistical Natural Language Processing , Manning and Schuetze for a good introduction 49
  50. 50. Decision Trees: Strengths Decision trees are able to generate understandable rules. Decision trees perform classification without requiring much computation. Decision trees are able to handle both continuous and categorical variables. Decision trees provide a clear indication of which features are most important for prediction or classification. http://dms.irb.hr/tutorial/tut_dtrees.php 50
  51. 51. Decision Trees: weaknesses Decision trees are prone to errors in classification problems with many classes and relatively small number of training examples. Decision tree can be computationally expensive to train. Need to compare all possible splits Pruning is also expensive Most decision-tree algorithms only examine a single field at a time. This leads to rectangular classification boxes that may not correspond well with the actual distribution of records in the decision space. http://dms.irb.hr/tutorial/tut_dtrees.php 51
  52. 52. Decision Trees Decision Trees in Weka 52
  53. 53. Naïve Bayes More powerful that Decision Trees Decision Trees Naïve Bayes 53
  54. 54. Naïve Bayes Models Graphical Models: graph theory plus probability theory Nodes are variables Edges are conditional probabilities A B C P(A) P(B|A) P(C|A) 54
  55. 55. Naïve Bayes Models Graphical Models: graph theory plus probability theory Nodes are variables Edges are conditional probabilities Absence of an edge between nodes implies independence between the variables of the nodes A B C P(A) P(B|A) P(C|A)  P(C| A,B) 55
  56. 56. Naïve Bayes for text classification Foundations of Statistical Natural Language Processing, Manning and Schuetze 56
  57. 57. Naïve Bayes for text classification earn Shr 34 cts vs per shr 57
  58. 58. Naïve Bayes for text classification Topic w1 w2 w3 w4 wn-1 wn The words depend on the topic: P(wi| Topic) P(cts|earn) > P(tennis| earn) Naïve Bayes assumption: all words are independent given the topic From training set we learn the probabilities P(w i| Topic) for each word and for each topic in the training set 58
  59. 59. Naïve Bayes for text classification Topic w1 w2 w3 w4 wn-1 wn To: Classify new example Calculate P(Topic | w1, w2, … wn) for each topic Bayes decision rule: Choose the topic T’ for which P(T’ | w1, w2, … wn) > P(T | w1, w2, … wn) for each T≠ T’ 59
  60. 60. Naïve Bayes: Math Naïve Bayes define a joint probability distribution: P(Topic , w1, w2, … wn) = P(Topic)∏ P(wi| Topic) We learn P(Topic) and P(wi| Topic) in training Test: we need P(Topic | w1, w2, … wn) P(Topic | w1, w2, … wn) = P(Topic , w1, w2, … wn) / P(w1, w2, … wn) 60
  61. 61. Naïve Bayes: Strengths Very simple model Easy to understand Very easy to implement Very efficient, fast training and classification Modest space storage Widely used because it works really well for text categorization Linear, but non parallel decision boundaries 61
  62. 62. Naïve Bayes: weaknesses Naïve Bayes independence assumption has two consequences: The linear ordering of words is ignored (bag of words model) The words are independent of each other given the class: False – President is more likely to occur in a context that contains election than in a context that contains poet Naïve Bayes assumption is inappropriate if there are strong conditional dependencies between the variables (But even if the model is not “right”, Naïve Bayes models do well in a surprisingly large number of cases because often we are interested in classification accuracy and not in accurate probability estimations) 62
  63. 63. Naïve Bayes Naïve Bayes in Weka 63
  64. 64. k Nearest Neighbor Classification Nearest Neighbor classification rule: to classify a new object, find the object in the training set that is most similar. Then assign the category of this nearest neighbor K Nearest Neighbor (KNN): consult k nearest neighbors. Decision based on the majority category of these neighbors. More robust than k = 1 Example of similarity measure often used in NLP is cosine similarity 64
  65. 65. 1-Nearest Neighbor 65
  66. 66. 1-Nearest Neighbor 66
  67. 67. 3-Nearest Neighbor 67
  68. 68. 3-Nearest Neighbor But this is closer.. We can weight neighbors according to their similarity Assign the category of the majority of the neighbors 68
  69. 69. k Nearest Neighbor Classification Strengths Robust Conceptually simple Often works well Powerful (arbitrary decision boundaries) Weaknesses Performance is very dependent on the similarity measure used (and to a lesser extent on the number of neighbors k used) Finding a good similarity measure can be difficult Computationally expensive 69
  70. 70. Summary Algorithms for Classification Linear versus non linear classification Binary classification Perceptron Winnow Support Vector Machines (SVM) Kernel Methods Multi-Class classification Decision Trees Naïve Bayes K nearest neighbor On Wednesday: Weka 70

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