Semi-supervised Learning Rong Jin

Semi-supervised learning  Label propagation  Transductive learning  Co-training  Active learning

Label Propagation  A toy problem Each node in the graph is an example  Two examples are labeled  Most examples are unlabeled Compute the similarity between examples S ij Connect examples to their most similar examples  How to predicate labels for unlabeled nodes using this graph? Unlabeled example Two labeled examples w ij

Label Propagation  Forward propagation

Label Propagation  Forward propagation

Label Propagation  Forward propagation How to resolve conflicting cases What label should be given to this node ?

Label Propagation  Let S be the similarity matrix S=[S i,j ] nxn  Let D be a diagonal matrix where D i =  i  j S i,j  Compute normalized similarity matrix S’ S’=D -1/2 SD -1/2  Let Y be the initial assignment of class labels Y i = 1 when the i-th node is assigned to the positive class Y i = -1 when the i-th node is assigned to the negative class Y i = 0 when the I-th node is not initially labeled  Let F be the predicted class labels The i-th node is assigned to the positive class if F i >0 The i-th node is assigned to the negative class if F i < 0

Label Propagation  Let S be the similarity matrix S=[S i,j ] nxn  Let D be a diagonal matrix where D i =  i  j S i,j  Compute normalized similarity matrix S’ S’=D -1/2 SD -1/2  Let Y be the initial assignment of class labels Y i = 1 when the i-th node is assigned to the positive class Y i = -1 when the i-th node is assigned to the negative class Y i = 0 when the i-th node is not initially labeled  Let F be the predicted class labels The i-th node is assigned to the positive class if F i >0 The i-th node is assigned to the negative class if F i < 0

Label Propagation  One iteration F = Y +  S’Y = (I +  S’)Y  weights the propagation values  Two iteration F =Y +  S’Y +  2 S’ 2 Y = (I +  S’ +  2 S’ 2 )Y  How about the infinite iteration F = (  n=0 1  n S’ n )Y = (I -  S’) -1 Y  Any problems with such an approach?

Label Consistency Problem  Predicted vector F may not be consistent with the initially assigned class labels Y

Energy Minimization  Using the same notation S i,j : similarity between the I-th node and j-th node Y: initially assigned class labels F: predicted class labels  Energy: E(F) =  i,j S i,j (F i – F j ) 2  Goal: find label assignment F that is consistent with labeled examples Y and meanwhile minimizes the energy function E(F)

Harmonic Function  E(F) =  i,j S i,j (F i – F j ) 2 = F T (D-S)F  Thus, the minimizer for E(F) should be (D-S)F = 0, and meanwhile F should be consistent with Y.  F T = (F l T, F u T ), Y T = (Y l T, Y u T ) F l = Y l 

Optical Character Recognition  Given an image of a digit letter, determine its value 1 2  Create a graph for images of digit letters

Optical Character Recognition  #Labeled_Examples+#Unlabeled_Examples = 4000  CMN: label propagation  1NN: for each unlabeled example, using the label of its closest neighbor

Spectral Graph Transducer  Problem with harmonic function  Why this could happen ?  The condition (D-S)F = 0 does not hold for constrained cases

Spectral Graph Transducer  Problem with harmonic function  Why this could happen ?  The condition (D-S)F = 0 does not hold for constrained cases

Spectral Graph Transducer min F F T LF + c (F-Y) T C(F-Y) s.t. F T F=n, F T e = 0  C is the diagonal cost matrix, C i,i = 1 if the i-th node is initially labeled, zero otherwise  Parameter c controls the balance between the consistency requirement and the requirement of energy minimization  Can be solved efficiently through the computation of eigenvector

Empirical Studies

Problems with Spectral Graph Transducer min F F T LF + c (F-Y) T C(F-Y) s.t. F T F=n, F T e = 0  The obtained solution is different from the desirable one: minimize the energy function and meanwhile is consistent with labeled examples Y  It is difficult to extend the approach to multi-class classification

Green’s Function  The problem of minimizing energy and meanwhile being consistent with initially assigned class labels can be formulated into Green’s function problem  Minimizing E(F) = F T LF  LF = 0 Turns out L can be viewed as Laplacian operator in the discrete case LF = 0  r 2 F=0  Thus, our problem is find solution F r 2 F=0, s.t. F = Y for labeled examples We can treat the constraint that F = Y for labeled examples as boundary condition (Von Neumann boundary condition) A standard Green function problem

Why Energy Minimization? Final classification results

Label Propagation  How the unlabeled data help classification?  Consider a smaller number of unlabeled example

Label Propagation  How the unlabeled data help classification?  Consider a smaller number of unlabeled example  Classification results can be very different

Cluster Assumption  Cluster assumption Decision boundary should pass low density area  Unlabeled data provide more accurate estimation of local density

Cluster Assumption vs. Maximum Margin  Maximum margin classifier (e.g. SVM) denotes +1 denotes -1 w  x+b  Maximum margin  low density around decision boundary  Cluster assumption  Any thought about utilizing the unlabeled data in support vector machine?

Transductive SVM  Decision boundary given a small number of labeled examples

Transductive SVM  Decision boundary given a small number of labeled examples  How will the decision boundary change given both labeled and unlabeled examples?

Transductive SVM  Decision boundary given a small number of labeled examples  Move the decision boundary to place with low local density

Transductive SVM  Decision boundary given a small number of labeled examples  Move the decision boundary to place with low local density  Classification results  How to formulate this idea?

Transductive SVM: Formulation  Labeled data L:  Unlabeled data D:  Maximum margin principle for mixture of labeled and unlabeled data For each label assignment of unlabeled data, compute its maximum margin Find the label assignment whose maximum margin is maximized

Tranductive SVM Different label assignment for unlabeled data  different maximum margin

Transductive SVM: Formulation Original SVM Transductive SVM Constraints for unlabeled data A binary variables for label of each example

Computational Issue  No longer convex optimization problem. (why?)  How to optimize transductive SVM?  Alternating optimization

Alternating Optimization  Step 1: fix y n+1,…, y n+m, learn weights w  Step 2: fix weights w, try to predict y n+1,…, y n+m (How?)

Empirical Study with Transductive SVM  10 categories from the Reuter collection  3299 test documents  1000 informative words selected using MI criterion

Co-training for Semi-supervised Learning  Consider the task of classifying web pages into two categories: category for students and category for professors  Two aspects of web pages should be considered Content of web pages  “I am currently the second year Ph.D. student …” Hyperlinks  “My advisor is …”  “Students: …”

Co-training for Semi-Supervised Learning

It is easy to classify the type of this web page based on its content It is easier to classify this web page using hyperlinks

Co-training  Two representation for each web page Content representation: (doctoral, student, computer, university…) Hyperlink representation: Inlinks: Prof. Cheng Oulinks: Prof. Cheng

Co-training: Classification Scheme 1. Train a content-based classifier using labeled web pages 2. Apply the content-based classifier to classify unlabeled web pages 3. Label the web pages that have been confidently classified 4. Train a hyperlink based classifier using the web pages that are initially labeled and labeled by the classifier 5. Apply the hyperlink-based classifier to classify the unlabeled web pages 6. Label the web pages that have been confidently classified

Co-training  Train a content-based classifier

Co-training  Train a content-based classifier using labeled examples  Label the unlabeled examples that are confidently classified

Co-training  Train a content-based classifier using labeled examples  Label the unlabeled examples that are confidently classified  Train a hyperlink-based classifier Prof. : outlinks to students

Co-training  Train a content-based classifier using labeled examples  Label the unlabeled examples that are confidently classified  Train a hyperlink-based classifier Prof. : outlinks to students  Label the unlabeled examples that are confidently classified

Co-training  Train a content-based classifier using labeled examples  Label the unlabeled examples that are confidently classified  Train a hyperlink-based classifier Prof. : outlinks to  Label the unlabeled examples that are confidently classified