Semi-supervised%20Learning

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Semi-supervised Learning

• Rong Jin

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Semi-supervised learning

• Label propagation
• Transductive learning
• Co-training
• Active learning

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Label Propagation
Two labeled examples

• A toy problem
• Each node in the graph is an example
• Two examples are labeled
• Most examples are unlabeled
• Compute the similarity between examples Sij
• Connect examples to their most similar examples
• How to predicate labels for unlabeled nodes using
  this graph?

wij
Unlabeled example
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Label Propagation

• Forward propagation

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Label Propagation

• Forward propagation
• Forward propagation

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Label Propagation

• Forward propagation
• Forward propagation
• Forward propagation
• How to resolve conflicting cases

What label should be given to this node ?
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Label Propagation

• Let S be the similarity matrix SSi,jnxn
• Let D be a diagonal matrix where Di åi ¹ j Si,j
• Compute normalized similarity matrix S
• SD-1/2SD-1/2
• Let Y be the initial assignment of class labels
• Yi 1 when the i-th node is assigned to the
  positive class
• Yi -1 when the i-th node is assigned to the
  negative class
• Yi 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 Fi gt0
• The i-th node is assigned to the negative class
  if Fi lt 0

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Label Propagation

• Let S be the similarity matrix SSi,jnxn
• Let D be a diagonal matrix where Di åi ¹ j Si,j
• Compute normalized similarity matrix S
• SD-1/2SD-1/2
• Let Y be the initial assignment of class labels
• Yi 1 when the i-th node is assigned to the
  positive class
• Yi -1 when the i-th node is assigned to the
  negative class
• Yi 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 Fi gt0
• The i-th node is assigned to the negative class
  if Fi lt 0

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Label Propagation

• One iteration
• F Y aSY (I aS)Y
• a weights the propagation values
• Two iteration
• F Y aSY a2S2Y (I aS a2S2)Y
• How about the infinite iteration
• F (ån01anSn)Y (I - a S)-1Y
• Any problems with such an approach?

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Label Consistency Problem

• Predicted vector F may not be consistent with the
  initially assigned class labels Y

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Energy Minimization

• Using the same notation
• Si,j similarity between the I-th node and j-th
  node
• Y initially assigned class labels
• F predicted class labels
• Energy E(F) åi,jSi,j(Fi Fj)2
• Goal find label assignment F that is consistent
  with labeled examples Y and meanwhile minimizes
  the energy function E(F)

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Harmonic Function

• E(F) åi,jSi,j (Fi Fj)2 FT(D-S)F
• Thus, the minimizer for E(F) should be (D-S)F
  0, and meanwhile F should be consistent with Y.
• FT (FlT, FuT), YT (YlT, YuT)
• Fl Yl

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Optical Character Recognition

• Given an image of a digit letter, determine its
  value

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Optical Character Recognition

• Labeled_ExamplesUnlabeled_Examples 4000

• CMN label propagation
• 1NN for each unlabeled example, using the label
  of its closest neighbor

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Spectral Graph Transducer

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

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Spectral Graph Transducer

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

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Spectral Graph Transducer

• minF FTLF c (F-Y)TC(F-Y)
• s.t. FTFn, FTe 0
• C is the diagonal cost matrix, Ci,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

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Empirical Studies
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Problems with Spectral Graph Transducer

• minF FTLF c (F-Y)TC(F-Y)
• s.t. FTFn, FTe 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

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Greens Function

• The problem of minimizing energy and meanwhile
  being consistent with initially assigned class
  labels can be formulated into Greens function
  problem
• Minimizing E(F) FTLF ? LF 0
• Turns out L can be viewed as Laplacian operator
  in the discrete case
• LF 0 ? r2F0
• Thus, our problem is find solution F
• r2F0, 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

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Why Energy Minimization?
Final classification results
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Label Propagation

• How the unlabeled data help classification?

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Label Propagation

• How the unlabeled data help classification?

• Consider a smaller number of unlabeled example

• Classification results can be very different

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Cluster Assumption

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

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Cluster Assumption vs. Maximum Margin

• Maximum margin classifier (e.g. SVM)

w?xb

• Maximum margin
• ? low density around decision boundary
• ? Cluster assumption

• Any thought about utilizing the unlabeled data in
  support vector machine?

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Transductive SVM

• Decision boundary given a small number of labeled
  examples

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Transductive SVM

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

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Transductive SVM

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

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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?

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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

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Tranductive SVM
Different label assignment for unlabeled data ?
different maximum margin
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Transductive SVM Formulation
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Computational Issue

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

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Alternating Optimization

• Step 1 fix yn1,, ynm, learn weights w
• Step 2 fix weights w, try to predict yn1,,
  ynm (How?)

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Empirical Study with Transductive SVM

• 10 categories from the Reuter collection
• 3299 test documents
• 1000 informative words selected using MI criterion

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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

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Co-training for Semi-Supervised Learning
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Co-training for Semi-Supervised Learning
It is easier to classify this web page using
hyperlinks
It is easy to classify the type of this web page
based on its content
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Co-training

• Two representation for each web page

Content representation (doctoral, student,
computer, university)
Hyperlink representation Inlinks Prof.
Cheng Oulinks Prof. Cheng
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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

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Co-training

• Train a content-based classifier

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Co-training

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

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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

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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

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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