Ming Ji
Department of Computer Science
University of Illinois at Urbana-Champaign
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Information Networks: the Data
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Information networks
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Abstraction: graphs
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Data instances connected by edges representing certain relationships
Examples
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Telephone account networks linked by calls
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Email user networks linked by emails
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Social networks linked by friendship relations
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Twitter users linked by the ``follow” relation
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Webpage networks interconnected by hyperlinks in the World Wide
Web …
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Active Learning: the Problem
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Classical task: classification of the nodes in a graph
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Applications: terrorist email detection, fraud detection …
Why active learning
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Training classification models requires labels that are often very
expensive to obtain
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Different labeled data will train different learners
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Given an email network containing millions of users, we can only
sample a few users and ask experts to investigate whether they are
suspicious or not, and then use the labeled data to predict which users
are suspicious among all the users
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Active Learning: the Problem
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Problem definition of active learning
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Input: data and a classification model
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Output: find out which data examples (e.g., which users) should be
labeled such that the classifier could achieve higher prediction accuracy
over the unlabeled data as compared to random label selection
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Goal: maximize the learner's ability given a fixed budget of labeling
effort.
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Notations
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𝒱 = {𝑣1 , … , 𝑣𝑛 }: the set of nodes
𝒚 = 𝑦1 , … , 𝑦𝑛 𝑇 : the labels of the nodes
𝑊 = 𝑤𝑖𝑗 ∈ ℝ𝑛×𝑛 , where 𝑤𝑖𝑗 is the weight on the edge
between two nodes 𝑣𝑖 and 𝑣𝑗
Goal: find out a subset of nodes ℒ ⊂ 𝒱, such that the
classifier learned from the labels of ℒ could achieve the
smallest expected prediction error on the unlabeled data 𝒰 =
𝒱\ℒ, measured by 𝑣𝑖 ∈𝒰 𝑦𝑖 − 𝑦𝑖∗ 2 , where 𝑦𝑖∗ is the label
prediction for 𝑣𝑖
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Classification Model
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Gaussian random field
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exp
𝑍𝛽
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𝑃 𝒚 =

𝐸 𝒚 =
𝑤 𝑦 − 𝑦𝑗 : energy function measuring the
2 𝑖,𝑗 𝑖𝑗 𝑖
smoothness of a label assignment 𝒚 = 𝑦1 , … , 𝑦𝑛 𝑇
1
−𝛽𝐸 𝒚
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Label prediction
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Without loss of generality, we can arrange the data points chosen to
be labeled to be the first 𝑙 instances, i.e., ℒ = {𝑣1 , … , 𝑣𝑙 }
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Design constraint 𝒚∗ℒ = 𝑦1 , … , 𝑦𝑙 𝑇 , we want to predict 𝒚∗𝒰 with the
highest probability
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
𝐿
Let 𝐿 = 𝐷 − 𝑊 be the graph Laplacian, split 𝐿 as: 𝐿 = ( 𝑙𝑙
𝐿𝑢𝑙
Prediction: 𝒚∗𝒰 = −𝐿−1
𝑢𝑢 𝐿𝑢𝑙 𝒚ℒ
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𝐿𝑙𝑢
)
𝐿𝑢𝑢
The Variance Minimization Criterion
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Recall the goal of active learning
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Analyze the distribution of the Gaussian field conditioned on
the labeled data
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𝒚𝒰 ∼ 𝒩(𝒚∗𝒰 , 𝐿−1
𝑢𝑢 )
Compute the expected prediction error on the unlabeled
nodes
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Maximize the learner's ability ⟺ Minimize the error
E
𝑣𝑖 ∈𝒰
𝑦𝑖 − 𝑦𝑖∗
2
= Tr Cov 𝒚𝒰
= Tr 𝐿−1
𝑢𝑢
Choose the nodes to label such that the expected error (=
total variance) is minimized

argminℒ⊂𝒱 Tr 𝐿−1
𝑢𝑢
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Experimental Results on the Co-author Network
# of labels
VM
ERM
Random
LSC
Uncertainty
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50.4
47.0
41.7
30.0
39.4
50
62.2
54.7
50.7
54.3
54.1
Classification accuracy (%) comparison
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Experimental Results on the Isolet Data Set
Classification accuracy vs. the number of labels used
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Conclusions
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Publication: Ming Ji and Jiawei Han, “A Variance Minimization
Criterion to Active Learning on Graphs”, Proc. 2012 Int. Conf.
on Artificial Intelligence and Statistics (AISTAT'12), La Palma,
Canary Islands, April 2012.
Main advantages of the novel criterion proposed
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The first work to theoretically minimize the expected prediction error
of a classification model on networks/graphs
The only information used: the graph structure
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Do not need to know any label information
The data points do not need to have feature representation
Future work
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Test the assumptions and applicability of the criterion on real data
Study the expected error of other classification models
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