In label ranking, the problem is to learn a mapping from instances to rankings over a finite set of predefined class labels. In this
paper, we consider a generalization of this problem, namely label ranking with a reject option. Just like in conventional classification, where a classifier can refuse a presumably unreliable prediction, the idea is to concede a label ranker the possibility to abstain. More specifically, the label ranker is allowed to make a weaker prediction in the form of a partial instead of a total order. Thus, unlike a conventional classifier which either makes a prediction or not, a label ranker can abstain to a certain degree. To realize label ranking with a reject option, we propose a method based on ensemble learning techniques. First empirical results are presented showing great promise for the usefulness of the approach.
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Label Ranking with Partial Abstention using Ensemble Learning
1. Weiwei Cheng & Eyke Hüllermeier
Knowledge Engineering & Bioinformatics Lab
Department of Mathematics and Computer Science
University of Marburg, Germany
2. Label Ranking (an example)
Learning geeks’ preferences on hotels
Golf ≻ Park ≻ Krim
label ranking
Krim ≻ Golf ≻ Park
geek 1
Krim ≻ Park ≻ Golf
geek 2
Park ≻ Golf ≻ Krim
geek 3
geek 4
new geek ???
where the geek could be described by feature vectors,
e.g., (gender, age, place of birth, is a professor, …)
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3. Label Ranking (an example)
Learning geeks’ preferences on hotels
Golf Park Krim
geek 1 1 2 3
geek 2 2 3 1
geek 3 3 2 1
geek 4 2 1 3
new geek ? ? ?
π(i) = position of the i-th label in the ranking
1: Golf 2: Park 3: Krim
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4. Label Ranking (more formally)
Given:
a set of training instances
a set of labels
for each training instance : a set of pairwise preferences
of the form (for some of the labels)
Find:
A ranking function ( mapping) that maps each
to a ranking of (permutation ) and generalizes well
in terms of a loss function on rankings (e.g., Kendall’s tau)
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5. Existing Approaches
Constraint classification
Har-Peled , Roth, and Zimak, NIPS-03
Log linear models for label ranking
Dekel, Manning, and Singer, NIPS-03
Label ranking by learning pairwise preferences
Hüllermeier, Fürnkranz, Cheng, and Brinker, Artificial Intelligence
Decision tree and instance-based learning for label ranking
Cheng, Hühn, and Hüllermeier, ICML-09
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6. Learning with Reject Option
usnews.com To train a learner that is able to say
“I don’t know”.
laptoplogic.com
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7. Label Ranking with Reject Option
predict a≻b or b≻a, or
For each pair of labels a and b, the learner can
abstain from prediction (reject option).
The learner should be consistent (transitivity).
partial orders
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9. Label Ranking Ensemble
≻1 , ≻2 , ……. , ≻k .
For a query, setup a label ranking ensemble of size k
Define a partial order with
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10. Two Problems
If a≻b and b≻c, then a≻c. If a≻b and b≻c, then not c≻a.
problem Transitivity No cycle
Get transitive closure with
solution to be solved
Marshall’s algorithm.
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11. Proposition
Given a set of total orders on a finite set , denote by Pab
any triple of elements a, b, c ∈ , we have
the proportion of orders in which a precedes b. Then, for
Pca ≤ 2 − Pab − Pbc .
S.t. (Pab ≥ 2/3) ⋀ (Pbc ≥ 2/3) ⟹ (Pca ≤ 2/3)
Choosing t > 2/3, we can guarantee acyclic.
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14. Our Contributions
A first attempt on label ranking with reject option;
Output a reliably partial ranking with ensemble
learning.
Follow up!
The End