Learning to Rank A Brief Review

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Learning to Rank --A Brief Review

• Yunpeng Xu

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Ranking and sorting

• Rank only has K structured categories
• Sorting each sample has a distinct rank
• Generally, no need to differentiate them

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Overview

• Rank aggregation
• Label ranking
• Query and rank by example
• Preference learning
• Problems left, what we can do?

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Ranking aggregation

• Needs of combining different ranking results
• Voting systems, welfare economics, decision
  making
• 1. Hillary Clinton gt John Edwards gt Barack Obama
• 2. Barack Obama gtJohn Edwards gt Hillary Clinton
• gt ?

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Ranking aggregation (cont.)

• Arrows impossibility theorem
• Kenneth Arrow, 1951

If the decision-making body has at least two
members and at least three options to decide
among, then it is impossible to design a social
welfare function that satisfies all these
conditions at once.
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Ranking aggregation (cont.)

• Arrows impossibility theorem
• 5 fair assumptions
• non-dictatorship, unrestricted domain or
  universality, independence of irrelevant
  alternatives, positive association of social and
  individual values or monotonicity, non-imposition
  or citizen sovereignty
• Cannot be satisfied simultaneously

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Ranking aggregation (cont.)

• Bordas method (1971)
• Given lists , each has n items
• For each
• Define as the number of items rank below
  j in
• Rank all items by
• Hillary Clinton 2, John Edwards 2, Barack
  Obama 2

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Ranking aggregation (cont.) -- Border

• Condorcet Criteria
• If the majority prefers x to y, then x must be
  ranked above y
• Borders method does not satisfy CC, neither any
  method that assigns weights to each rank position

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Ranking aggregation (cont.)

• Assumption relaxation
• Maximize consensus criteria
• Equivalent to minimize disagreement (Kemeny,
  Social Choice Theorem)
• NP Hard!
• Sub-optimal solutions using heuristics

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Ranking aggregation (cont.)

• Basic idea
• Assign different weights to different experts
• Supervised aggregation
• Weighting according to a final judger (ground
  truth)
• Unsupervised aggregation
• Aims to minimize the disagreement measured by
  certain distances

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Ranking aggregation (cont.)

• Distance measure
• Spearman footrule distance
• Kendal tau distance
• Kendal tau distance for multiple lists
• Scaled footrule distance

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Ranking aggregation (cont.) -Distance Measure

• Kemeny optimal ranking
• Minimizing Kendal distance
• Still NP-Hard to compute
• Local Kemenization (local optimal aggregation)
• Can be computed in O(knlogn)

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Ranking aggregation (cont.)

• Supervised Ranking Aggregation (SRA WWW07)
• Ground truth preference matrix H
• Example
• Goal rank by the score
• It can be seen that , or with
  relaxation

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Ranking aggregation (cont.) -- SRA

• Method
• Use Bordas score
• Objective

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Ranking aggregation (cont.)

• Markov Chain Rank Aggregation (MCRA, WWW05)
• Map a ranked list to a Markov Chain M
• Compute the stationary distribution of M
• Rank items based on
• Example
• B gt C gt D
• A gt D gt E
• A gt B gt E

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Ranking aggregation (cont.) - MCRA

• Different transition strategies
• MC1
• all out-degree edges have uniform
  probabilities
• MC2
• choose a list, then choose next item on the
  list
• For disconnected graph, define transition
  probability based on measure item similarity

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Ranking aggregation (cont.)

• Unsupervised Learning Algorithm for Rank
  Aggregation (ULARA Dan Roth ECML07)
• Goal
• Method maximize agreement

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Ranking aggregation (cont.) - UCLRA

• Method
• Algorithm iterative gradient decent
• Initially, w is uniform, then updated iteratively

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Overview

• Rank aggregation
• Label ranking
• Query and rank by example
• Preference learning
• Problems left, what we can do?

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Label Ranking

• Goal Map from the input space to the set of
  total order over a finite set of labels
• Related to multi-label or multi-class problems

Input Customer information Output Porsche gt
Toyota gt Ford
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Label Ranking (cont.)

• Pairwise ranking (ECML03)
• Train a classifier for each pair of labels
• When judge on an example
• If the classifier predicts , then
  count it as a vote on
• Then rank all labels according to their votes
• Total classifiers

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Label Ranking (cont.)

• Constraint Classification (NIPS 02)
• Consider a linear sorting function
• Goal learn the values of
• rank all labels by the score

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Label Ranking (cont.) -- CC

• Expand the feature vector
• Generate positive/ negative samples in

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Label Ranking (cont.) -- CC

• Learn a separating hyper plane
• Can be solved by SVM

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Overview

• Rank aggregation
• Label ranking
• Query and rank by example
• Preference learning
• Problems left, what we can do?

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Query and rank by example

• Given one query, rank retrieved items according
  to their relevancy w.r.t the query.

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Query and rank by example (cont.)

• Rank on manifold
• Convergence form
• Essentially, this is an one-class semi-supervised
  method

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Preference learning

• Given a set of items, and a set of user
  preference over these items, to rank all items
  according to the user preference.
• Motivated by the needs of personalized search.

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Preference learning

• Input
• preference a set of partial order
  on X
• Output a total order on X
• or, map X onto a structured label space Y
• Preference function

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Existing methods

• Learning to order things W. Cohen 98
• Large margin ordinal regression R. Herbrich 98
• PRanking with Ranking K Crammer 01
• Optimizing Search Engines using Clickthrough Data
  T Joachims 02
• Efficient boosting algorithm for combining
  preferences Yoav Freund 03
• Classification Approach towards Ranking and
  Sorting Problems S Rajaram 03

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Existing methods

• Learning to Rank using Gradient Descent C Burges
  05
• Stability and Generalization of Bipartite Ranking
  S Agarwal 05
• Generalization Bounds for k-Partite RankingS
  Rajaram 05
• Ranking with a p-norm push C Rudin 05
• Magnitutde-Preserving Ranking Algorithms C
  Cortes 07
• From Pairwise Approach to Listwise Z Cao 07

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Large Margin Ordinal Regression

• Mapping to an axis using inner product

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Large Margin Ordinal Regression

• Consider
• Then
• Introduce soft margin
• Solve using SVM

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Learn to order things

• A greedy ordering algorithm to order things

Calculate a score for each item
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Learn to order things (cont.)

• Combine different ranking functions
• To learn the weight iteratively

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Learn to order things
Combine preference functions
Do ranking aggregation
Update weights based on feedbacks
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• Initially, w is uniform
• At each step
• Compute a combined ranking function
• Produce a ranking aggregation
• Measure the loss

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RankBoost

• Bipartite ranking problems
• Combine weaker rankers
• Sort based on values of H(x)

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RankBoost (cont.)
Sampling distribution Initialization

• Bipartite ranking problem

Learn weak ranker
Sampling distribution updation
normalization
Combine weak rankers
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Stability and Generalization

• Bipartite ranking problems
• Expected rank error
• Empirical rank error

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Stability and Generalization (cont.)

• Stability
• Remove one training sample, how much changes
• Generalization
• Generalize to k-partite ranking problem

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Rank on graph data

• Objective

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P-norm push

• Focus on the topmost ranked items
• The top left region is the most important

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P-norm push (cont.)

• Height of k (k is a negative sample)
• Cost of sample k
• g is convex, monotonically
  incresasing

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p-norm push

• Run RankBoost to solve the problem

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Thanks!