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Ranking and Machine Learning The Lov´ asz-Bregman divergences Properties of the Lov´ asz-Bregman Applications Summary The Lov´ asz-Bregman Divergence and Connections to Rank Aggregation, Clustering, and Web Ranking Rishabh Iyer Jeff Bilmes University of Washington, Seattle UAI-2013 MELODI M achin E L earning, O ptimization, & D ata I nterpretation @ UW Iyer & Bilmes, 2013 Lov´ asz Bregman Divergences page 1 / 24

Ranking and Machine Learning The Lov´ asz-Bregman divergences Properties of the Lov´ asz-Bregman Applications Summary Outline Ranking and Machine Learning 1 The Lov´ asz-Bregman divergences 2 Properties of the Lov´ asz-Bregman 3 Applications 4 Summary 5 Iyer & Bilmes, 2013 Lov´ asz Bregman Divergences page 2 / 24

Ranking and Machine Learning The Lov´ asz-Bregman divergences Properties of the Lov´ asz-Bregman Applications Summary Combining Scores and Rankings Occur in a number of Machine Learning applications: Iyer & Bilmes, 2013 Lov´ asz Bregman Divergences page 3 / 24

Ranking and Machine Learning The Lov´ asz-Bregman divergences Properties of the Lov´ asz-Bregman Applications Summary Combining Scores and Rankings Occur in a number of Machine Learning applications: Combining Classifiers (Lebanon & Lafferty, 2002) Iyer & Bilmes, 2013 Lov´ asz Bregman Divergences page 3 / 24

Ranking and Machine Learning The Lov´ asz-Bregman divergences Properties of the Lov´ asz-Bregman Applications Summary Combining Scores and Rankings Occur in a number of Machine Learning applications: 1) Munich 1) Seattle 1) Munich 2) Paris 2) Munich 2) Seattle 3) London 3) London 3) London 4) Seattle 4) Atlanta 4) Paris 5) Atlanta 5) Paris 5) Atlanta Aggregating Preferences Combining Classifiers (Murphy & Martin, (Lebanon & Lafferty, 2002) 2003) Iyer & Bilmes, 2013 Lov´ asz Bregman Divergences page 3 / 24

Ranking and Machine Learning The Lov´ asz-Bregman divergences Properties of the Lov´ asz-Bregman Applications Summary Combining Scores and Rankings Occur in a number of Machine Learning applications: 1) Munich 1) Seattle 1) Munich 2) Paris 2) Munich 2) Seattle 3) London 3) London 3) London 4) Seattle 4) Atlanta 4) Paris 5) Atlanta 5) Paris 5) Atlanta Aggregating Preferences Combining Classifiers (Murphy & Martin, (Lebanon & Lafferty, 2002) Web Ranking (Liu, 2009) 2003) Iyer & Bilmes, 2013 Lov´ asz Bregman Divergences page 3 / 24

Ranking and Machine Learning The Lov´ asz-Bregman divergences Properties of the Lov´ asz-Bregman Applications Summary Combining Scores and Rankings Denote σ as a permutation of { 1 , 2 , · · · , n } such that σ ( i ) denotes the item at rank i and σ − 1 ( i ) as the rank of item i . Iyer & Bilmes, 2013 Lov´ asz Bregman Divergences page 4 / 24

Ranking and Machine Learning The Lov´ asz-Bregman divergences Properties of the Lov´ asz-Bregman Applications Summary Combining Scores and Rankings Denote σ as a permutation of { 1 , 2 , · · · , n } such that σ ( i ) denotes the item at rank i and σ − 1 ( i ) as the rank of item i . Denote { σ 1 , σ 2 , . . . , σ k } as a set of k permutations. Iyer & Bilmes, 2013 Lov´ asz Bregman Divergences page 4 / 24

Ranking and Machine Learning The Lov´ asz-Bregman divergences Properties of the Lov´ asz-Bregman Applications Summary Combining Scores and Rankings Denote σ as a permutation of { 1 , 2 , · · · , n } such that σ ( i ) denotes the item at rank i and σ − 1 ( i ) as the rank of item i . Denote { σ 1 , σ 2 , . . . , σ k } as a set of k permutations. Some important problems concerning rankings: Iyer & Bilmes, 2013 Lov´ asz Bregman Divergences page 4 / 24

Ranking and Machine Learning The Lov´ asz-Bregman divergences Properties of the Lov´ asz-Bregman Applications Summary Combining Scores and Rankings Denote σ as a permutation of { 1 , 2 , · · · , n } such that σ ( i ) denotes the item at rank i and σ − 1 ( i ) as the rank of item i . Denote { σ 1 , σ 2 , . . . , σ k } as a set of k permutations. Some important problems concerning rankings: Combining Permutations: Given permutations σ 1 , σ 2 , · · · , σ k , find 1 a representative σ , which is “close“ to σ 1 , σ 2 , · · · , σ k . Iyer & Bilmes, 2013 Lov´ asz Bregman Divergences page 4 / 24

Ranking and Machine Learning The Lov´ asz-Bregman divergences Properties of the Lov´ asz-Bregman Applications Summary Combining Scores and Rankings Denote σ as a permutation of { 1 , 2 , · · · , n } such that σ ( i ) denotes the item at rank i and σ − 1 ( i ) as the rank of item i . Denote { σ 1 , σ 2 , . . . , σ k } as a set of k permutations. Some important problems concerning rankings: Combining Permutations: Given permutations σ 1 , σ 2 , · · · , σ k , find 1 a representative σ , which is “close“ to σ 1 , σ 2 , · · · , σ k . Combining Scores: Given a set of score vectors x 1 , x 2 , · · · , x k , find 2 a representative σ , which is “close“ to x 1 , x 2 , · · · , x k . Iyer & Bilmes, 2013 Lov´ asz Bregman Divergences page 4 / 24

Ranking and Machine Learning The Lov´ asz-Bregman divergences Properties of the Lov´ asz-Bregman Applications Summary Combining Scores and Rankings Denote σ as a permutation of { 1 , 2 , · · · , n } such that σ ( i ) denotes the item at rank i and σ − 1 ( i ) as the rank of item i . Denote { σ 1 , σ 2 , . . . , σ k } as a set of k permutations. Some important problems concerning rankings: Combining Permutations: Given permutations σ 1 , σ 2 , · · · , σ k , find 1 a representative σ , which is “close“ to σ 1 , σ 2 , · · · , σ k . Combining Scores: Given a set of score vectors x 1 , x 2 , · · · , x k , find 2 a representative σ , which is “close“ to x 1 , x 2 , · · · , x k . Clustering: Cluster the set of permutations σ 1 , σ 2 , · · · , σ k (or 3 equivalently score vectors x 1 , x 2 , · · · , x k ). Iyer & Bilmes, 2013 Lov´ asz Bregman Divergences page 4 / 24

Ranking and Machine Learning The Lov´ asz-Bregman divergences Properties of the Lov´ asz-Bregman Applications Summary Rank aggregation Combine a set of rankings σ 1 , σ 2 , · · · , σ k . Iyer & Bilmes, 2013 Lov´ asz Bregman Divergences page 5 / 24

Ranking and Machine Learning The Lov´ asz-Bregman divergences Properties of the Lov´ asz-Bregman Applications Summary Rank aggregation Combine a set of rankings σ 1 , σ 2 , · · · , σ k . Rank Aggregation . . . Iyer & Bilmes, 2013 Lov´ asz Bregman Divergences page 5 / 24

Ranking and Machine Learning The Lov´ asz-Bregman divergences Properties of the Lov´ asz-Bregman Applications Summary Rank aggregation Combine a set of rankings σ 1 , σ 2 , · · · , σ k . Rank Aggregation . . . Often done using permutation based distance metrics. Iyer & Bilmes, 2013 Lov´ asz Bregman Divergences page 5 / 24

Ranking and Machine Learning The Lov´ asz-Bregman divergences Properties of the Lov´ asz-Bregman Applications Summary Permutation based Distance Metrics d ( σ, π ) Metric on the space of permutations. Iyer & Bilmes, 2013 Lov´ asz Bregman Divergences page 6 / 24

Ranking and Machine Learning The Lov´ asz-Bregman divergences Properties of the Lov´ asz-Bregman Applications Summary Permutation based Distance Metrics d ( σ, π ) Metric on the space of permutations. Kendall τ , � I ( σ − 1 π ( i ) > σ − 1 π ( j )) d T ( σ, π ) = i , j , i < j and Spearman’s footrule: n � | σ − 1 ( i ) − π − 1 ( i ) | d S ( σ, π ) = i =1 Iyer & Bilmes, 2013 Lov´ asz Bregman Divergences page 6 / 24

Ranking and Machine Learning The Lov´ asz-Bregman divergences Properties of the Lov´ asz-Bregman Applications Summary Permutation based Distance Metrics d ( σ, π ) Metric on the space of permutations. Kendall τ , � I ( σ − 1 π ( i ) > σ − 1 π ( j )) d T ( σ, π ) = i , j , i < j and Spearman’s footrule: n � | σ − 1 ( i ) − π − 1 ( i ) | d S ( σ, π ) = i =1 Invariance with respect to re-orderings – i.e d ( πσ, πτ ) = d ( σ, τ ). Iyer & Bilmes, 2013 Lov´ asz Bregman Divergences page 6 / 24

Ranking and Machine Learning The Lov´ asz-Bregman divergences Properties of the Lov´ asz-Bregman Applications Summary Permutation based Distance Metrics d ( σ, π ) Metric on the space of permutations. Kendall τ , � I ( σ − 1 π ( i ) > σ − 1 π ( j )) d T ( σ, π ) = i , j , i < j and Spearman’s footrule: n � | σ − 1 ( i ) − π − 1 ( i ) | d S ( σ, π ) = i =1 Invariance with respect to re-orderings – i.e d ( πσ, πτ ) = d ( σ, τ ). Given a set of permutations σ 1 , σ 2 , · · · , σ k , find a permutation σ : k � σ = argmin d ( σ i , π ) (1) π i =1 Iyer & Bilmes, 2013 Lov´ asz Bregman Divergences page 6 / 24

Ranking and Machine Learning The Lov´ asz-Bregman divergences Properties of the Lov´ asz-Bregman Applications Summary Score Aggregation What if one has scores instead of just the orderings? For example, Iyer & Bilmes, 2013 Lov´ asz Bregman Divergences page 7 / 24