Convex Calibrated Surrogates for Low-Rank Loss Matrices with - PowerPoint PPT Presentation

Convex Calibrated Surrogates for Low-Rank Loss Matrices with Applications to Subset Ranking Losses Harish G. Ramaswamy 1 , Shivani Agarwal 1 and Ambuj Tewari 2 1 Indian Institute of Science 2 University of Michigan

Calibrated Surrogates General Multiclass Problem Binary Classification (classes) (predictions) Y = { 1 , . . . , n } ; � Y = { 1 , . . . , k } Y = � Y = {± 1 }   (predictions) � 0 � 0 1 2 1 1 L =   (classes) L = 1 0 3 2 1 0 4 5 0 1 Minimize surrogate loss (e.g. Minimize surrogate loss over R d ; learn f : X→ R d hinge) over R ; learn f : X→ R R d R 0 0 Final prediction in {± 1 } : Final prediction in { 1 , . . . , k } : h ( x ) = sign ( f ( x )) h ( x ) = pred ( f ( x ))

Convex Calibrated Surrogates for Low Rank Losses     � �    A  L = B + const d × k n × k n × d Calibrated Convex Surrogate for L d d � � ( u i − A yi ) 2 ψ ∗ pred ∗ L ( y , u ) = L ( u ) ∈ argmin t ∈ [ k ] u i B it i = 1 i = 1

Application to Subset Ranking Exponential sized loss matrices with low rank. Loss matrix Rank Efficient predictor NDCG r � Precision@q r � Expected Rank Utility r � ≤ r 2 Mean Average Precision X ≤ r 2 Pairwise Disagreement X r = No. of docs. to be ranked ˆ y σ 1 σ 2 . . . . . . σ r ! 00 . . . 00 00 . . . 01 . Poster Sat35 . . Today y . . . 11 . . . 11