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. Ramaswamy1, Shivani Agarwal1 and Ambuj Tewari2

1Indian Institute of Science 2University of Michigan

Calibrated Surrogates

Binary Classification Y = Y = {±1} L = 1 1

• Minimize surrogate loss (e.g.

hinge) over R; learn f : X→R R Final prediction in {±1} : h(x) = sign(f(x)) General Multiclass Problem

(classes) (predictions)

Y = {1, . . . , n}; Y = {1, . . . , k}

(predictions)

L =   1 2 1 1 3 2 4 5 1   (classes) Minimize surrogate loss over Rd; learn f : X→Rd Rd Final prediction in {1, . . . , k} : h(x) = pred(f(x))

Convex Calibrated Surrogates for Low Rank Losses

  L  

n×k

=   A  

n×d

• B
• d×k

+ const Calibrated Convex Surrogate for L ψ∗

L(y, u) = d

• i=1

(ui − Ayi)2 pred∗

L(u) ∈ argmint∈[k] d

• i=1

uiBit

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

• Mean Average Precision

≤ r 2 X Pairwise Disagreement ≤ r 2 X

r = No. of docs. to be ranked

σ1 σ2 . . . ˆ y . . . σr! 00 . . . 00 00 . . . 01 . . . y . . . 11 . . . 11

Poster Sat35 Today