Supplementary Material for: Learning Structured Models with the AUC Loss and Its Generalizations

We have shown that with our enhanced representation, the ranking problem for given weights w reduces to the one in Joachims (2005) in the case of a fully-factored model. Here we show a similar result for the learning problem. Recall that our learning objective is defined as: min w λ 2 w 2 + 1 M m max z∈Z w ϕ(x m , z) + ∆ AU C (z, y m) − maxˆz∈Zˆz∼y m w ϕ(x m , ˆ z) (1) We would like to consider this objective in the case of a fully-factored model, where the single element scores are simply: w x i ≡ a i , and therefore: w ϕ(x, z) = i a i k z ki. Our goal is to show that the loss in this case is a simpler function of w. For a specific training example, we denote: z * = argmax z∈Z w ϕ(x m , z) + ∆ AU C (z, y m) (2) ˆ z * = argmaxˆz∈Z,ˆ z∼y m w ϕ(x m , ˆ z) (3) Due to the decomposition of the score and the AUC loss, z * is obtained by sorting a vector which consists of elements a i − c for i ∈ pos and a j for j ∈ neg, with c = 1/(|pos| · |neg|). On the other hand, ˆ z * is obtained by sorting the elements a i for all i ∈ pos, then sorting the elements a j for j ∈ neg, and then concatenating the results with pos before neg. W.l.g. assume that the elements are indexed according to the rankingˆz *. We next define indicator variables for all pairs i ∈ pos, j ∈ neg: y * ij =