Learning bivariate scoring functions for ranking

Abstract State-of-the-art Learning-to-Rank algorithms, e.g., $$\lambda$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>λ</mml:mi> </mml:math> MART , rely on univariate scoring functions to score a list of items. Univariate scoring functions score each item independently, i.e., without considering the other available items in the list. Nevertheless, ranking deals with producing an effective ordering of the items and comparisons between items are helpful to achieve this task. Bivariate scoring functions allow the model to exploit dependencies between the items in the list as they work by scoring pairs of items. In this paper, we exploit item dependencies in a novel framework—we call it the Lambda Bivariate ( LB ) framework—that allows to learn effective bivariate scoring functions for ranking using gradient boosting trees. We discuss the three main ingredients of LB : ( i ) the invariance to permutations property, ( ii ) the function aggregating the scores of all pairs into the per-item scores, and ( iii ) the optimization process to learn bivariate scoring functions for ranking using any differentiable loss functions. We apply LB to the $$\lambda$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>λ</mml:mi> </mml:math> Rank loss and we show that it results in learning a bivariate version of $$\lambda$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>λ</mml:mi> </mml:math> MART —we call it Bi- $$\lambda$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>λ</mml:mi> </mml:math> MART —that significantly outperforms all neural-network-based and tree-based state-of-the-art algorithms for Learning-to-Rank. To show the generality of LB with respect to other loss functions, we also discuss its application to the Softmax loss.