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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\u2014we call it the <jats:italic>Lambda Bivariate<\/jats:italic> (<jats:sc>LB<\/jats:sc>) framework\u2014that allows to learn effective bivariate scoring functions for ranking using gradient boosting trees. We discuss the three main ingredients of <jats:sc>LB<\/jats:sc>: (<jats:italic>i<\/jats:italic>) the invariance to permutations property, (<jats:italic>ii<\/jats:italic>) the function aggregating the scores of all pairs into the per-item scores, and (<jats:italic>iii<\/jats:italic>) the optimization process to learn bivariate scoring functions for ranking using any differentiable loss functions. We apply <jats:sc>LB<\/jats:sc> to the <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\lambda$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>\u03bb<\/mml:mi>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula><jats:sc>Rank<\/jats:sc> loss and we show that it results in learning a bivariate version of <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\lambda$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>\u03bb<\/mml:mi>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula><jats:sc>MART<\/jats:sc>\u2014we call it <jats:sc>Bi-<\/jats:sc><jats:inline-formula><jats:alternatives><jats:tex-math>$$\\lambda$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>\u03bb<\/mml:mi>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula><jats:sc>MART<\/jats:sc>\u2014that significantly outperforms all neural-network-based and tree-based state-of-the-art algorithms for Learning-to-Rank. To show the generality of <jats:sc>LB<\/jats:sc> with respect to other loss functions, we also discuss its application to the <jats:sc>Softmax<\/jats:sc> loss.<\/jats:p>","DOI":"10.1007\/s10791-024-09444-7","type":"journal-article","created":{"date-parts":[[2024,9,27]],"date-time":"2024-09-27T12:02:04Z","timestamp":1727438524000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Learning bivariate scoring functions for ranking"],"prefix":"10.1007","volume":"27","author":[{"given":"Franco Maria","family":"Nardini","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Roberto","family":"Trani","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Rossano","family":"Venturini","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2024,9,27]]},"reference":[{"issue":"3","key":"9444_CR1","doi-asserted-by":"publisher","first-page":"225","DOI":"10.1561\/1500000016","volume":"3","author":"T Liu","year":"2009","unstructured":"Liu T. 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