To learn s, we then simply use standard stochastic gradient descent and minimize a binary cross-entropy loss between the probability estimates and the preference values in the training data. Since this loss is a function of the learned δ values only, to ensure identifiability of the scores s, and to guarantee that these are centered around the real origin, we use a regularization term L reg (ŝ; λ) := λ ∥ŝ∥ 2 , where ∥•∥ is the Euclidean norm and λ is a user-defined hyperparameter. Empirically, we found that setting small values λ ≃ 10 -6 is enough to encourage the desired score behaviour for our use case.