The Multiclass Score-Oriented Loss (MultiSOL) on the Simplex

In the supervised binary classification setting, score-oriented losses have been introduced with the aim of optimizing a chosen performance metric directly during the training phase, thus avoiding \textit{a posteriori} threshold tuning. To do this, in their construction, the decision threshold is treated as a random variable provided with a certain \textit{a priori} distribution. In this paper, we use a recently introduced multidimensional threshold-based classification framework to extend such score-oriented losses to multiclass classification, defining the Multiclass Score-Oriented Loss (MultiSOL) functions. As also demonstrated by several classification experiments, this proposed family of losses is designed to preserve the main advantages observed in the binary setting, such as the direct optimization of the target metric and the robustness to class imbalance, achieving performance comparable to other state-of-the-art loss functions and providing new insights into the interaction between simplex geometry and score-oriented learning.