Multiclass Learning on the Simplex

Binary classification naturally separates scoring from decision making: a model outputs a score, and a threshold turns it into a label. By contrast, in multiclass classification, softmax outputs are usually followed by argmax, which fixes one decision rule in advance. In this paper, we present a simplex-based view in which multiclass prediction is interpreted as choosing a family of class regions on the probability simplex Sm over m classes. A threshold vector τ∈Sm induces such regions on the simplex, with argmax recovered at the barycenter, and threshold tuning becomes post-hoc selection of better decision regions for a target score. The same geometric framework also supports threshold-aware training. We formalize this through a common score functional defined on probability measures over threshold vectors: Dirac measures recover fixed decision rules such as argmax and post-hoc threshold tuning, while general measures recover threshold-aware training. In this view, multiclass classification is not only prediction followed by argmax, but prediction on the simplex together with a choice of decision regions.