Fenchel-Young losses are a family of convex loss functions, encompassing the
squared, logistic and sparsemax losses, among others. Each Fenchel-Young loss
is implicitly associated with a link function, for mapping model outputs to
predictions. For instance, the logistic loss is associated with the soft argmax
link function. Can we build new loss functions associated with the same link
function as Fenchel-Young losses? In this paper, we introduce Fitzpatrick
losses, a new family of convex loss functions based on the Fitzpatrick
function. A well-known theoretical tool in maximal monotone operator theory,
the Fitzpatrick function naturally leads to a refined Fenchel-Young inequality,
making Fitzpatrick losses tighter than Fenchel-Young losses, while maintaining
the same link function for prediction. As an example, we introduce the
Fitzpatrick logistic loss and the Fitzpatrick sparsemax loss, counterparts of
the logistic and the sparsemax losses. This yields two new tighter losses
associated with the soft argmax and the sparse argmax, two of the most
ubiquitous output layers used in machine learning. We study in details the
properties of Fitzpatrick losses and in particular, we show that they can be
seen as Fenchel-Young losses using a modified, target-dependent generating
function. We demonstrate the effectiveness of Fitzpatrick losses for label
proportion estimation.

引入了 Fitzpatrick 损失函数，它是一种基于 Fitzpatrick 函数的新型凸损失函数，与 Fenchel-Young 损失函数具有相同的链接函数，但更紧致，同时保持预测的链接函数。