Learning with Fitzpatrick Losses

TL;DR

This paper introduces Fitzpatrick losses, a new family of convex loss functions based on the Fitzpatrick function, which are tighter than Fenchel-Young losses while preserving the same link functions, and demonstrates their effectiveness in label proportion estimation.

Contribution

The paper proposes Fitzpatrick losses, a novel family of convex losses that are tighter than Fenchel-Young losses and maintain the same link functions, expanding the toolkit for machine learning models.

Findings

Fitzpatrick losses are tighter than Fenchel-Young losses.

Fitzpatrick logistic and sparsemax losses are introduced as new counterparts.

Fitzpatrick losses improve label proportion estimation accuracy.

Abstract

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.