Geometric Insights into Focal Loss: Reducing Curvature for Enhanced Model Calibration

Abstract

The key factor in implementing machine learning algorithms in decision-making situations is not only the accuracy of the model but also its confidence level. The confidence level of a model in a classification problem is often given by the output vector of a softmax function for convenience. However, these values are known to deviate significantly from the actual expected model confidence. This problem is called model calibration and has been studied extensively. One of the simplest techniques to tackle this task is focal loss, a generalization of cross-entropy by introducing one positive parameter. Although many related studies exist because of the simplicity of the idea and its formalization, the theoretical analysis of its behavior is still insufficient. In this study, our objective is to understand the behavior of focal loss by reinterpreting this function geometrically. Our analysis suggests that focal loss reduces the curvature of the loss surface in training the model. This indicates that curvature may be one of the essential factors in achieving model calibration. We design numerical experiments to support this conjecture to reveal the behavior of focal loss and the relationship between calibration performance and curvature.

Research links

• arXiv

How to cite

Masanari Kimura, Hiroki Naganuma. “Geometric Insights into Focal Loss: Reducing Curvature for Enhanced Model Calibration.” Pattern Recognition Letters, 2025.

@article{Kimura2025Geometric,
  title = {Geometric Insights into Focal Loss: Reducing Curvature for Enhanced Model Calibration},
  author = {Masanari Kimura and Hiroki Naganuma},
  year = {2025},
  journal = {Pattern Recognition Letters},
  eprint = {2405.00442},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2405.00442}
}