# Functional Properties of the Focal-Entropy

arXiv: https://arxiv.org/abs/2603.02533 · HTML: https://mlsanity.com/paper/2603.02533/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 11 of its 31 references that are in the map.
- First posted on arXiv 3 Mar 2026.

## Builds on (most cited first, 11 of 11)

- [Class-Balanced Loss Based on Effective Number of Samples](https://mlsanity.com/paper/1901.05555/) (2019)
- [Learning Imbalanced Datasets with Label-Distribution-Aware Margin Loss](https://mlsanity.com/paper/1906.07413/) (2019)
- [Generalized Focal Loss: Learning Qualified and Distributed Bounding Boxes for Dense Object Detection](https://mlsanity.com/paper/2006.04388/) (2020)
- [Balanced Meta-Softmax for Long-Tailed Visual Recognition](https://mlsanity.com/paper/2007.10740/) (2020)
- [Calibrating Deep Neural Networks using Focal Loss](https://mlsanity.com/paper/2002.09437/) (2020)
- [Dice Loss for Data-imbalanced NLP Tasks](https://mlsanity.com/paper/1911.02855/) (2019)
- [Sparse Sequence-to-Sequence Models](https://mlsanity.com/paper/1905.05702/) (2019)
- [PolyLoss: A Polynomial Expansion Perspective of Classification Loss Functions](https://mlsanity.com/paper/2204.12511/) (2022)
- [AdaFocal: Calibration-aware Adaptive Focal Loss](https://mlsanity.com/paper/2211.11838/) (2022)
- [Unifying Heterogeneous Classifiers with Distillation](https://mlsanity.com/paper/1904.06062/) (2019)
- [On Focal Loss for Class-Posterior Probability Estimation: A Theoretical Perspective](https://mlsanity.com/paper/2011.09172/) (2020)

## Abstract (arXiv)

The focal-loss has become a widely used alternative to cross-entropy in class-imbalanced classification problems, particularly in computer vision. Despite its empirical success, a systematic information-theoretic study of the focal-loss remains incomplete. In this work, we adopt a distributional viewpoint and study the focal-entropy, a focal-loss analogue of the cross-entropy. Our analysis establishes conditions for finiteness, convexity, and continuity of the focal-entropy, and provides various asymptotic characterizations. We prove the existence and uniqueness of the focal-entropy minimizer, describe its structure, and show that it can depart significantly from the data distribution. In particular, we rigorously show that the focal-loss amplifies mid-range probabilities, suppresses high-probability outcomes, and, under extreme class imbalance, induces an over-suppression regime in which very small probabilities are further diminished. These results, which are also experimentally validated, offer a theoretical foundation for understanding the focal-loss and clarify the trade-offs that it introduces when applied to imbalanced learning tasks.