Functional Properties of the Focal-Entropy
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866914364866428928 |
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| author | Shah, Jaimin Cardone, Martina Dytso, Alex |
| author_facet | Shah, Jaimin Cardone, Martina Dytso, Alex |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_02533 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Functional Properties of the Focal-Entropy Shah, Jaimin Cardone, Martina Dytso, Alex Information Theory Computer Vision and Pattern Recognition Machine Learning Statistics Theory 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. |
| title | Functional Properties of the Focal-Entropy |
| topic | Information Theory Computer Vision and Pattern Recognition Machine Learning Statistics Theory |
| url | https://arxiv.org/abs/2603.02533 |