Focal Calibration Loss: Controlling Posterior Distortion in Deep Neural Classifiers

[2410.18321] Focal Calibration Loss: Controlling Posterior Distortion in Deep Neural Classifiers Skip to main content Search Submit Donate Log in Computer Science > Machine Learning arXiv:2410.18321 (cs) [Submitted on 23 Oct 2024 ( v1 ), last revised 25 Aug 2026 (this version, v3)] Title: Focal Calibration Loss: Controlling Posterior Distortion in Deep Neural Classifiers Authors: Wenhao Liang , Liangwei Zheng , Wei Zhang , Weitong Chen View PDF HTML (experimental) Abstract: Confidence calibration matters wherever a classifier's probabilities, not just its labels, are consumed downstream. We study Focal Calibration Loss (FCL), which adds a squared probability-error (multiclass Brier) anchor to the focal objective, $\mathcal{L}{\mathrm{FCL}}^{\gamma,\lambda} = \mathcal{L}{\mathrm{focal}}^{\gamma} + \lambda |\hat{p}(x) - e_y|_2^2$. Our analysis separates two properties that are easily conflated: FCL is classification-calibrated for every $\gamma, \lambda \ge 0$, preserving the Bayes decision rule, yet for $\gamma > 0$ it is generally not proper, so its Bayes-optimal probability vector is displaced from the true posterior. The main result quantifies that displacement and shows the anchor controls it: bounded by $\sqrt{\log K / \lambda}$ for every posterior and minimizer without regularity assumptions, improving to $O(1/\lambda)$ for interior posteriors, with an exact first-order expansion identifying the bias and corresponding population $\ell_2$ calibration guarantees. We verify these population statements directly, minimizing the conditional risk on the simplex with no network involved: the posterior-distortion rate matches its prediction to a median fitted slope of $-0.994$, and exact population squared calibration error follows the predicted $\lambda^{-2}$ law (slopes $\approx -1.99$). Across CIFAR-10/100, Tiny-ImageNet, text and medical multi-label tasks, FCL is competitive rather than dominant, and the picture is regime- and metric-dependent: under a common validation-split protocol the validation-adaptive AdaFocal attains lower binned calibration error, while FCL attains lower NLL, Brier and error on two of three settings. On transformers its calibration advantage is absent, and a from-scratch experiment tested and did not support the conjecture that pretraining explains this. We report both the gains and the failure regimes. Comments: V3 Subjects: Machine Learning (cs.LG) ; Computer Vision and Pattern Recognition (cs.CV); Machine Learning (stat.ML) Cite as: arXiv:2410.18321 [cs.LG] (or arXiv:2410.18321v3 [cs.LG] for this version) https://doi.org/10.48550/arXiv.2410.18321 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Wenhao Liang [ view email ] [v1] Wed, 23 Oct 2024 23:06:50 UTC (14,932 KB) [v2] Thu, 7 Aug 2025 02:15:03 UTC (12,071 KB) [v3] Tue, 25 Aug 2026 05:57:00 UTC (1,079 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Additional Features Audio Summary Current browse context: cs.LG < prev | next > new | recent | 2024-10 Change to browse by: cs cs.CV stat stat.ML References & Citations SciX Google Scholar Semantic Scholar export BibTeX citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) IArxiv recommender toggle IArxiv Recommender ( What is IArxiv? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. 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