Focal Calibration Loss: Controlling Posterior Distortion in Deep Neural Classifiers

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arXiv:2410.18321v3 Announce Type: replace 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 …

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