Focal Calibration Loss: Controlling Posterior Distortion in Deep Neural Classifiers

arXiv:2410.18321v3 Announce Type: replace-cross 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, L{FCL}^{gamma,lambda} = L{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.

Source: arXiv cs.CV | 2026-08-26

Related