Focal Calibration Loss: Controlling Posterior Distortion in Deep Neural Classifiers

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, LFCLγ,λ = Lfocalγ + λ |hatp(x) - ey|22. Our analysis separates two properties that are easily conflated: FCL is classification-calibrated for every γ, λ ≥ 0, preserving the Bayes decision rule, yet for γ > 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 √log K / λ for every posterior and minimizer without regularity assumptions, improving to O(1/λ) for interior posteriors, with an exact first-order expansion identifying the bias and corresponding population ℓ2 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 λ-2 law (slopes ≈ -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.