The Minimax Rate of Perturbed Second-Order Calibration

Second-order calibration error quantifies how closely a higher-order predictor's epistemic-uncertainty estimate matches the conditional variance of the label probability on its level sets. We characterize the minimax rate of estimating the second-order calibration error for binary classification in the regime where a small perturbation is applied to the classifier outputs. Our procedure is simple: add independent bandwidth-h sech noise to the score coordinates, then regress Y(1) and Y(1)Y(2) on the perturbed score using low-degree polynomials. Crucially, the sech perturbation makes the calibration functions analytic in a suitable strip. The resulting estimator has error Oh(log3/2n/√ n), with explicit constants. In the same setting, a matching Omega(1/√n) lower bound establishes minimax optimality up to logarithmic factors. As a corollary, we give a finite-sample guarantee for perturbed second-order Platt scaling, yielding a post-hoc procedure that recalibrates both the mean prediction and the epistemic-variance estimate of the perturbed higher-order predictor. Along the way, we give an explicit two-moment formulation of second-order calibration error and relate it quantitatively to the bucketed formulation of Ahdritz et al. [2025]. Our experiments confirm the predicted rate and the quality of the recalibrated uncertainties.