The Minimax Rate of Perturbed Second-Order Calibration

תקציר מקורי באנגליתarXiv:2605.07808v2 Announce Type: replace Abstract: 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 $O_h(\log^{3/2}n/\sqrt n)$, with explicit constants. I