# The Minimax Rate of Perturbed Second-Order Calibration

arXiv: https://arxiv.org/abs/2605.07808 · HTML: https://mlsanity.com/paper/2605.07808/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 20 of its 34 references that are in the map.
- First posted on arXiv 8 May 2026.

## Builds on (most cited first, 20 of 20)

- [On Calibration of Modern Neural Networks](https://mlsanity.com/paper/1706.04599/) (2017)
- [Simple and Scalable Predictive Uncertainty Estimation using Deep Ensembles](https://mlsanity.com/paper/1612.01474/) (2016)
- [What Uncertainties Do We Need in Bayesian Deep Learning for Computer Vision?](https://mlsanity.com/paper/1703.04977/) (2017)
- [Aleatoric and Epistemic Uncertainty in Machine Learning: An Introduction to Concepts and Methods](https://mlsanity.com/paper/1910.09457/) (2019)
- [Evidential Deep Learning to Quantify Classification Uncertainty](https://mlsanity.com/paper/1806.01768/) (2018)
- [Predictive Uncertainty Estimation via Prior Networks](https://mlsanity.com/paper/1802.10501/) (2018)
- [Verified Uncertainty Calibration](https://mlsanity.com/paper/1909.10155/) (2019)
- [Evaluating model calibration in classification](https://mlsanity.com/paper/1902.06977/) (2019)
- [Quantifying Aleatoric and Epistemic Uncertainty in Machine Learning: Are Conditional Entropy and Mutual Information Appropriate Measures?](https://mlsanity.com/paper/2209.03302/) (2022)
- [Epistemic Neural Networks](https://mlsanity.com/paper/2107.08924/) (2021)
- [Mitigating Bias in Calibration Error Estimation](https://mlsanity.com/paper/2012.08668/) (2020)
- [Calibration tests in multi-class classification: A unifying framework](https://mlsanity.com/paper/1910.11385/) (2019)
- [Distribution-free binary classification: prediction sets, confidence intervals and calibration](https://mlsanity.com/paper/2006.10564/) (2020)
- [Smooth ECE: Principled Reliability Diagrams via Kernel Smoothing](https://mlsanity.com/paper/2309.12236/) (2023)
- [A Unifying Theory of Distance from Calibration](https://mlsanity.com/paper/2211.16886/) (2022)
- [Second-Order Uncertainty Quantification: A Distance-Based Approach](https://mlsanity.com/paper/2312.00995/) (2023)
- [T-Cal: An optimal test for the calibration of predictive models](https://mlsanity.com/paper/2203.01850/) (2022)
- [Quantifying Aleatoric and Epistemic Uncertainty with Proper Scoring Rules](https://mlsanity.com/paper/2404.12215/) (2024)
- [Provable Uncertainty Decomposition via Higher-Order Calibration](https://mlsanity.com/paper/2412.18808/) (2024)
- [Measuring Uncertainty Calibration](https://mlsanity.com/paper/2512.13872/) (2025)

## Abstract (arXiv)

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.