提出二阶校准误差的最优估计率，实现更精准的不确定性评估。
The Minimax Rate of Second-Order Calibration
- 用双曲正割扰动核构造解析校准函数，提升估计精度
- 理论证明最优收敛速率达 $\tilde{O}(1/\sqrt{n})$，优于传统方法
- 首个有限样本保证的二阶Platt校准，适合高阶预测器后处理
本文刻画了二分类中二阶校准误差估计的极小极大率，用于衡量高阶预测器的主观不确定性估计是否与标签概率在水平集上的条件方差一致。关键发现是，此前仅用于保证校准函数光滑性的双曲正割扰动核，实际上使校准函数在宽度为 $hπ/2$ 的带状区域内解析。基于多项式回归的估计方法达到 $\tilde{O}(1/\sqrt{n})$ 的收敛速率，且常数显式，定性优于桶化或核平滑的 $O(n^{-1/4})$ 速率。匹配的 $Ω(1/\sqrt{n})$ 下界表明该速率在对数因子内达到极小极大最优。作为推论，首次给出二阶Platt缩放的有限样本保证，提供一种可后处理任意高阶预测器均值与主观方差估计的校准方法。过程中还给出了无需分桶的二阶校准定义，并定量关联了Ahdritz等[2025]的桶化形式。实验验证了预测速率和校准后不确定性的质量。
原文摘要 · Abstract (English)
We characterize the minimax rate of estimating the second-order calibration error for binary classification, which quantifies whether a higher-order predictor's epistemic-uncertainty estimate matches the conditional variance of the label probability on its level sets. Our key observation is that the sech perturbation kernel, previously used only to enforce smoothness of calibration functions, in fact makes them analytic in a strip of half-width $hπ/2$. Polynomial regression then estimates the calibration error at rate $\tilde{O}(1/\sqrt{n})$, with explicit constants, a qualitative improvement over the $O(n^{-1/4})$ rate achievable by bucketing or kernel smoothing. A matching $Ω(1/\sqrt{n})$ lower bound establishes minimax optimality up to logarithmic factors. As a corollary, we give the first finite-sample guarantee for second-order Platt scaling, yielding a post-hoc procedure that recalibrates both the mean prediction and the epistemic-variance estimate of any higher-order predictor. Along the way, we provide a bucket-free definition of second-order calibration 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.
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