揭示了二值等倾回归的自由度上限，为概率校准提供理论保障。
An analysis of binary isotonic regression: degrees of freedom and implications for calibration
- 通过解析数论推导出自由度的紧致上界，主导项为n^{2/3}阶。
- 首次给出无需分布假设的概率校准误差（ECE）非平凡界。
- 适用于任意二值标签的预测校准，无需模型假设。
等倾回归是估计单调函数和校准概率预测的经典工具。本文对二值样本下其最坏情况下的自由度给出了完全精确的有限样本刻画，识别出使拟合值数量最多的二值序列。利用解析数论，我们推导出一个紧致的自由度上界，主导项为 $\frac{3}{(4π^2)^{1/3}} n^{2/3}$，优于此前结果。基于此，我们构建了首个无需分布假设的概率校准误差（ECE）非平凡保证。该边界完全模型无关且分布无关，仅假设标签 $Y \in \{0,1\}$。
原文摘要 · Abstract (English)
Isotonic regression is a canonical tool for estimating monotone functions and calibrating probabilistic predictors. We provide a fully sharp finite-sample characterization of its worst-case degrees of freedom on binary samples. Specifically, we identify the binary sequences that maximize the number of distinct fitted values produced by isotonic regression. We develop a sharp bound on the degrees of freedom with a leading term of $\frac{3}{(4π^2)^{1/3}} n^{2/3}$ using analytic number theory, improving on previous bounds. We then apply this result to calibration. Calibration is a central requirement for probabilistic prediction, and isotonic regression is a widely used post-processing method for improving calibration. Building on deterministic degrees-of-freedom bounds, we derive, to our knowledge, the first nontrivial distribution-free guarantee on the Expected Calibration Error (ECE) of isotonic regression. This ECE bound is fully model-free and distribution-free, only assuming $Y \in \{0,1\}$.
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