发表机构
University of Chicago; University of Pennsylvania; University of Michigan（芝加哥大学; 宾夕法尼亚大学; 密歇根大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究分析二元保序回归的自由度，推导改进的精确自由度界，得出首个无分布的保序回归预期校准误差保证，为概率预测校准提供理论支撑。
AI 中文摘要
保序回归是估计单调函数和校准概率预测器的经典工具。我们针对二元样本给出了其最坏情况下自由度的完全精确有限样本刻画，明确了能最大化保序回归产生的不同拟合值数量的二元序列。我们运用解析数论推导出自由度的精确界，其主项为$\frac{3}{(4\pi^2)^{1/3}} n^{2/3}$，改进了此前的界。随后将该结果应用于校准：校准是概率预测的核心要求，保序回归是广泛用于提升校准的后处理方法。基于确定性自由度界，我们得出了据我们所知首个非平凡的无分布保序回归预期校准误差（ECE）保证，该ECE界完全无模型、无分布，仅假设$Y \in \{0,1\}$。
英文摘要
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 $Y \in \{0,1\}$.
