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arXiv · 2607.27301

An analysis of binary isotonic regression: degrees of freedom and implications for calibration

Raphael Rossellini·Rina Foygel Barber·Zhimei Ren·Jake A. Soloff
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Abstract

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 $\frac{3}{(4\pi^2)^{1/3}} n^{2/3}$ 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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Raphael Rossellini, Rina Foygel Barber, Zhimei Ren, Jake A. Soloff. 2026-07-29. An analysis of binary isotonic regression: degrees of freedom and implications for calibration. https://arxiv.org/abs/2607.27301

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