# CAIRO: Decoupling Order from Scale in Regression

arXiv: https://arxiv.org/abs/2602.14440 · HTML: https://mlsanity.com/paper/2602.14440/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 11 of its 41 references that are in the map.
- First posted on arXiv 16 Feb 2026.

## Builds on (most cited first, 11 of 11)

- [Fast Differentiable Sorting and Ranking](https://mlsanity.com/paper/2002.08871/) (2020)
- [Differentiable Ranks and Sorting using Optimal Transport](https://mlsanity.com/paper/1905.11885/) (2019)
- [SoftSort: A Continuous Relaxation for the argsort Operator](https://mlsanity.com/paper/2006.16038/) (2020)
- [Regression Diagnostics meets Forecast Evaluation: Conditional Calibration, Reliability Diagrams, and Coefficient of Determination](https://mlsanity.com/paper/2108.03210/) (2021)
- [You Are the Best Reviewer of Your Own Papers: An Owner-Assisted Scoring Mechanism](https://mlsanity.com/paper/2110.14802/) (2021)
- [Generic Conditions for Forecast Dominance](https://mlsanity.com/paper/1805.09902/) (2018)
- [Nonparametric Shape-restricted Regression](https://mlsanity.com/paper/1709.05707/) (2017)
- [Isotonic Recalibration under a Low Signal-to-Noise Ratio](https://mlsanity.com/paper/2301.02692/) (2023)
- [Model selection with Gini indices under auto-calibration](https://mlsanity.com/paper/2207.14372/) (2022)
- [Isotonic Mechanism for Exponential Family Estimation in Machine Learning Peer Review](https://mlsanity.com/paper/2304.11160/) (2023)
- [An Isotonic Mechanism for Overlapping Ownership](https://mlsanity.com/paper/2306.11154/) (2023)

## Abstract (arXiv)

Standard regression methods typically optimize a single pointwise objective, such as mean squared error, which conflates the learning of ordering with the learning of scale. This coupling renders models vulnerable to outliers and heavy-tailed noise. We propose CAIRO (Calibrate After Initial Rank Ordering), a framework that decouples regression into two distinct stages. In the first stage, we learn a scoring function by minimizing a scale-invariant ranking loss; in the second, we recover the target scale via isotonic regression. We theoretically characterize a class of "Optimal-in-Rank-Order" objectives -- including variants of RankNet and Gini covariance -- and prove that they recover the ordering of the true conditional mean under mild assumptions. We further show that subsequent monotone calibration recovers the true regression function at the population level and mathematically guarantees that finite-sample predictions are strictly auto-calibrated. Empirically, CAIRO combines the representation learning of neural networks with the robustness of rank-based statistics. It matches the performance of state-of-the-art tree ensembles on tabular benchmarks and significantly outperforms standard regression objectives in regimes with heavy-tailed or heteroskedastic noise.