# Distributionally robust risk evaluation with an isotonic constraint

arXiv: https://arxiv.org/abs/2407.06867 · HTML: https://mlsanity.com/paper/2407.06867/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 47 of its 116 references that are in the map.
- First posted on arXiv 9 Jul 2024.

## Builds on (most cited first, 30 of 47)

- [Distribution-Free Predictive Inference For Regression](https://mlsanity.com/paper/1604.04173/) (2016)
- [Conformal Prediction Under Covariate Shift](https://mlsanity.com/paper/1904.06019/) (2019)
- [Learning Models with Uniform Performance via Distributionally Robust Optimization](https://mlsanity.com/paper/1810.08750/) (2018)
- [Data-driven Distributionally Robust Optimization Using the Wasserstein Metric: Performance Guarantees and Tractable Reformulations](https://mlsanity.com/paper/1505.05116/) (2015)
- [Statistics of Robust Optimization: A Generalized Empirical Likelihood Approach](https://mlsanity.com/paper/1610.03425/) (2016)
- [Variance-based regularization with convex objectives](https://mlsanity.com/paper/1610.02581/) (2016)
- [Conformal Inference of Counterfactuals and Individual Treatment Effects](https://mlsanity.com/paper/2006.06138/) (2020)
- [Distributionally Robust Logistic Regression](https://mlsanity.com/paper/1509.09259/) (2015)
- [A survey on domain adaptation theory: learning bounds and theoretical guarantees](https://mlsanity.com/paper/2004.11829/) (2020)
- [Transfer Learning for High-dimensional Linear Regression: Prediction, Estimation, and Minimax Optimality](https://mlsanity.com/paper/2006.10593/) (2020)
- [Support and Invertibility in Domain-Invariant Representations](https://mlsanity.com/paper/1903.03448/) (2019)
- [Robust Validation: Confident Predictions Even When Distributions Shift](https://mlsanity.com/paper/2008.04267/) (2020)
- [Transfer Learning under High-dimensional Generalized Linear Models](https://mlsanity.com/paper/2105.14328/) (2021)
- [Transfer Learning for Nonparametric Classification: Minimax Rate and Adaptive Classifier](https://mlsanity.com/paper/1906.02903/) (2019)
- [On the Value of Target Data in Transfer Learning](https://mlsanity.com/paper/2002.04747/) (2020)
- [Sensitivity Analysis of Individual Treatment Effects: A Robust Conformal Inference Approach](https://mlsanity.com/paper/2111.12161/) (2021)
- [Optimally tackling covariate shift in RKHS-based nonparametric regression](https://mlsanity.com/paper/2205.02986/) (2022)
- [Conformalized Survival Analysis](https://mlsanity.com/paper/2103.09763/) (2021)
- [Doubly Robust Calibration of Prediction Sets under Covariate Shift](https://mlsanity.com/paper/2203.01761/) (2022)
- [Bounds on the conditional and average treatment effect with unobserved confounding factors](https://mlsanity.com/paper/1808.09521/) (2018)
- [SLOPE - Adaptive variable selection via convex optimization](https://mlsanity.com/paper/1407.3824/) (2014)
- [Sensitivity analysis for inverse probability weighting estimators via the percentile bootstrap](https://mlsanity.com/paper/1711.11286/) (2017)
- [A new similarity measure for covariate shift with applications to nonparametric regression](https://mlsanity.com/paper/2202.02837/) (2022)
- [Isotonic Distributional Regression](https://mlsanity.com/paper/1909.03725/) (2019)
- [Prediction Sets Adaptive to Unknown Covariate Shift](https://mlsanity.com/paper/2203.06126/) (2022)
- [Classifier Calibration with ROC-Regularized Isotonic Regression](https://mlsanity.com/paper/2311.12436/) (2023)
- [Conformalized survival analysis with adaptive cutoffs](https://mlsanity.com/paper/2211.01227/) (2022)
- [Causal isotonic calibration for heterogeneous treatment effects](https://mlsanity.com/paper/2302.14011/) (2023)
- [Not all distributional shifts are equal: Fine-grained robust conformal inference](https://mlsanity.com/paper/2402.13042/) (2024)
- [Maximum Likelihood Estimation is All You Need for Well-Specified Covariate Shift](https://mlsanity.com/paper/2311.15961/) (2023)

## Built on by (most cited first, 3 of 3)

- [Conformal Prediction: A Data Perspective](https://mlsanity.com/paper/2410.06494/) (2024)
- [Conformal Inference under High-Dimensional Covariate Shifts via Likelihood-Ratio Regularization](https://mlsanity.com/paper/2502.13030/) (2025)
- [Representation-Aware Distributionally Robust Optimization: A Knowledge Transfer Framework](https://mlsanity.com/paper/2509.09371/) (2025)

## Abstract (arXiv)

Statistical learning under distribution shift is challenging when neither prior knowledge nor fully accessible data from the target distribution is available. Distributionally robust learning (DRL) aims to control the worst-case statistical performance within an uncertainty set of candidate distributions, but how to properly specify the set remains challenging. To enable distributional robustness without being overly conservative, in this paper, we propose a shape-constrained approach to DRL, which incorporates prior information about the way in which the unknown target distribution differs from its estimate. More specifically, we assume the unknown density ratio between the target distribution and its estimate is isotonic with respect to some partial order. At the population level, we provide a solution to the shape-constrained optimization problem that does not involve the isotonic constraint. At the sample level, we provide consistency results for an empirical estimator of the target in a range of different settings. Empirical studies on both synthetic and real data examples demonstrate the improved accuracy of the proposed shape-constrained approach.