# When No-Rejection Learning is Consistent for Regression with Rejection

arXiv: https://arxiv.org/abs/2307.02932 · HTML: https://mlsanity.com/paper/2307.02932/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 56 references that are in the map.
- First posted on arXiv 6 Jul 2023.

## Builds on (most cited first, 14 of 14)

- [SelectiveNet: A Deep Neural Network with an Integrated Reject Option](https://mlsanity.com/paper/1901.09192/) (2019)
- [Consistent Estimators for Learning to Defer to an Expert](https://mlsanity.com/paper/2006.01862/) (2020)
- [Learning to Complement Humans](https://mlsanity.com/paper/2005.00582/) (2020)
- [Differentiable Learning Under Triage](https://mlsanity.com/paper/2103.08902/) (2021)
- [Who Should Predict? Exact Algorithms For Learning to Defer to Humans](https://mlsanity.com/paper/2301.06197/) (2023)
- [Calibrated Learning to Defer with One-vs-All Classifiers](https://mlsanity.com/paper/2202.03673/) (2022)
- [Learning to Defer to Multiple Experts: Consistent Surrogate Losses, Confidence Calibration, and Conformal Ensembles](https://mlsanity.com/paper/2210.16955/) (2022)
- [Sample Efficient Learning of Predictors that Complement Humans](https://mlsanity.com/paper/2207.09584/) (2022)
- [Classification Under Human Assistance](https://mlsanity.com/paper/2006.11845/) (2020)
- [On the Calibration of Multiclass Classification with Rejection](https://mlsanity.com/paper/1901.10655/) (2019)
- [Regression Under Human Assistance](https://mlsanity.com/paper/1909.02963/) (2019)
- [Selective Regression Under Fairness Criteria](https://mlsanity.com/paper/2110.15403/) (2021)
- [Regression with reject option and application to kNN](https://mlsanity.com/paper/2006.16597/) (2020)
- [Distribution-Free Model-Agnostic Regression Calibration via Nonparametric Methods](https://mlsanity.com/paper/2305.12283/) (2023)

## Built on by (most cited first, 2 of 2)

- [Learning to Route LLMs with Confidence Tokens](https://mlsanity.com/paper/2410.13284/) (2024)
- [Calibrating conditional risk](https://mlsanity.com/paper/2604.20409/) (2026)

## Abstract (arXiv)

Learning with rejection has been a prototypical model for studying the human-AI interaction on prediction tasks. Upon the arrival of a sample instance, the model first uses a rejector to decide whether to accept and use the AI predictor to make a prediction or reject and defer the sample to humans. Learning such a model changes the structure of the original loss function and often results in undesirable non-convexity and inconsistency issues. For the classification with rejection problem, several works develop consistent surrogate losses for the joint learning of the predictor and the rejector, while there have been fewer works for the regression counterpart. This paper studies the regression with rejection (RwR) problem and investigates a no-rejection learning strategy that uses all the data to learn the predictor. We first establish the consistency for such a strategy under the weak realizability condition. Then for the case without the weak realizability, we show that the excessive risk can also be upper bounded with the sum of two parts: prediction error and calibration error. Lastly, we demonstrate the advantage of such a proposed learning strategy with empirical evidence.