# Regression with Cost-based Rejection

arXiv: https://arxiv.org/abs/2311.04550 · HTML: https://mlsanity.com/paper/2311.04550/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 47 references that are in the map.
- First posted on arXiv 8 Nov 2023.

## Builds on (most cited first, 16 of 16)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [On Calibration of Modern Neural Networks](https://mlsanity.com/paper/1706.04599/) (2017)
- [Selective Classification for Deep Neural Networks](https://mlsanity.com/paper/1705.08500/) (2017)
- [A Survey of Deep Learning Techniques for Autonomous Driving](https://mlsanity.com/paper/1910.07738/) (2019)
- [SelectiveNet: A Deep Neural Network with an Integrated Reject Option](https://mlsanity.com/paper/1901.09192/) (2019)
- [AI Fairness 360: An Extensible Toolkit for Detecting, Understanding, and Mitigating Unwanted Algorithmic Bias](https://mlsanity.com/paper/1810.01943/) (2018)
- [Neural Legal Judgment Prediction in English](https://mlsanity.com/paper/1906.02059/) (2019)
- [A vector-contraction inequality for Rademacher complexities](https://mlsanity.com/paper/1605.00251/) (2016)
- [Classification with Rejection Based on Cost-sensitive Classification](https://mlsanity.com/paper/2010.11748/) (2020)
- [On the Calibration of Multiclass Classification with Rejection](https://mlsanity.com/paper/1901.10655/) (2019)
- [Selective Classification via One-Sided Prediction](https://mlsanity.com/paper/2010.07853/) (2020)
- [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)
- [An Embedding Framework for Consistent Polyhedral Surrogates](https://mlsanity.com/paper/1907.07330/) (2019)
- [Consistent Algorithms for Multiclass Classification with a Reject Option](https://mlsanity.com/paper/1505.04137/) (2015)
- [Large-scale Heteroscedastic Regression via Gaussian Process](https://mlsanity.com/paper/1811.01179/) (2018)

## Built on by (most cited first, 11 of 11)

- [Regression with Multi-Expert Deferral](https://mlsanity.com/paper/2403.19494/) (2024)
- [Realizable H-Consistent and Bayes-Consistent Loss Functions for Learning to Defer](https://mlsanity.com/paper/2407.13732/) (2024)
- [Mastering Multiple-Expert Routing: Realizable H-Consistency and Strong Guarantees for Learning to Defer](https://mlsanity.com/paper/2506.20650/) (2025)
- [Budgeted Multiple-Expert Deferral](https://mlsanity.com/paper/2510.26706/) (2025)
- [A Unifying Post-Processing Framework for Multi-Objective Learn-to-Defer Problems](https://mlsanity.com/paper/2407.12710/) (2024)
- [Learning to Partially Defer for Sequences](https://mlsanity.com/paper/2502.01459/) (2025)
- [Partial-Label Learning with a Reject Option](https://mlsanity.com/paper/2402.00592/) (2024)
- [PRISM: Festina Lente Proactivity -- Risk-Sensitive, Uncertainty-Aware Deliberation for Proactive Agents](https://mlsanity.com/paper/2602.01532/) (2026)
- [Optimized Deferral for Imbalanced Settings](https://mlsanity.com/paper/2604.27723/) (2026)
- [Calibrating conditional risk](https://mlsanity.com/paper/2604.20409/) (2026)
- [When Does Synthetic CT Transfer? A Label-Free Donor/Host Diagnostic for Medical Vision-Language Model Routing on Real Lung CT](https://mlsanity.com/paper/2606.29232/) (2026)

## Abstract (arXiv)

Learning with rejection is an important framework that can refrain from making predictions to avoid critical mispredictions by balancing between prediction and rejection. Previous studies on cost-based rejection only focused on the classification setting, which cannot handle the continuous and infinite target space in the regression setting. In this paper, we investigate a novel regression problem called regression with cost-based rejection, where the model can reject to make predictions on some examples given certain rejection costs. To solve this problem, we first formulate the expected risk for this problem and then derive the Bayes optimal solution, which shows that the optimal model should reject to make predictions on the examples whose variance is larger than the rejection cost when the mean squared error is used as the evaluation metric. Furthermore, we propose to train the model by a surrogate loss function that considers rejection as binary classification and we provide conditions for the model consistency, which implies that the Bayes optimal solution can be recovered by our proposed surrogate loss. Extensive experiments demonstrate the effectiveness of our proposed method.