# Regression Under Human Assistance

arXiv: https://arxiv.org/abs/1909.02963 · HTML: https://mlsanity.com/paper/1909.02963/

## Key facts

- Cited by 40 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 6 Sep 2019.

## Built on by (most cited first, 30 of 40)

- [Differentiable Learning Under Triage](https://mlsanity.com/paper/2103.08902/) (2021)
- [Towards Unbiased and Accurate Deferral to Multiple Experts](https://mlsanity.com/paper/2102.13004/) (2021)
- [Who Should Predict? Exact Algorithms For Learning to Defer to Humans](https://mlsanity.com/paper/2301.06197/) (2023)
- [Improving Expert Predictions with Conformal Prediction](https://mlsanity.com/paper/2201.12006/) (2022)
- [Principled Approaches for Learning to Defer with Multiple Experts](https://mlsanity.com/paper/2310.14774/) (2023)
- [Regression with Multi-Expert Deferral](https://mlsanity.com/paper/2403.19494/) (2024)
- [Designing Decision Support Systems Using Counterfactual Prediction Sets](https://mlsanity.com/paper/2306.03928/) (2023)
- [Realizable H-Consistent and Bayes-Consistent Loss Functions for Learning to Defer](https://mlsanity.com/paper/2407.13732/) (2024)
- [Learning to Defer with Limited Expert Predictions](https://mlsanity.com/paper/2304.07306/) (2023)
- [Towards Human-AI Complementarity with Prediction Sets](https://mlsanity.com/paper/2405.17544/) (2024)
- [Uncertainty Quantification on Clinical Trial Outcome Prediction](https://mlsanity.com/paper/2401.03482/) (2024)
- [Mastering Multiple-Expert Routing: Realizable H-Consistency and Strong Guarantees for Learning to Defer](https://mlsanity.com/paper/2506.20650/) (2025)
- [Reinforcement Learning Under Algorithmic Triage](https://mlsanity.com/paper/2109.11328/) (2021)
- [Training Data Subset Selection for Regression with Controlled Generalization Error](https://mlsanity.com/paper/2106.12491/) (2021)
- [A Causal Framework for Evaluating Deferring Systems](https://mlsanity.com/paper/2405.18902/) (2024)
- [Efficient Data Subset Selection to Generalize Training Across Models: Transductive and Inductive Networks](https://mlsanity.com/paper/2409.12255/) (2024)
- [Budgeted Multiple-Expert Deferral](https://mlsanity.com/paper/2510.26706/) (2025)
- [A2C: A Modular Multi-stage Collaborative Decision Framework for Human-AI Teams](https://mlsanity.com/paper/2401.14432/) (2024)
- [Learning Complementary Policies for Human-AI Teams](https://mlsanity.com/paper/2302.02944/) (2023)
- [AI, Meet Human: Learning Paradigms for Hybrid Decision Making Systems](https://mlsanity.com/paper/2402.06287/) (2024)
- [Towards Human-AI Complementarity in Matching Tasks](https://mlsanity.com/paper/2508.13285/) (2025)
- [Neural Estimation of Submodular Functions with Applications to Differentiable Subset Selection](https://mlsanity.com/paper/2210.11033/) (2022)
- [A Unifying Post-Processing Framework for Multi-Objective Learn-to-Defer Problems](https://mlsanity.com/paper/2407.12710/) (2024)
- [Confounding-Robust Policy Improvement with Human-AI Teams](https://mlsanity.com/paper/2310.08824/) (2023)
- [Learning To Guide Human Decision Makers With Vision-Language Models](https://mlsanity.com/paper/2403.16501/) (2024)
- [How to Enable Effective Cooperation Between Humans and NLP Models: A Survey of Principles, Formalizations, and Beyond](https://mlsanity.com/paper/2501.05714/) (2025)
- [Unraveling Human-AI Teaming: A Review and Outlook](https://mlsanity.com/paper/2504.05755/) (2025)
- [Narrowing Action Choices with AI Improves Human Sequential Decisions](https://mlsanity.com/paper/2510.16097/) (2025)
- [When No-Rejection Learning is Consistent for Regression with Rejection](https://mlsanity.com/paper/2307.02932/) (2023)
- [Submodular Maximization under the Intersection of Matroid and Knapsack Constraints](https://mlsanity.com/paper/2307.09487/) (2023)

## Abstract (arXiv)

Decisions are increasingly taken by both humans and machine learning models. However, machine learning models are currently trained for full automation -- they are not aware that some of the decisions may still be taken by humans. In this paper, we take a first step towards the development of machine learning models that are optimized to operate under different automation levels. More specifically, we first introduce the problem of ridge regression under human assistance and show that it is NP-hard. Then, we derive an alternative representation of the corresponding objective function as a difference of nondecreasing submodular functions. Building on this representation, we further show that the objective is nondecreasing and satisfies α-submodularity, a recently introduced notion of approximate submodularity. These properties allow a simple and efficient greedy algorithm to enjoy approximation guarantees at solving the problem. Experiments on synthetic and real-world data from two important applications -- medical diagnosis and content moderation-demonstrate that our algorithm outsources to humans those samples in which the prediction error of the ridge regression model would have been the highest if it had to make a prediction, it outperforms several competitive baselines, and its performance is robust with respect to several design choices and hyperparameters used in the experiments.