# Learning from a Biased Sample

arXiv: https://arxiv.org/abs/2209.01754 · HTML: https://mlsanity.com/paper/2209.01754/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- Builds on 26 of its 74 references that are in the map.
- First posted on arXiv 5 Sep 2022.

## Builds on (most cited first, 26 of 26)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [Distributionally Robust Neural Networks for Group Shifts: On the Importance of Regularization for Worst-Case Generalization](https://mlsanity.com/paper/1911.08731/) (2019)
- [Nonparametric regression using deep neural networks with ReLU activation function](https://mlsanity.com/paper/1708.06633/) (2017)
- [Smart "Predict, then Optimize"](https://mlsanity.com/paper/1710.08005/) (2017)
- [Learning Models with Uniform Performance via Distributionally Robust Optimization](https://mlsanity.com/paper/1810.08750/) (2018)
- [Quasi-Oracle Estimation of Heterogeneous Treatment Effects](https://mlsanity.com/paper/1712.04912/) (2017)
- [Data-driven Distributionally Robust Optimization Using the Wasserstein Metric: Performance Guarantees and Tractable Reformulations](https://mlsanity.com/paper/1505.05116/) (2015)
- [Multitask learning and benchmarking with clinical time series data](https://mlsanity.com/paper/1703.07771/) (2017)
- [Robust Estimators in High Dimensions without the Computational Intractability](https://mlsanity.com/paper/1604.06443/) (2016)
- [Large-Scale Methods for Distributionally Robust Optimization](https://mlsanity.com/paper/2010.05893/) (2020)
- [Orthogonal Statistical Learning](https://mlsanity.com/paper/1901.09036/) (2019)
- [From Predictive to Prescriptive Analytics](https://mlsanity.com/paper/1402.5481/) (2014)
- [Distributionally Robust Language Modeling](https://mlsanity.com/paper/1909.02060/) (2019)
- [MIMIC-Extract: A Data Extraction, Preprocessing, and Representation Pipeline for MIMIC-III](https://mlsanity.com/paper/1907.08322/) (2019)
- [Learning from Untrusted Data](https://mlsanity.com/paper/1611.02315/) (2016)
- [Offline Multi-Action Policy Learning: Generalization and Optimization](https://mlsanity.com/paper/1810.04778/) (2018)
- [Distributionally Robust Losses for Latent Covariate Mixtures](https://mlsanity.com/paper/2007.13982/) (2020)
- [Bounds on the conditional and average treatment effect with unobserved confounding factors](https://mlsanity.com/paper/1808.09521/) (2018)
- [Doubly-Valid/Doubly-Sharp Sensitivity Analysis for Causal Inference with Unmeasured Confounding](https://mlsanity.com/paper/2112.11449/) (2021)
- [Regularizing towards Causal Invariance: Linear Models with Proxies](https://mlsanity.com/paper/2103.02477/) (2021)
- [Distributionally robust stochastic programs with side information based on trimmings -- Extended version](https://mlsanity.com/paper/2009.10592/) (2020)
- [Distributionally Robust Models with Parametric Likelihood Ratios](https://mlsanity.com/paper/2204.06340/) (2022)
- [Evaluating Robustness to Dataset Shift via Parametric Robustness Sets](https://mlsanity.com/paper/2205.15947/) (2022)
- [Covariate Balancing Sensitivity Analysis for Extrapolating Randomized Trials across Locations](https://mlsanity.com/paper/2112.04723/) (2021)

## Built on by (most cited first, 14 of 14)

- [Diagnosing Model Performance Under Distribution Shift](https://mlsanity.com/paper/2303.02011/) (2023)
- [Domain constraints improve risk prediction when outcome data is missing](https://mlsanity.com/paper/2312.03878/) (2023)
- [Distributionally Robust Policy Learning under Concept Drifts](https://mlsanity.com/paper/2412.14297/) (2024)
- [Efficient and robust transfer learning of optimal individualized treatment regimes with right-censored survival data](https://mlsanity.com/paper/2301.05491/) (2023)
- [DRO: A Python Library for Distributionally Robust Optimization in Machine Learning](https://mlsanity.com/paper/2505.23565/) (2025)
- [Distributionally robust risk evaluation with an isotonic constraint](https://mlsanity.com/paper/2407.06867/) (2024)
- [A Directional Rockafellar-Uryasev Regression](https://mlsanity.com/paper/2411.02557/) (2024)
- [High-Dimensional Gaussian Mean Estimation under Realizable Contamination](https://mlsanity.com/paper/2603.16798/) (2026)
- [On Prediction Feature Assignment in the Heckman Selection Model](https://mlsanity.com/paper/2309.08043/) (2023)
- [A Semiparametric Instrumented Difference-in-Differences Approach to Policy Learning](https://mlsanity.com/paper/2310.09545/) (2023)
- [Data Fusion for High-Resolution Estimation](https://mlsanity.com/paper/2508.14858/) (2025)
- [Policy Learning with Observational Data: The Case of Hepatitis C Treatment for HIV/HCV Co-Infected Patients](https://mlsanity.com/paper/2605.16593/) (2026)
- [Achieving First-Order Statistical Improvements in Data-Driven Optimization: From No-Free-Lunch to Amplified Decision Perturbation](https://mlsanity.com/paper/2608.04312/) (2026)
- [Learning Risk Scores Robust to Unobserved Confounders](https://mlsanity.com/paper/2609.27144/) (2026)

## Abstract (arXiv)

The empirical risk minimization approach to data-driven decision making requires access to training data drawn under the same conditions as those that will be faced when the decision rule is deployed. However, in a number of settings, we may be concerned that our training sample is biased in the sense that some groups (characterized by either observable or unobservable attributes) may be under- or over-represented relative to the general population; and in this setting empirical risk minimization over the training set may fail to yield rules that perform well at deployment. We propose a model of sampling bias called conditional Gamma-biased sampling, where observed covariates can affect the probability of sample selection arbitrarily much but the amount of unexplained variation in the probability of sample selection is bounded by a constant factor. Applying the distributionally robust optimization framework, we propose a method for learning a decision rule that minimizes the worst-case risk incurred under a family of test distributions that can generate the training distribution under Gamma-biased sampling. We apply a result of Rockafellar and Uryasev to show that this problem is equivalent to an augmented convex risk minimization problem. We give statistical guarantees for learning a model that is robust to sampling bias via the method of sieves, and propose a deep learning algorithm whose loss function captures our robust learning target. We empirically validate our proposed method in a case study on prediction of mental health scores from health survey data and a case study on ICU length of stay prediction.