# Causal Inference Under Unmeasured Confounding With Negative Controls: A Minimax Learning Approach

arXiv: https://arxiv.org/abs/2103.14029 · HTML: https://mlsanity.com/paper/2103.14029/

## Key facts

- Cited by 44 papers in the map (data through 14 Oct 2026).
- Builds on 22 of its 65 references that are in the map.
- First posted on arXiv 25 Mar 2021.

## Builds on (most cited first, 22 of 22)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Training GANs with Optimism](https://mlsanity.com/paper/1711.00141/) (2017)
- [Doubly Robust Policy Evaluation and Optimization](https://mlsanity.com/paper/1503.02834/) (2015)
- [Orthogonal Statistical Learning](https://mlsanity.com/paper/1901.09036/) (2019)
- [Minimax Weight and Q-Function Learning for Off-Policy Evaluation](https://mlsanity.com/paper/1910.12809/) (2019)
- [Identifying Causal Effects With Proxy Variables of an Unmeasured Confounder](https://mlsanity.com/paper/1609.08816/) (2016)
- [Non-convex Optimization for Machine Learning](https://mlsanity.com/paper/1712.07897/) (2017)
- [Kernel Instrumental Variable Regression](https://mlsanity.com/paper/1906.00232/) (2019)
- [An Introduction to Proximal Causal Learning](https://mlsanity.com/paper/2009.10982/) (2020)
- [Deep Generalized Method of Moments for Instrumental Variable Analysis](https://mlsanity.com/paper/1905.12495/) (2019)
- [Off-Policy Evaluation via the Regularized Lagrangian](https://mlsanity.com/paper/2007.03438/) (2020)
- [Minimax Estimation of Conditional Moment Models](https://mlsanity.com/paper/2006.07201/) (2020)
- [Off-Policy Evaluation in Partially Observable Environments](https://mlsanity.com/paper/1909.03739/) (2019)
- [Semiparametric proximal causal inference](https://mlsanity.com/paper/2011.08411/) (2020)
- [A Kernel Loss for Solving the Bellman Equation](https://mlsanity.com/paper/1905.10506/) (2019)
- [Adversarial Estimation of Riesz Representers](https://mlsanity.com/paper/2101.00009/) (2020)
- [Adversarial Generalized Method of Moments](https://mlsanity.com/paper/1803.07164/) (2018)
- [Augmented Minimax Linear Estimation](https://mlsanity.com/paper/1712.00038/) (2017)
- [Provably Efficient Neural Estimation of Structural Equation Model: An Adversarial Approach](https://mlsanity.com/paper/2007.01290/) (2020)
- [Nonparametric causal effects based on incremental propensity score interventions](https://mlsanity.com/paper/1704.00211/) (2017)
- [Kernel Methods for Unobserved Confounding: Negative Controls, Proxies, and Instruments](https://mlsanity.com/paper/2012.10315/) (2020)
- [The Variational Method of Moments](https://mlsanity.com/paper/2012.09422/) (2020)

## Built on by (most cited first, 30 of 44)

- [Proximal Reinforcement Learning: Efficient Off-Policy Evaluation in Partially Observed Markov Decision Processes](https://mlsanity.com/paper/2110.15332/) (2021)
- [A Minimax Learning Approach to Off-Policy Evaluation in Confounded Partially Observable Markov Decision Processes](https://mlsanity.com/paper/2111.06784/) (2021)
- [Provably Efficient Reinforcement Learning in Partially Observable Dynamical Systems](https://mlsanity.com/paper/2206.12020/) (2022)
- [Off-Policy Confidence Interval Estimation with Confounded Markov Decision Process](https://mlsanity.com/paper/2202.10589/) (2022)
- [Applied Causal Inference Powered by ML and AI](https://mlsanity.com/paper/2403.02467/) (2024)
- [Long-term Causal Inference Under Persistent Confounding via Data Combination](https://mlsanity.com/paper/2202.07234/) (2022)
- [Pessimism in the Face of Confounders: Provably Efficient Offline Reinforcement Learning in Partially Observable Markov Decision Processes](https://mlsanity.com/paper/2205.13589/) (2022)
- [Proximal Learning for Individualized Treatment Regimes Under Unmeasured Confounding](https://mlsanity.com/paper/2105.01187/) (2021)
- [Future-Dependent Value-Based Off-Policy Evaluation in POMDPs](https://mlsanity.com/paper/2207.13081/) (2022)
- [Augmented balancing weights as linear regression](https://mlsanity.com/paper/2304.14545/) (2023)
- [Minimax Kernel Machine Learning for a Class of Doubly Robust Functionals with Application to Proximal Causal Inference](https://mlsanity.com/paper/2104.02929/) (2021)
- [An Instrumental Variable Approach to Confounded Off-Policy Evaluation](https://mlsanity.com/paper/2212.14468/) (2022)
- [Reinforcement Learning from Partial Observation: Linear Function Approximation with Provable Sample Efficiency](https://mlsanity.com/paper/2204.09787/) (2022)
- [Deep Learning Methods for Proximal Inference via Maximum Moment Restriction](https://mlsanity.com/paper/2205.09824/) (2022)
- [Minimax Instrumental Variable Regression and L2 Convergence Guarantees without Identification or Closedness](https://mlsanity.com/paper/2302.05404/) (2023)
- [Inference on Strongly Identified Functionals of Weakly Identified Functions](https://mlsanity.com/paper/2208.08291/) (2022)
- [Spectral Representation Learning for Conditional Moment Models](https://mlsanity.com/paper/2210.16525/) (2022)
- [Adapting to Latent Subgroup Shifts via Concepts and Proxies](https://mlsanity.com/paper/2212.11254/) (2022)
- [The Proximal ID Algorithm](https://mlsanity.com/paper/2108.06818/) (2021)
- [Falsification of Internal and External Validity in Observational Studies via Conditional Moment Restrictions](https://mlsanity.com/paper/2301.13133/) (2023)
- [A Simple and General Debiased Machine Learning Theorem with Finite Sample Guarantees](https://mlsanity.com/paper/2105.15197/) (2021)
- [Source Condition Double Robust Inference on Functionals of Inverse Problems](https://mlsanity.com/paper/2307.13793/) (2023)
- [Proximal Causal Learning of Conditional Average Treatment Effects](https://mlsanity.com/paper/2301.10913/) (2023)
- [Proximal Causal Inference With Text Data](https://mlsanity.com/paper/2401.06687/) (2024)
- [Regularized DeepIV with Model Selection](https://mlsanity.com/paper/2403.04236/) (2024)
- [Causal Inference with Hidden Mediators](https://mlsanity.com/paper/2111.02927/) (2021)
- [Optimal Treatment Regimes for Proximal Causal Learning](https://mlsanity.com/paper/2212.09494/) (2022)
- [Kernel Single Proxy Control for Deterministic Confounding](https://mlsanity.com/paper/2308.04585/) (2023)
- [Policy Learning with Distributional Welfare](https://mlsanity.com/paper/2311.15878/) (2023)
- [Many Proxy Controls](https://mlsanity.com/paper/2110.03973/) (2021)

## Abstract (arXiv)

We study the estimation of causal parameters when not all confounders are observed and instead negative controls are available. Recent work has shown how these can enable identification and efficient estimation via two so-called bridge functions. In this paper, we tackle the primary challenge to causal inference using negative controls: the identification and estimation of these bridge functions. Previous work has relied on completeness conditions on these functions to identify the causal parameters and required uniqueness assumptions in estimation, and they also focused on parametric estimation of bridge functions. Instead, we provide a new identification strategy that avoids the completeness condition. And, we provide new estimators for these functions based on minimax learning formulations. These estimators accommodate general function classes such as Reproducing Kernel Hilbert Spaces and neural networks. We study finite-sample convergence results both for estimating bridge functions themselves and for the final estimation of the causal parameter under a variety of combinations of assumptions. We avoid uniqueness conditions on the bridge functions as much as possible.