# Deep Learning Methods for Proximal Inference via Maximum Moment Restriction

arXiv: https://arxiv.org/abs/2205.09824 · HTML: https://mlsanity.com/paper/2205.09824/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 37 references that are in the map.
- First posted on arXiv 19 May 2022.

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

- [Causal Effect Inference with Deep Latent-Variable Models](https://mlsanity.com/paper/1705.08821/) (2017)
- [Identifying Causal Effects With Proxy Variables of an Unmeasured Confounder](https://mlsanity.com/paper/1609.08816/) (2016)
- [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)
- [Minimax Estimation of Conditional Moment Models](https://mlsanity.com/paper/2006.07201/) (2020)
- [Learning Deep Features in Instrumental Variable Regression](https://mlsanity.com/paper/2010.07154/) (2020)
- [Semiparametric proximal causal inference](https://mlsanity.com/paper/2011.08411/) (2020)
- [Proximal Causal Learning with Kernels: Two-Stage Estimation and Moment Restriction](https://mlsanity.com/paper/2105.04544/) (2021)
- [Causal Inference Under Unmeasured Confounding With Negative Controls: A Minimax Learning Approach](https://mlsanity.com/paper/2103.14029/) (2021)
- [Deep Proxy Causal Learning and its Application to Confounded Bandit Policy Evaluation](https://mlsanity.com/paper/2106.03907/) (2021)
- [Adversarial Generalized Method of Moments](https://mlsanity.com/paper/1803.07164/) (2018)
- [Kernel Methods for Unobserved Confounding: Negative Controls, Proxies, and Instruments](https://mlsanity.com/paper/2012.10315/) (2020)
- [Minimax Kernel Machine Learning for a Class of Doubly Robust Functionals with Application to Proximal Causal Inference](https://mlsanity.com/paper/2104.02929/) (2021)
- [Kernel Conditional Moment Test via Maximum Moment Restriction](https://mlsanity.com/paper/2002.09225/) (2020)
- [Learning Causal Models from Conditional Moment Restrictions by Importance Weighting](https://mlsanity.com/paper/2108.01312/) (2021)

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

- [Spectral Representation for Causal Estimation with Hidden Confounders](https://mlsanity.com/paper/2407.10448/) (2024)
- [DeCaFlow: A deconfounding causal generative model](https://mlsanity.com/paper/2503.15114/) (2025)
- [Proximal Causal Learning of Conditional Average Treatment Effects](https://mlsanity.com/paper/2301.10913/) (2023)
- [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)
- [Doubly Robust Proximal Causal Learning for Continuous Treatments](https://mlsanity.com/paper/2309.12819/) (2023)
- [Assumption-lean falsification tests of rate double-robustness of double-machine-learning estimators](https://mlsanity.com/paper/2306.10590/) (2023)
- [DAG-aware Transformer for Causal Effect Estimation](https://mlsanity.com/paper/2410.10044/) (2024)
- [Density Ratio-Free Doubly Robust Proxy Causal Learning](https://mlsanity.com/paper/2505.19807/) (2025)
- [DNA-SE: Towards Deep Neural-Nets Assisted Semiparametric Estimation](https://mlsanity.com/paper/2408.02045/) (2024)
- [DISCO: Mitigating Bias in Deep Learning with Conditional Distance Correlation](https://mlsanity.com/paper/2506.11653/) (2025)
- [Coupling Generative Modeling and an Autoencoder with the Causal Bridge](https://mlsanity.com/paper/2509.25599/) (2025)
- [Decision-Aware Proximal Bridge Learning for Optimal Treatment Selection](https://mlsanity.com/paper/2605.16989/) (2026)
- [Proximal Causal Learning under Unmeasured Confounding](https://mlsanity.com/paper/2610.04519/) (2026)

## Abstract (arXiv)

The No Unmeasured Confounding Assumption is widely used to identify causal effects in observational studies. Recent work on proximal inference has provided alternative identification results that succeed even in the presence of unobserved confounders, provided that one has measured a sufficiently rich set of proxy variables, satisfying specific structural conditions. However, proximal inference requires solving an ill-posed integral equation. Previous approaches have used a variety of machine learning techniques to estimate a solution to this integral equation, commonly referred to as the bridge function. However, prior work has often been limited by relying on pre-specified kernel functions, which are not data adaptive and struggle to scale to large datasets. In this work, we introduce a flexible and scalable method based on a deep neural network to estimate causal effects in the presence of unmeasured confounding using proximal inference. Our method achieves state of the art performance on two well-established proximal inference benchmarks. Finally, we provide theoretical consistency guarantees for our method.