# Proximal Causal Learning with Kernels: Two-Stage Estimation and Moment Restriction

arXiv: https://arxiv.org/abs/2105.04544 · HTML: https://mlsanity.com/paper/2105.04544/

## Key facts

- Cited by 55 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 80 references that are in the map.
- First posted on arXiv 10 May 2021.

## Builds on (most cited first, 14 of 14)

- [Kernel Mean Embedding of Distributions: A Review and Beyond](https://mlsanity.com/paper/1605.09522/) (2016)
- [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 Theory for Distribution Regression](https://mlsanity.com/paper/1411.2066/) (2014)
- [Provably Efficient Neural Estimation of Structural Equation Model: An Adversarial Approach](https://mlsanity.com/paper/2007.01290/) (2020)
- [Kernel Methods for Unobserved Confounding: Negative Controls, Proxies, and Instruments](https://mlsanity.com/paper/2012.10315/) (2020)
- [Kernel Conditional Moment Test via Maximum Moment Restriction](https://mlsanity.com/paper/2002.09225/) (2020)
- [Ivy: Instrumental Variable Synthesis for Causal Inference](https://mlsanity.com/paper/2004.05316/) (2020)
- [Typing linear algebra: A biproduct-oriented approach](https://mlsanity.com/paper/1312.4818/) (2013)
- [Fixing an error in Caponnetto and de Vito (2007)](https://mlsanity.com/paper/1702.02982/) (2017)
- [A New Method for Partial Correction of Residual Confounding in Time-Series and other Observational Studies](https://mlsanity.com/paper/1510.06905/) (2015)

## Built on by (most cited first, 30 of 55)

- [Deep Proxy Causal Learning and its Application to Confounded Bandit Policy Evaluation](https://mlsanity.com/paper/2106.03907/) (2021)
- [Causal Effect Inference for Structured Treatments](https://mlsanity.com/paper/2106.01939/) (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)
- [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)
- [Towards Optimal Sobolev Norm Rates for the Vector-Valued Regularized Least-Squares Algorithm](https://mlsanity.com/paper/2312.07186/) (2023)
- [Practical Kernel Tests of Conditional Independence](https://mlsanity.com/paper/2402.13196/) (2024)
- [Deep Learning Methods for Proximal Inference via Maximum Moment Restriction](https://mlsanity.com/paper/2205.09824/) (2022)
- [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)
- [Optimal Rates for Vector-Valued Spectral Regularization Learning Algorithms](https://mlsanity.com/paper/2405.14778/) (2024)
- [Spectral Representation for Causal Estimation with Hidden Confounders](https://mlsanity.com/paper/2407.10448/) (2024)
- [Causal Inference with Treatment Measurement Error: A Nonparametric Instrumental Variable Approach](https://mlsanity.com/paper/2206.09186/) (2022)
- [Nonparametric Instrumental Regression via Kernel Methods is Minimax Optimal](https://mlsanity.com/paper/2411.19653/) (2024)
- [DeCaFlow: A deconfounding causal generative model](https://mlsanity.com/paper/2503.15114/) (2025)
- [Partial Identification of Dose Responses with Hidden Confounders](https://mlsanity.com/paper/2204.11206/) (2022)
- [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)
- [On the Hardness of Conditional Independence Testing In Practice](https://mlsanity.com/paper/2512.14000/) (2025)
- [Proxy Methods for Domain Adaptation](https://mlsanity.com/paper/2403.07442/) (2024)
- [An Overview of Causal Inference using Kernel Embeddings](https://mlsanity.com/paper/2410.22754/) (2024)
- [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)
- [Towards Principled Causal Effect Estimation by Deep Identifiable Models](https://mlsanity.com/paper/2109.15062/) (2021)
- [Doubly Robust Proximal Causal Learning for Continuous Treatments](https://mlsanity.com/paper/2309.12819/) (2023)
- [Stochastic Optimization Algorithms for Instrumental Variable Regression with Streaming Data](https://mlsanity.com/paper/2405.19463/) (2024)
- [Adjustment for Confounding using Pre-Trained Representations](https://mlsanity.com/paper/2506.14329/) (2025)
- [Quasi-Bayesian Dual Instrumental Variable Regression](https://mlsanity.com/paper/2106.08750/) (2021)
- [Causal Inference from Small High-dimensional Datasets](https://mlsanity.com/paper/2205.09281/) (2022)

## Abstract (arXiv)

We address the problem of causal effect estimation in the presence of unobserved confounding, but where proxies for the latent confounder(s) are observed. We propose two kernel-based methods for nonlinear causal effect estimation in this setting: (a) a two-stage regression approach, and (b) a maximum moment restriction approach. We focus on the proximal causal learning setting, but our methods can be used to solve a wider class of inverse problems characterised by a Fredholm integral equation. In particular, we provide a unifying view of two-stage and moment restriction approaches for solving this problem in a nonlinear setting. We provide consistency guarantees for each algorithm, and we demonstrate these approaches achieve competitive results on synthetic data and data simulating a real-world task. In particular, our approach outperforms earlier methods that are not suited to leveraging proxy variables.