# Doubly Robust Proxy Causal Learning with Neural Mean Embeddings

arXiv: https://arxiv.org/abs/2605.09514 · HTML: https://mlsanity.com/paper/2605.09514/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 8 of its 54 references that are in the map.
- First posted on arXiv 10 May 2026.

## Builds on (most cited first, 8 of 8)

- [Semiparametric doubly robust targeted double machine learning: a review](https://mlsanity.com/paper/2203.06469/) (2022)
- [Causal Inference Under Unmeasured Confounding With Negative Controls: A Minimax Learning Approach](https://mlsanity.com/paper/2103.14029/) (2021)
- [Kernel Methods for Unobserved Confounding: Negative Controls, Proxies, and Instruments](https://mlsanity.com/paper/2012.10315/) (2020)
- [Nonparametric Instrumental Regression via Kernel Methods is Minimax Optimal](https://mlsanity.com/paper/2411.19653/) (2024)
- [Counterfactual Prediction with Deep Instrumental Variables Networks](https://mlsanity.com/paper/1612.09596/) (2016)
- [Closed-Form Last Layer Optimization](https://mlsanity.com/paper/2510.04606/) (2025)
- [Nonparametric Instrumental Variable Regression with Observed Covariates](https://mlsanity.com/paper/2511.19404/) (2025)
- [Towards a Unified Analysis of Neural Networks in Nonparametric Instrumental Variable Regression: Optimization and Generalization](https://mlsanity.com/paper/2511.14710/) (2025)

## Abstract (arXiv)

We propose a novel doubly robust neural mean-embedding framework for proxy causal learning (PCL) for continuous and structured treatments. PCL uses proxy variables and bridge functions to identify causal effects under unobserved confounding: either an outcome bridge or a treatment bridge can identify the target, while doubly robust methods combine both. Our framework learns both bridges with data-adaptive neural features and combines them into doubly robust estimators of population, treatment-conditional, and heterogeneous dose-response functions. Two features of our result are particularly notable: the treatment bridge can be learned without density ratios, and the same treatment bridge yields both population and heterogeneous responses, eliminating the need for a separate bridge fit. Our analysis shows that the doubly robust error is controlled by final-stage error and the smaller of the two projected bridge errors, with no separate density-ratio estimation error. Across challenging continuous- and structured-treatment benchmarks, our method outperforms neural single-bridge estimators and fixed-feature doubly robust estimators. Our implementation is available at https://github.com/BariscanBozkurt/DoublyRobust-PCL-Network.