# Source Condition Double Robust Inference on Functionals of Inverse Problems

arXiv: https://arxiv.org/abs/2307.13793 · HTML: https://mlsanity.com/paper/2307.13793/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- Builds on 21 of its 61 references that are in the map.
- First posted on arXiv 25 Jul 2023.

## Builds on (most cited first, 21 of 21)

- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [Orthogonal Statistical Learning](https://mlsanity.com/paper/1901.09036/) (2019)
- [Kernel Instrumental Variable Regression](https://mlsanity.com/paper/1906.00232/) (2019)
- [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)
- [Double/Debiased/Neyman Machine Learning of Treatment Effects](https://mlsanity.com/paper/1701.08687/) (2017)
- [Off-Policy Evaluation in Partially Observable Environments](https://mlsanity.com/paper/1909.03739/) (2019)
- [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)
- [Causal Inference Under Unmeasured Confounding With Negative Controls: A Minimax Learning Approach](https://mlsanity.com/paper/2103.14029/) (2021)
- [Proximal Reinforcement Learning: Efficient Off-Policy Evaluation in Partially Observed Markov Decision Processes](https://mlsanity.com/paper/2110.15332/) (2021)
- [Machine Learning For Elliptic PDEs: Fast Rate Generalization Bound, Neural Scaling Law and Minimax Optimality](https://mlsanity.com/paper/2110.06897/) (2021)
- [Adversarial Estimation of Riesz Representers](https://mlsanity.com/paper/2101.00009/) (2020)
- [Adversarial Generalized Method of Moments](https://mlsanity.com/paper/1803.07164/) (2018)
- [Provably Efficient Neural Estimation of Structural Equation Model: An Adversarial Approach](https://mlsanity.com/paper/2007.01290/) (2020)
- [The Variational Method of Moments](https://mlsanity.com/paper/2012.09422/) (2020)
- [Off-Policy Evaluation for Episodic Partially Observable Markov Decision Processes under Non-Parametric Models](https://mlsanity.com/paper/2209.10064/) (2022)
- [On Deep Instrumental Variables Estimate](https://mlsanity.com/paper/2004.14954/) (2020)
- [Minimax Instrumental Variable Regression and L2 Convergence Guarantees without Identification or Closedness](https://mlsanity.com/paper/2302.05404/) (2023)
- [Learning Causal Models from Conditional Moment Restrictions by Importance Weighting](https://mlsanity.com/paper/2108.01312/) (2021)
- [Nonparametric inference about mean functionals of nonignorable nonresponse data without identifying the joint distribution](https://mlsanity.com/paper/2110.05776/) (2021)

## Built on by (most cited first, 9 of 9)

- [Spectral Representation for Causal Estimation with Hidden Confounders](https://mlsanity.com/paper/2407.10448/) (2024)
- [Nonparametric Instrumental Regression via Kernel Methods is Minimax Optimal](https://mlsanity.com/paper/2411.19653/) (2024)
- [Automatic Debiased Machine Learning for Smooth Functionals of Nonparametric M-Estimands](https://mlsanity.com/paper/2501.11868/) (2025)
- [Optimality and Adaptivity of Deep Neural Features for Instrumental Variable Regression](https://mlsanity.com/paper/2501.04898/) (2025)
- [Detecting clinician implicit biases in diagnoses using proximal causal inference](https://mlsanity.com/paper/2501.16399/) (2025)
- [Fitted Occupancy-Ratio Evaluation without Bellman Completeness](https://mlsanity.com/paper/2607.05375/) (2026)
- [Learning Treatment Representations for Downstream Instrumental Variable Regression](https://mlsanity.com/paper/2506.02200/) (2025)
- [Adaptive Estimation and Inference in Conditional Moment Models via the Discrepancy Principle](https://mlsanity.com/paper/2603.01337/) (2026)
- [Nonparametric Instrumental Variable Analysis Without Structural Equations: Debiased Inference on Functionals of Inverse Problems with No Solutions](https://mlsanity.com/paper/2604.24660/) (2026)

## Abstract (arXiv)

We consider estimation of parameters defined as linear functionals of solutions to linear inverse problems. Any such parameter admits a doubly robust representation that depends on the solution to a dual linear inverse problem, where the dual solution can be thought as a generalization of the inverse propensity function. We provide the first source condition double robust inference method that ensures asymptotic normality around the parameter of interest as long as either the primal or the dual inverse problem is sufficiently well-posed, without knowledge of which inverse problem is the more well-posed one. Our result is enabled by novel guarantees for iterated Tikhonov regularized adversarial estimators for linear inverse problems, over general hypothesis spaces, which are developments of independent interest.