# Adaptive Estimation and Inference in Conditional Moment Models via the Discrepancy Principle

arXiv: https://arxiv.org/abs/2603.01337 · HTML: https://mlsanity.com/paper/2603.01337/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 16 of its 42 references that are in the map.
- First posted on arXiv 2 Mar 2026.

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

- [Orthogonal Statistical Learning](https://mlsanity.com/paper/1901.09036/) (2019)
- [Kernel Instrumental Variable Regression](https://mlsanity.com/paper/1906.00232/) (2019)
- [An Introduction to Proximal Causal Learning](https://mlsanity.com/paper/2009.10982/) (2020)
- [Minimax Estimation of Conditional Moment Models](https://mlsanity.com/paper/2006.07201/) (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)
- [Deep Proxy Causal Learning and its Application to Confounded Bandit Policy Evaluation](https://mlsanity.com/paper/2106.03907/) (2021)
- [A confounding bridge approach for double negative control inference on causal effects](https://mlsanity.com/paper/1808.04945/) (2018)
- [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)
- [Instrumental Variable Regression via Kernel Maximum Moment Loss](https://mlsanity.com/paper/2010.07684/) (2020)
- [Inference on Strongly Identified Functionals of Weakly Identified Functions](https://mlsanity.com/paper/2208.08291/) (2022)
- [Source Condition Double Robust Inference on Functionals of Inverse Problems](https://mlsanity.com/paper/2307.13793/) (2023)
- [Regularized DeepIV with Model Selection](https://mlsanity.com/paper/2403.04236/) (2024)
- [Nonparametric inference about mean functionals of nonignorable nonresponse data without identifying the joint distribution](https://mlsanity.com/paper/2110.05776/) (2021)

## Abstract (arXiv)

We study adaptive estimation and inference in ill-posed linear inverse problems defined by conditional moment restrictions. Existing regularized estimators such as Regularized DeepIV (RDIV) require prior knowledge of the smoothness of the nuisance function, typically encoded by a beta source condition to tune their regularization parameters. In practice, this smoothness is unknown, and misspecified hyperparameters can lead to suboptimal convergence or instability. We introduce a discrepancy-principle-based framework for adaptive hyperparameter selection that automatically balances bias and variance without relying on the unknown smoothness parameter. Our framework applies to both RDIV (Li et al. [2024]) and the Tikhonov Regularized Adversarial Estimator (TRAE) (Bennett et al. [2023a]) and achieves the same rates in both weak and strong metrics. Building on this, we construct a fully adaptive doubly robust estimator for linear functionals that attains the optimal rate of the better-conditioned primal or dual problem, providing a practical, theoretically grounded approach for adaptive inference in ill-posed econometric models.