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

2026-03-01

AI Summary

This work addresses the challenge of selecting regularization hyperparameters in conditional moment models when the smoothness of the unknown perturbation function is unavailable. We propose the first adaptive framework based on the bias principle, which automatically balances bias and variance without requiring prior knowledge of the smoothness level. The method is applicable to both Regularized DeepIV (RDIV) and Tikhonov-regularized Adversarial Estimators (TRAE), yielding a fully adaptive doubly robust estimator. Theoretical analysis demonstrates that the proposed approach achieves the same optimal convergence rates—both in weak and strong norms—as if the smoothness were known a priori, thereby establishing the first instance of adaptive optimal inference for linear functionals in this setting.

Technology Category

• Machine Learning: Adversarial Learning & Robustness
• Intelligent Robots: State Estimation
• Reasoning under Uncertainty: Graphical Models

Application Category

• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

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.

Problem

• conditional moment models
• ill-posed inverse problems
• adaptive estimation
• regularization
• smoothness

Innovation

• discrepancy principle
• adaptive estimation
• conditional moment models
• ill-posed inverse problems
• doubly robust inference