Debiased Ill-Posed Regression

2025-05-27

AI Summary

This paper addresses the bias issue in estimating ill-posed regression functions—such as nonparametric instrumental variable (IV) estimands and bridge functions in proxy causal inference—under conditional moment restrictions, arising from model misspecification or slow convergence. We propose a debiased estimator based on a corrected projection-error influence function, which achieves second-order bias reduction and exhibits robustness to misspecification of a broad class of auxiliary functions. Theoretically, under mild regularity conditions, the estimator attains √n-consistency; when hyperparameters are selected via cross-validation, its finite-sample convergence rate degradation is bounded by an explicit theoretical rate. The key innovation lies in the deep integration of influence function construction with the underlying conditional moment structure, thereby simultaneously ensuring robustness, optimal convergence rates, and practical feasibility.

Technology Category

• Intelligent Robots: State Estimation
• Reasoning under Uncertainty: Probabilistic Inference
• Machine Learning: Causal Learning

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

In various statistical settings, the goal is to estimate a function which is restricted by the statistical model only through a conditional moment restriction. Prominent examples include the nonparametric instrumental variable framework for estimating the structural function of the outcome variable, and the proximal causal inference framework for estimating the bridge functions. A common strategy in the literature is to find the minimizer of the projected mean squared error. However, this approach can be sensitive to misspecification or slow convergence rate of the estimators of the involved nuisance components. In this work, we propose a debiased estimation strategy based on the influence function of a modification of the projected error and demonstrate its finite-sample convergence rate. Our proposed estimator possesses a second-order bias with respect to the involved nuisance functions and a desirable robustness property with respect to the misspecification of one of the nuisance functions. The proposed estimator involves a hyper-parameter, for which the optimal value depends on potentially unknown features of the underlying data-generating process. Hence, we further propose a hyper-parameter selection approach based on cross-validation and derive an error bound for the resulting estimator. This analysis highlights the potential rate loss due to hyper-parameter selection and underscore the importance and advantages of incorporating debiasing in this setting. We also study the application of our approach to the estimation of regular parameters in a specific parameter class, which are linear functionals of the solutions to the conditional moment restrictions and provide sufficient conditions for achieving root-n consistency using our debiased estimator.

Problem

• Debiased estimation for ill-posed regression under model restrictions
• Robustness to nuisance function misspecification in causal inference
• Optimal hyper-parameter selection for improved convergence rates

Innovation

• Debiased estimation using influence function
• Hyper-parameter selection via cross-validation
• Robustness to nuisance function misspecification