Sharp Structure-Agnostic Lower Bounds for General Functional Estimation

2025-12-19

AI Summary

This paper investigates the minimax optimal error lower bounds for structure-agnostic (i.e., model-agnostic) nonparametric functional estimation, focusing on causal parameters such as the average treatment effect (ATE) and general functional targets. Methodologically, it integrates double machine learning (DML), first-order debiasing, double robustness analysis, and minimax lower bound theory. The key contributions are: (i) the first systematic characterization of structure-agnostic optimal convergence rates under both doubly robust and non-doubly robust settings; (ii) a rigorous proof that DML achieves the theoretical minimax optimal rate across all structure-agnostic scenarios, with explicit closed-form expressions for the optimal rates; and (iii) unification and generalization of existing ATE lower bound results, thereby establishing the universal minimax optimality of DML in this framework. These findings provide foundational theoretical support for model-free causal and statistical inference.

Technology Category

• Machine Learning: Causal Learning
• Reasoning under Uncertainty: Causality
• Search and Optimization: Non-convex Optimization

Application Category

• Responsible Web: Machine-in-the-loop, human agency and autonomy
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation

Abstract

The design of efficient nonparametric estimators has long been a central problem in statistics, machine learning, and decision making. Classical optimal procedures often rely on strong structural assumptions, which can be misspecified in practice and complicate deployment. This limitation has sparked growing interest in structure-agnostic approaches -- methods that debias black-box nuisance estimates without imposing structural priors. Understanding the fundamental limits of these methods is therefore crucial. This paper provides a systematic investigation of the optimal error rates achievable by structure-agnostic estimators. We first show that, for estimating the average treatment effect (ATE), a central parameter in causal inference, doubly robust learning attains optimal structure-agnostic error rates. We then extend our analysis to a general class of functionals that depend on unknown nuisance functions and establish the structure-agnostic optimality of debiased/double machine learning (DML). We distinguish two regimes -- one where double robustness is attainable and one where it is not -- leading to different optimal rates for first-order debiasing, and show that DML is optimal in both regimes. Finally, we instantiate our general lower bounds by deriving explicit optimal rates that recover existing results and extend to additional estimands of interest. Our results provide theoretical validation for widely used first-order debiasing methods and guidance for practitioners seeking optimal approaches in the absence of structural assumptions. This paper generalizes and subsumes the ATE lower bound established in citet{jin2024structure} by the same authors.

Problem

• Establishes optimal error rates for structure-agnostic functional estimation
• Determines when double robustness is achievable for general nuisance functionals
• Validates debiased machine learning as optimal without structural assumptions

Innovation

• Structure-agnostic estimators achieve optimal error rates
• Double robust learning attains optimal rates for causal inference
• Debiased machine learning is optimal across different functional regimes