Constructing Confidence Intervals for Infinite-Dimensional Functional Prameters by Highly Adaptive Lasso

2025-07-14

AI Summary

This paper addresses nonparametric estimation of conditional mean functions with infinite-dimensional functional parameters—such as càdlàg or bounded-variation functions—under potential model misspecification. To mitigate regularization bias that undermines nominal coverage of confidence intervals, we propose a robust inference framework based on the Highly Adaptive Lasso (HAL). Our method introduces debiased targeted HAL and relaxed HAL estimators, coupled with an adaptive model selection strategy that integrates global and local undersmoothing. Estimation employs higher-order tensor-product spline bases and ℓ₁-constrained empirical risk minimization; variance is estimated via the delta method. Simulation studies demonstrate substantial bias reduction, with 95% confidence intervals achieving coverage rates close to the nominal level. The approach extends naturally to complex target parameters, including the conditional average treatment effect.

Technology Category

• Intelligent Robots: State Estimation
• Machine Learning: Calibration & Uncertainty Quantification
• Reasoning under Uncertainty: Relational Probabilistic Models

Application Category

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• 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

Estimating the conditional mean function is a central task in statistical learning. In this paper, we consider estimation and inference for a nonparametric class of real-valued càdlàg functions with bounded sectional variation (Gill et al., 1995), using the Highly Adaptive Lasso (HAL) (van der Laan, 2015; Benkeser and van der Laan, 2016; van der Laan, 2023), a flexible empirical risk minimizer over linear combinations of tensor products of zero- or higher-order spline basis functions under an L1 norm constraint. Building on recent theoretical advances in asymptotic normality and uniform convergence rates for higher-order spline HAL estimators (van der Laan, 2023), this work focuses on constructing robust confidence intervals for HAL-based conditional mean estimators. To address regularization bias, we propose a targeted HAL with a debiasing step to remove bias for the conditional mean, and also consider a relaxed HAL estimator to reduce bias. We also introduce both global and local undersmoothing strategies to adaptively select the working model, reducing bias relative to variance. Combined with delta-method-based variance estimation, we construct confidence intervals for conditional means based on HAL. Through simulations, we evaluate combinations of estimation and model selection strategies, showing that our methods substantially reduce bias and yield confidence intervals with coverage rates close to nominal levels across scenarios. We also provide recommendations for different estimation objectives and illustrate the generality of our framework by applying it to estimate conditional average treatment effect (CATE) functions, highlighting how HAL-based inference extends to other infinite-dimensional, non-pathwise differentiable parameters.

Problem

• Estimating nonparametric conditional mean functions with bounded variation
• Constructing robust confidence intervals for HAL-based estimators
• Addressing regularization bias via debiasing and undersmoothing strategies

Innovation

• Uses Highly Adaptive Lasso for nonparametric estimation
• Debiasing step reduces conditional mean bias
• Global and local undersmoothing adapts model selection