Targeted Highly Adaptive Lasso Minimum Loss Estimation of Target Functions

2026-07-04

AI Summary

This work addresses the challenge of efficiently estimating non-pathwise differentiable functionals—such as dose–response curves under continuous exposure—by proposing a novel approach that integrates higher-order highly adaptive lasso (HAL), spline basis projection, and Targeted Maximum Likelihood Estimation (TMLE). The method projects the target functional onto a finite-dimensional space spanned by higher-order spline bases to construct a pathwise differentiable approximation, and embeds LASSO regularization within the TMLE framework to enable fully data-adaptive inference without requiring pre-specified sieves or parametric models. The resulting estimator exhibits pointwise asymptotic normality, with convergence rates determined solely by the dimensionality and smoothness of the target functional, and demonstrates markedly superior performance over conventional HAL plug-in estimators in simulations.

Abstract

We propose a Targeted Highly Adaptive Lasso for estimation of non-pathwise differentiable functional parameters such as the dose-response curve (DRC) for continuous exposure. We assume the target function lies in the $k$-th order smoothness class used to define the $k$-th order Highly Adaptive Lasso (HAL), which can be well approximated by linear spans of $k$-th order spline basis functions. We construct a projection of the true target function onto a large finite dimensional working model spanned by an initial set of $k$-th order spline basis functions, which defines a pathwise differentiable approximation of the target functional parameter. A standard TMLE is then applied with a data-adaptive initial fit, replacing the MLE targeting step with a LASSO step over HAL spline basis functions that span the target function. We prove that the resulting Targeted HAL-MLE is pointwise asymptotically normally distributed and achieves a convergence rate determined solely by the dimension and smoothness of the target function, giving dimension free rates up till $\log n$-factors. Through a simulation study for the DRC, we show that the Targeted HAL outperforms a HAL plug-in estimator in terms of bias and mean squared error. Targeted HAL offers a fully data-adaptive approach to inference on functional parameters without requiring sieve specification or parametric assumptions.

Problem

• non-pathwise differentiable
• dose-response curve
• functional parameter estimation
• continuous exposure
• data-adaptive inference

Innovation

• Targeted Highly Adaptive Lasso
• non-pathwise differentiable functionals
• dose-response curve
• asymptotic normality
• data-adaptive inference