Regularized Targeted Maximum Likelihood Estimation in Highly Adaptive Lasso Implied Working Models

TL;DR

This paper introduces two novel HAL-TMLE methods that improve computational stability and efficiency in complex nonparametric and semiparametric models, especially in survival and mediation analyses.

Contribution

We develop two new HAL-TMLE approaches—Delta-method regHAL-TMLE and Projection-based regHAL-TMLE—that address computational instability and high costs of existing methods.

Findings

Both methods show improved stability over existing HAL-TMLEs.

Simulation results demonstrate better performance in complex models.

Proposed methods reduce computational burden significantly.

Abstract

We address the challenge of performing Targeted Maximum Likelihood Estimation (TMLE) after an initial Highly Adaptive Lasso (HAL) fit. Existing approaches that utilize the data-adaptive working model selected by HAL-such as the relaxed HAL update-can be simple and versatile but may become computationally unstable when the HAL basis expansions introduce collinearity. Undersmoothed HAL may fail to solve the efficient influence curve (EIC) at the desired level without overfitting, particularly in complex settings like survival-curve estimation. A full HAL-TMLE, which treats HAL as the initial estimator and then targets in the nonparametric or semiparametric model, typically demands costly iterative clever-covariate calculations in complex set-ups like survival analysis and longitudinal mediation analysis. To overcome these limitations, we propose two new HAL-TMLEs that operate within the finite-dimensional working model implied by HAL: Delta-method regHAL-TMLE and Projection-based regHAL-TMLE. We conduct extensive simulations to demonstrate the performance of our proposed methods.