# Efficient Estimation of Pathwise Differentiable Target Parameters with the Undersmoothed Highly Adaptive Lasso

arXiv: https://arxiv.org/abs/1908.05607 · HTML: https://mlsanity.com/paper/1908.05607/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 14 Aug 2019.

## Built on by (most cited first, 13 of 13)

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- [Kernel Debiased Plug-in Estimation: Simultaneous, Automated Debiasing without Influence Functions for Many Target Parameters](https://mlsanity.com/paper/2306.08598/) (2023)
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- [Estimation of population size based on capture recapture designs and evaluation of the estimation reliability](https://mlsanity.com/paper/2105.05373/) (2021)
- [Causal Effect Estimation with TMLE: Handling Missing Data and Near-Violations of Positivity](https://mlsanity.com/paper/2510.22202/) (2025)
- [Targeted Synthetic Control Method](https://mlsanity.com/paper/2602.04611/) (2026)
- [Kernel Debiased Plug-in Estimation based on the Universal Least Favorable Submodel](https://mlsanity.com/paper/2603.08945/) (2026)
- [When Sample-Splitting Choices Matter in TMLE and AIPW: Cross-Fitting, Targeting, and Repetition](https://mlsanity.com/paper/2610.16192/) (2026)

## Abstract (arXiv)

We consider estimation of a functional parameter of a realistically modeled data distribution based on observing independent and identically distributed observations. We define an m-th order Spline Highly Adaptive Lasso Minimum Loss Estimator (Spline HAL-MLE) of a functional parameter that is defined by minimizing the empirical risk function over an m-th order smoothness class of functions. We show that this m-th order smoothness class consists of all functions that can be represented as an infinitesimal linear combination of tensor products of ≤ m-th order spline-basis functions, and involves assuming m-derivatives in each coordinate. By selecting m with cross-validation we obtain a Spline-HAL-MLE that is able to adapt to the underlying unknown smoothness of the true function, while guaranteeing a rate of convergence faster than n-1/4, as long as the true function is cadlag (right-continuous with left-hand limits) and has finite sectional variation norm. The m=0-smoothness class consists of all cadlag functions with finite sectional variation norm and corresponds with the original HAL-MLE defined in van der Laan (2015). In this article we establish that this Spline-HAL-MLE yields an asymptotically efficient estimator of any smooth feature of the functional parameter under an easily verifiable global undersmoothing condition. A sufficient condition for the latter condition is that the minimum of the empirical mean of the selected basis functions is smaller than a constant times n-1/2, which is not parameter specific and enforces the selection of the L1-norm in the lasso to be large enough to include sparsely supported basis. We demonstrate our general result for the m=0-HAL-MLE of the average treatment effect and of the integral of the square of the data density. We also present simulations for these two examples confirming the theory.