The Scalable Highly Adaptive Lasso

Authors: Jeremy Coyle, Nima Hejazi, and Mark van der Laan

What’s hal9001?

hal9001 is an R package providing an implementation of the scalable highly adaptive lasso (HAL), a nonparametric regression estimator that applies L1-regularized lasso regression to a design matrix composed of indicator functions corresponding to the support of the functional over a set of covariates and interactions thereof. HAL regression allows for arbitrarily complex functional forms to be estimated at fast (near-parametric) convergence rates under only global smoothness assumptions (van der Laan 2017a; Bibaut and van der Laan 2019). For detailed theoretical discussions of the highly adaptive lasso estimator, consider consulting, for example, van der Laan (2017a), van der Laan (2017b), and van der Laan and Bibaut (2017). For a computational demonstration of the versatility of HAL regression, see Benkeser and van der Laan (2016). Recent theoretical works have demonstrated success in building efficient estimators of complex parameters when particular variations of HAL regression are used to estimate nuisance parameters (e.g., van der Laan, Benkeser, and Cai 2019; Ertefaie, Hejazi, and van der Laan 2020).


For standard use, we recommend installing the package from CRAN via


To contribute, install the development version of hal9001 from GitHub via remotes:



If you encounter any bugs or have any specific feature requests, please file an issue.


Consider the following minimal example in using hal9001 to generate predictions via Highly Adaptive Lasso regression:

# load the package and set a seed
#> Loading required package: Rcpp
#> hal9001 v0.2.8: The Scalable Highly Adaptive Lasso

# simulate data
n <- 100
p <- 3
x <- matrix(rnorm(n * p), n, p)
y <- x[, 1] * sin(x[, 2]) + rnorm(n, mean = 0, sd = 0.2)

# fit the HAL regression
hal_fit <- fit_hal(X = x, Y = y)
#> [1] "I'm sorry, Dave. I'm afraid I can't do that."
#>                   user.self sys.self elapsed user.child sys.child
#> enumerate_basis       0.003    0.000   0.003          0         0
#> design_matrix         0.003    0.000   0.002          0         0
#> reduce_basis          0.000    0.000   0.000          0         0
#> remove_duplicates     0.010    0.000   0.011          0         0
#> lasso                 0.289    0.012   0.304          0         0
#> total                 0.306    0.012   0.321          0         0

# training sample prediction
preds <- predict(hal_fit, new_data = x)
mean(hal_mse <- (preds - y)^2)
#> [1] 0.006991539


Contributions are very welcome. Interested contributors should consult our contribution guidelines prior to submitting a pull request.


After using the hal9001 R package, please cite both of the following:

      author = {Coyle, Jeremy R and Hejazi, Nima S and {van der Laan}, Mark
      title = {{hal9001}: The scalable highly adaptive lasso},
      year  = {2020},
      url = {},
      doi = {10.5281/zenodo.3558313}
      note = {{R} package version 0.2.7}

      author = {Hejazi, Nima S and Coyle, Jeremy R and {van der Laan}, Mark
      title = {{hal9001}: Scalable highly adaptive lasso regression in
      year  = {2020},
      url = {},
      doi = {10.21105/joss.02526},
      journal = {Journal of Open Source Software},
      publisher = {The Open Journal}


© 2017-2021 Jeremy R. Coyle & Nima S. Hejazi

The contents of this repository are distributed under the GPL-3 license. See file LICENSE for details.


Benkeser, David, and Mark J van der Laan. 2016. “The Highly Adaptive Lasso Estimator.” In 2016 IEEE International Conference on Data Science and Advanced Analytics (DSAA). IEEE.

Bibaut, Aurélien F, and Mark J van der Laan. 2019. “Fast Rates for Empirical Risk Minimization over Càdlàg Functions with Bounded Sectional Variation Norm.”

Ertefaie, Ashkan, Nima S Hejazi, and Mark J van der Laan. 2020. “Nonparametric Inverse Probability Weighted Estimators Based on the Highly Adaptive Lasso.”

van der Laan, Mark J. 2017a. “A Generally Efficient Targeted Minimum Loss Based Estimator Based on the Highly Adaptive Lasso.” The International Journal of Biostatistics.

———. 2017b. “Finite Sample Inference for Targeted Learning.”

van der Laan, Mark J, David Benkeser, and Weixin Cai. 2019. “Efficient Estimation of Pathwise Differentiable Target Parameters with the Undersmoothed Highly Adaptive Lasso.”

van der Laan, Mark J, and Aurélien F Bibaut. 2017. “Uniform Consistency of the Highly Adaptive Lasso Estimator of Infinite-Dimensional Parameters.”