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L0Learn: Fast Best Subset Selection

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Hussein Hazimeh, Rahul Mazumder, and Tim Nonet

Massachusetts Institute of Technology

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L0Learn is a highly efficient framework for solving L0-regularized learning problems. It can (approximately) solve the following three problems, where the empirical loss is penalized by combinations of the L0, L1, and L2 norms:

We support both regression (using squared error loss) and classification (using logistic or squared hinge loss). Optimization is done using coordinate descent and local combinatorial search over a grid of regularization parameter(s) values. Several computational tricks and heuristics are used to speed up the algorithms and improve the solution quality. These heuristics include warm starts, active set convergence, correlation screening, greedy cycling order, and efficient methods for updating the residuals through exploiting sparsity and problem dimensions. Moreover, we employed a new computationally efficient method for dynamically selecting the regularization parameter λ in the path. We describe the details of the algorithms in our paper: Fast Best Subset Selection: Coordinate Descent and Local Combinatorial Optimization Algorithms (link).

The toolkit is implemented in C++11 and can often run faster than popular sparse learning toolkits (see our experiments in the paper above). We also provide an easy-to-use R interface; see the section below for installation and usage of the R package.

NEW: Version 2 (03/2021) adds support for sparse matrices and box constraints on the coefficients.

R Package Installation

The latest version (v2.1.0) can be installed from CRAN as follows:

install.packages("L0Learn", repos = "")

Alternatively, L0Learn can also be installed from Github as follows:


L0Learn's changelog can be accessed from here.


For a tutorial, please refer to L0Learn's Vignette. For a detailed description of the API, check the Reference Manual.


Which penalty to use?

Pure L0 regularization can overfit when the signal strength in the data is relatively low. Adding L2 regularization can alleviate this problem and lead to competitive models (see the experiments in our paper). Thus, in practice, we strongly recommend using the L0L2 penalty. Ideally, the parameter gamma (for L2 regularization) should be tuned over a sufficiently large interval, and this can be performed using L0Learn's built-in cross-validation method.

Which algorithm to use?

By default, L0Learn uses a coordinate descent-based algorithm, which achieves competitive run times compared to popular sparse learning toolkits. This can work well for many applications. We also offer a local search algorithm that is guaranteed to return higher quality solutions, at the expense of an increase in the run time. We recommend using the local search algorithm if the problem has highly correlated features or the number of samples is much smaller than the number of features---see the local search section of the Vignette for how to use this algorithm.

How to certify optimality?

While for many challenging statistical instances L0Learn can lead to optimal or near-optimal solutions, it cannot provide certificates of optimality. Such certificates can be provided via Integer Programming. Our toolkit L0BnB is a scalable integer programming framework for L0-regularized regression, which can provide such certificates and potentially improve upon the solutions of L0Learn (if they are sub-optimal). We recommend using L0Learn first to obtain a candidate solution (or a pool of solutions) and then checking optimality using L0BnB.

Citing L0Learn

If you find L0Learn useful in your research, please consider citing the following papers.

Paper 1 (Toolkit):

  title={L0learn: A scalable package for sparse learning using l0 regularization},
  author={Hazimeh, Hussein and Mazumder, Rahul and Nonet, Tim},
  journal={Journal of Machine Learning Research},

Paper 2 (Regression):

author = {Hazimeh, Hussein and Mazumder, Rahul},
title = {Fast Best Subset Selection: Coordinate Descent and Local Combinatorial Optimization Algorithms},
journal = {Operations Research},
volume = {68},
number = {5},
pages = {1517-1537},
year = {2020},
doi = {10.1287/opre.2019.1919},
URL = {},
eprint = {}

Paper 3 (Classification):

  author  = {Antoine Dedieu and Hussein Hazimeh and Rahul Mazumder},
  title   = {Learning Sparse Classifiers: Continuous and Mixed Integer Optimization Perspectives},
  journal = {Journal of Machine Learning Research},
  year    = {2021},
  volume  = {22},
  number  = {135},
  pages   = {1-47},
  url     = {}