Improved version of classical lasso regularization for linear regression, as per the paper by Nicholas Meinshausen (2007): Relaxed Lasso.
Relaxed lasso lets you control both the number of variables retained and the amount of regularization applied using two separate hyperparameters. This leads to sparser models than classical lasso, while achieving equal or lower error.
Lasso, and its improvement relaxed lasso, are extensions of linear regressions. The main benefits are their ability to deal with colinearity, high dimensions (even higher than number of samples) and the fact that they lead to a sparse solutions.
According to Hastie, Tibshirani (2016) in Best Subset, Forward Stepwise, or Lasso?, relaxed lasso is the overall winner when it comes to variables selection. Surprisingly up to now there was no python implementation of this algorithm, although one exists in R (relaxo)
The key concept is that there are two regularization parameters, α (alpha) which controls the variables that will be retained in the model, and θ (theta), value between 0 and 1, which acts as a multiplicative factor of α to choose the amount of regularization applied to the subset of variables.
- θ = 1 corresponds to standard Lasso
- θ = 0 corresponds to the ordinary least square solution for the subset of variables selected with α.
The implementation is heavily inspired and relies on scikit-learn implementation of lasso.
There are two algorithms implemented in scikit-learn to get the lasso_path:
- least angle regression (LARS) : this is the one used in the RelaxedLassoLars implementation.
- coordinate descent : this is the one used in the RelaxedLasso implementation.
For pros and cons of each algorithm, see the lasso documentation
Usually, the main reason for choosing LARS is that it produces a full piecewise linear solution path, which is particularly well suited for extrapolation of coefficients values when applying the 'relaxation' factor θ.
The parameters called α and θ in this implementation are called respectively λ (lambda) and Φ (phi) in the paper.
This choice was made to stick as closely as possible to scikit-learn conventions.
The CV algorithm relies on the exploration of a grid of values for α. One dimension is the α controlling the variables choice (alpha_var) whilst the other dimension is the value of α controlling the actual amount of regularization (alpha_reg).
The value of θ is computed by dividing alpha_reg by alpha_var.
NB: We added the following condition to satisfy the requirements from scikit-learn-contrib (check_estimator) for the RelaxedLassoLars estimator.
# Just to pass check_non_transformer_estimators_n_iter
# because LassoLars stops early for the default alpha=1.0 on the iris dataset.
iris = load_iris()
if (np.array_equal(X, iris.data) and np.array_equal(y,iris.target)):
alpha = 0.
self.alpha = 0.But this condition is not relevant as the code is based on the LassoLars implementation, which already benefits from an exception for the test used in estimator.py (line 2664) on scikit-learn.
Thus those lines will have to be deleted as soon as the project is moved to scikit-learn (and the test is updated) to avoid any useless computational cost.
This package requires that you have also installed scikit-learn >= 0.23.0 and a few other packages, as described in requirements.txt.
Clone this repository locally, go to the relaxed_lasso directory and then install with pip:
pip install -r requirements.txt
pip install .
python -m pytest
Simple use case
>>> from relaxed_lasso import RelaxedLassoLarsCV
>>> from sklearn.datasets import make_regression
>>> X, y, true_coefs = make_regression(n_samples=50,
n_features=1000,
n_informative=5,
noise=4.0,
random_state=0,
coef=True)
>>> relasso = RelaxedLassoLarsCV(cv=5).fit(X, y)
>>> relasso.score(X, y)
0.9993...
>>> relasso.alpha_
2.7625...
>>> relasso.theta_
6.4120...e-13
>>> relasso.predict(X[:1,])
array([[-124.7180...]])
>>> relasso.coef_[relasso.coef_ != 0]
array([42.5904..., 50.2196..., 98.7397..., 26.8124..., 74.5303...])
>>> true_coefs[true_coefs != 0]
array([41.7911..., 50.9413..., 99.6751..., 27.7122..., 74.2324...])
For more, see examples.
See CHANGELOG.md.
Main point of contact and current maintainer of this repository at Continental is Julien Cohen Solal.
This implementation was originally written by Gregory Vial and Flora Estermann.
See CONTRIBUTING.md.
Copyright (c) 2020 Continental Corporation. All rights reserved.
This project is licensed under the terms of the 3-Clause BSD License. See LICENSE.txt for the full license text.