# Efficient Group Lasso in Python¶

This library provides efficient computation of sparse group lasso regularise linear and logistic regression.

## What is group lasso?¶

It is often the case that we have a dataset where the covariates form natural groups. These groups can represent biological function in gene expression data or maybe sensor location in climate data. We then wish to find a sparse subset of these covariate groups that describe the relationship in the data. Let us look at an example to crystalise the usefulness of this further.

Say that we work as data scientists for a large Norwegian food supplier and wish to make a prediction model for the amount of that will be sold based on weather data. We have weather data from cities in Norway and need to know how the fruit should be distributed across different warehouses. From each city, we have information about temperature, precipitation, wind strength, wind direction and how cloudy it is. Multiplying the number of cities with the number of covariates per city, we get 1500 different covariates in total. It is unlikely that we need all these covariates in our model, so we seek a sparse set of these to do our predictions with.

Let us now assume that the weather data API that we use charge money by the number of cities we query, but the amount of information we get per city. We therefore wish to create a regression model that predicts fruit demand based on a sparse set of city observations. One way to achieve such sparsity is through the framework of group lasso regularisation [1].

## What is sparse group lasso¶

The sparse group lasso regulariser [2] is an extension of the group lasso regulariser that also promotes parameter-wise sparsity. It is the combination of the group lasso penalty and the normal lasso penalty. If we consider the example above, then the sparse group lasso penalty will yield a sparse set of groups and also a sparse set of covariates in each selected group. An example of where this is useful is if each city query has a set price that increases based on the number of measurements we want from each city.

## A quick mathematical interlude¶

Let us now briefly describe the mathematical problem solved in group lasso regularised machine learning problems. Originally, group lasso algorithm [1] was defined as regularised linear regression with the following loss function

$\text{arg} \min_{\mathbf{\beta}_g \in \mathbb{R^{d_g}}} || \sum_{g \in \mathcal{G}} \left[\mathbf{X}_g\mathbf{\beta}_g\right] - \mathbf{y} ||_2^2 + \lambda_1 ||\mathbf{\beta}||_1 + \lambda_2 \sum_{g \in \mathcal{G}} \sqrt{d_g}||\mathbf{\beta}_g||_2,$

where $$\mathbf{X}_g \in \mathbb{R}^{n \times d_g}$$ is the data matrix corresponding to the covariates in group $$g$$, $$\mathbf{\beta}_g$$ is the regression coefficients corresponding to group $$g$$, $$\mathbf{y} \in \mathbf{R}^n$$ is the regression target, $$n$$ is the number of measurements, $$d_g$$ is the dimensionality of group $$g$$, $$\lambda_1$$ is the parameter-wise regularisation penalty, $$\lambda_2$$ is the group-wise regularisation penalty and $$\mathcal{G}$$ is the set of all groups.

Notice, in the equation above, that the 2-norm is not squared. A consequence of this is that the regulariser has a “kink” at zero, uninformative covariate groups to have zero-valued regression coefficients. Later, it has been popular to use this methodology to regularise other machine learning algorithms, such as logistic regression. The “only” thing neccesary to do this is to exchange the squared norm term, $$|| \sum_{g \in \mathcal{G}} \left[\mathbf{X}_g\mathbf{\beta}_g\right] - \mathbf{y} ||_2^2$$, with a general loss term, $$L(\mathbf{\beta}; \mathbf{X}, \mathbf{y})$$, where $$\mathbf{\beta}$$ and $$\mathbf{X}$$ is the concatenation of all group coefficients and group data matrices, respectively.

## API design¶

The group-lasso python library is modelled after the scikit-learn API and should be fully compliant with the scikit-learn ecosystem. Consequently, the group-lasso library depends on numpy, scipy and scikit-learn.

Currently, the only supported algorithm is group-lasso regularised linear and multiple regression, which is available in the group_lasso.GroupLasso class. However, I am working on an experimental class with group lasso regularised logistic regression, which is available in the group_lasso.LogisticGroupLasso class. Currently, this class only supports binary classification problems through a sigmoidal transformation, but I am working on a multiple classification algorithm with the softmax transformation.

All classes in this library is implemented as both scikit-learn transformers and their regressors or classifiers (dependent on their use case). The reason for this is that to use lasso based models for variable selection, the regularisation coefficient should be quite high, resulting in sub-par performance on the actual task of interest. Therefore, it is common to first use a lasso-like algorithm to select the relevant features before using another another algorithm (say ridge regression) for the task at hand. Therefore, the transform method of group_lasso.GroupLasso to remove the columns of the input dataset corresponding to zero-valued coefficients.

## References¶

 [1] (1, 2) Yuan M, Lin Y. Model selection and estimation in regression with grouped variables. Journal of the Royal Statistical Society: Series B (Statistical Methodology). 2006 Feb;68(1):49-67.
 [2] Simon, N., Friedman, J., Hastie, T., & Tibshirani, R. (2013). A sparse-group lasso. Journal of Computational and Graphical Statistics, 22(2), 231-245.