Fast group lasso regularised linear models in a sklearn-style API.

## Project description   The group lasso  regulariser is a well known method to achieve structured sparsity in machine learning and statistics. The idea is to create non-overlapping groups of covariates, and recover regression weights in which only a sparse set of these covariate groups have non-zero components.

There are several reasons for why this might be a good idea. Say for example that we have a set of sensors and each of these sensors generate five measurements. We don’t want to maintain an unneccesary number of sensors. If we try normal LASSO regression, then we will get sparse components. However, these sparse components might not correspond to a sparse set of sensors, since they each generate five measurements. If we instead use group LASSO with measurements grouped by which sensor they were measured by, then we will get a sparse set of sensors.

This project is developed by Yngve Mardal Moe and released under an MIT lisence.

## Installation guide

Currently, the code only works with Python 3.6+, but I aim to support Python 3.5 in the future. To install group-lasso via pip, simply run the command:

```pip install group-lasso
```

Alternatively, you can manually pull this repository and run the setup.py file:

```git clone https://github.com/yngvem/group-lasso.git
cd group-lasso
python setup.py
```

## Examples

### Group lasso regression

The group lasso regulariser is implemented following the scikit-learn API, making it easy to use for those familiar with the Python ML ecosystem.

```import numpy as np
from group_lasso import GroupLasso

# Dataset parameters
num_data_points = 10_000
num_features = 500
num_groups = 25
assert num_features % num_groups == 0

# Generate data matrix
X = np.random.standard_normal((num_data_points, num_features))

# Generate coefficients and intercept
w = np.random.standard_normal((500, 1))
intercept = 2

# Generate groups and randomly set coefficients to zero
groups = np.array([[group]*20 for group in range(25)]).ravel()
for group in range(num_groups):
w[groups == group] *= np.random.random() < 0.8

# Generate target vector:
y = X@w + intercept
noise = np.random.standard_normal(y.shape)
noise /= np.linalg.norm(noise)
noise *= 0.3*np.linalg.norm(y)
y += noise

# Generate group lasso object and fit the model
gl = GroupLasso(groups=groups, reg=.05)
gl.fit(X, y)
estimated_w = gl.coef_
estimated_intercept = gl.intercept_

# Evaluate the model
coef_correlation = np.corrcoef(w.ravel(), estimated_w.ravel())[0, 1]
print(f"True intercept: {intercept:.2f}. Estimated intercept: {estimated_intercept:.2f}")
print(f"Correlation between true and estimated coefficients: {coef_correlation:.2f}")
```
```True intercept: 2.00. Estimated intercept: 1.53
Correlation between true and estimated coefficients: 0.98
```

### Group lasso as a transformer

Group lasso regression can also be used as a transformer

```import numpy as np
from sklearn.pipeline import Pipeline
from sklearn.linear_model import Ridge
from group_lasso import GroupLasso

# Dataset parameters
num_data_points = 10_000
num_features = 500
num_groups = 25
assert num_features % num_groups == 0

# Generate data matrix
X = np.random.standard_normal((num_data_points, num_features))

# Generate coefficients and intercept
w = np.random.standard_normal((500, 1))
intercept = 2

# Generate groups and randomly set coefficients to zero
groups = np.array([[group]*20 for group in range(25)]).ravel()
for group in range(num_groups):
w[groups == group] *= np.random.random() < 0.8

# Generate target vector:
y = X@w + intercept
noise = np.random.standard_normal(y.shape)
noise /= np.linalg.norm(noise)
noise *= 0.3*np.linalg.norm(y)
y += noise

# Generate group lasso object and fit the model
# We use an artificially high regularisation coefficient since
#  we want to use group lasso as a variable selection algorithm.
gl = GroupLasso(groups=groups, reg=.1)
gl.fit(X, y)
new_X = gl.transform(X)

# Evaluate the model
predicted_y = gl.predict(X)
R_squared = 1 - np.sum((y - predicted_y)**2)/np.sum(y**2)

print("The rows with zero-valued coefficients have now been removed from the dataset.")
print("The new shape is:", new_X.shape)
print(f"The R^2 statistic for the group lasso model is: {R_squared:.2f}")
print("This is very low since the regularisation is so high."

# Use group lasso in a scikit-learn pipeline
pipe = Pipeline(
memory=None,
steps=[
('variable_selection', GroupLasso(groups=groups, reg=.1)),
('regressor', Ridge(alpha=0.1))
]
)
pipe.fit(X, y)
predicted_y = pipe.predict(X)
R_squared = 1 - np.sum((y - predicted_y)**2)/np.sum(y**2)

print(f"The R^2 statistic for the pipeline is: {R_squared:.2f}")
```
```The rows with zero-valued coefficients have now been removed from the dataset.
The new shape is: (10000, 280)
The R^2 statistic for the group lasso model is: 0.17
This is very low since the regularisation is so high.
The R^2 statistic for the pipeline is: 0.72
```

## API

The class is modelled after the scikit-learn API and should seamlessly integrate with the Python ML ecosystem. There is currently one supported class, the GroupLasso class. However, the experimental LogisticGroupLasso class is also readily available.

The GroupLasso class implements group lasso regularised linear regression with a mean squared error penalty function. Likewise, the experimental LogisticGroupLasso implements one-class logistic regression with a sigmoidal non-linearity function and cross entropy loss.

### GroupLasso

The GroupLasso class has one mandatory and several optional arguments.

#### Arguments

groups : Iterable
Iterable that specifies which group each column corresponds to. For columns that should not be regularised, the corresponding group index should either be None or negative. For example, the list [1, 1, 1, 2, 2, -1] specifies that the first three columns of the data matrix belong to the first group, the next two columns belong to the second group and the last column should not be regularised.
reg : float or iterable (default=0.05)
The regularisation coefficient(s). If reg is an iterable, then it should have the same length as groups.
n_iter : int (default=100)
The maximum number of iterations to perform
tol : float (default=1e-5)
The convergence tolerance. The optimisation algorithm will stop once ||x_{n+1} - x_n|| < tol.
subsampling_scheme : None, float, int or str (default=None)
The subsampling rate used for the gradient and singular value computations. If it is a float, then it specifies the fraction of rows to use in the computations. If it is an int, it specifies the number of rows to use in the computation and if it is a string, then it must be ‘sqrt’ and the number of rows used in the computations is the square root of the number of rows in X.
frobenius_lipschitz : bool (default=False)
Use the Frobenius norm to estimate the lipschitz coefficient of the MSE loss. This works well for systems whose power iterations converge slowly. If False, then subsampled power iterations are used. Using the Frobenius approximation for the Lipschitz coefficient might fail, and end up with all-zero weights.
fit_intercept : bool (default=True)
Whether to fit an intercept or not.

## Furher work

The todos are, in decreasing order of importance

• Better description of Group LASSO
2. Write more docstrings
3. Sphinx documentation
4. Python 3.5 compatibility
5. Better ScikitLearn compatibility
• Use Mixins?
6. Classification problems
• I have an experimental implementation one-class logistic regression, but it is not yet fully validated.

Unfortunately, the most interesting parts are the least important ones, so expect the list to be worked on from both ends simultaneously.

## Implementation details

The problem is solved using the FISTA optimiser  with a gradient-based adaptive restarting scheme . No line search is currently implemented, but I hope to look at that later.

Although fast, the FISTA optimiser does not achieve as low loss values as the significantly slower second order interior point methods. This might, at first glance, seem like a problem. However, it does recover the sparsity patterns of the data, which can be used to train a new model with the given subset of the features.

Also, even though the FISTA optimiser is not meant for stochastic optimisation, it has to my experience not suffered a large fall in performance when the mini batch was large enough. I have therefore implemented mini-batch optimisation using FISTA, and thus been able to fit models based on data with ~500 columns and 10 000 000 rows on my moderately priced laptop.

Finally, we note that since FISTA uses Nesterov acceleration, is not a descent algorithm. We can therefore not expect the loss to decrease monotonically.

## Project details

This version 0.1.3 0.1.2 0.1.1 0.1.0