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

Project description

The group lasso [1] 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.

An extension of the group lasso regulariser is the sparse group lasso regulariser [2], which imposes both group-wise sparsity and coefficient-wise sparsity. This is done by combining the group lasso penalty with the traditional lasso penalty. In this library, I have implemented an efficient sparse group lasso solver being fully scikit-learn API compliant.

This project is developed by Yngve Mardal Moe and released under an MIT lisence. I am still working out a few things so changes might come rapidly.

Installation guide

Group-lasso requires Python 3.5+, numpy and scikit-learn. 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_[0]

# Evaluate the model
coef_correlation = np.corrcoef(w.ravel(), estimated_w.ravel())[0, 1]
print(
"True intercept: {intercept:.2f}. Estimated intercept: {estimated_intercept:.2f}".format(
intercept=intercept,
estimated_intercept=estimated_intercept
)
)
print(
"Correlation between true and estimated coefficients: {coef_correlation:.2f}".format(
coef_correlation=coef_correlation
)
)
```
```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, group_reg=0.1, l1_reg=0.05)
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("The R^2 statistic for the group lasso model is: {R_squared:.2f}".format(R_squared=R_squared))
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("The R^2 statistic for the pipeline is: {R_squared:.2f}".format(R_squared=R_squared))
```
```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
```

Further work

1. Fully test with sparse arrays and make examples
2. Make it easier to work with categorical data
3. Poisson regression

Implementation details

The problem is solved using the FISTA optimiser [3] with a gradient-based adaptive restarting scheme [4]. 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.

References

 [1] Yuan, M. and Lin, Y. (2006), Model selection and estimation in regression with grouped variables. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 68: 49-67. doi:10.1111/j.1467-9868.2005.00532.x
 [2] Simon, N., Friedman, J., Hastie, T., & Tibshirani, R. (2013). A sparse-group lasso. Journal of Computational and Graphical Statistics, 22(2), 231-245.
 [3] Beck, A. and Teboulle, M. (2009), A Fast Iterative Shrinkage-Thresholding Algorithm for Linear Inverse Problems. SIAM Journal on Imaging Sciences 2009 2:1, 183-202. doi:10.1137/080716542
 [4] O’Donoghue, B. & Candès, E. (2015), Adaptive Restart for Accelerated Gradient Schemes. Found Comput Math 15: 715. doi:10.1007/s10208-013-9150-