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Is Sklearn Lasso regression several orders of magnitude worse than Ridge Regression?

I am currently implementing regression of Ridge and Lasso using a module sklearn.linear_model.

However, the Lasso regression seems to be 3 orders of magnitude worse in the same dataset!

I'm not sure what is wrong, because mathematically this should not happen. Here is my code:

def ridge_regression(X_train, Y_train, X_test, Y_test, model_alpha):
    clf = linear_model.Ridge(model_alpha)
    clf.fit(X_train, Y_train)
    predictions = clf.predict(X_test)
    loss = np.sum((predictions - Y_test)**2)
    return loss

def lasso_regression(X_train, Y_train, X_test, Y_test, model_alpha):
    clf = linear_model.Lasso(model_alpha)
    clf.fit(X_train, Y_train)
    predictions = clf.predict(X_test)
    loss = np.sum((predictions - Y_test)**2)
    return loss


X_train, X_test, Y_train, Y_test = cross_validation.train_test_split(X, Y, test_size=0.1, random_state=0)
for alpha in [0, 0.01, 0.1, 0.5, 1, 2, 5, 10, 100, 1000, 10000]:
    print("Lasso loss for alpha=" + str(alpha) +": " + str(lasso_regression(X_train, Y_train, X_test, Y_test, alpha)))

for alpha in [1, 1.25, 1.5, 1.75, 2, 5, 10, 100, 1000, 10000, 100000, 1000000]:
    print("Ridge loss for alpha=" + str(alpha) +": " + str(ridge_regression(X_train, Y_train, X_test, Y_test, alpha)))

And here is my conclusion:

Lasso loss for alpha=0: 20575.7121727
Lasso loss for alpha=0.01: 19762.8763969
Lasso loss for alpha=0.1: 17656.9926418
Lasso loss for alpha=0.5: 15699.2014387
Lasso loss for alpha=1: 15619.9772649
Lasso loss for alpha=2: 15490.0433166
Lasso loss for alpha=5: 15328.4303197
Lasso loss for alpha=10: 15328.4303197
Lasso loss for alpha=100: 15328.4303197
Lasso loss for alpha=1000: 15328.4303197
Lasso loss for alpha=10000: 15328.4303197
Ridge loss for alpha=1: 61.6235890425
Ridge loss for alpha=1.25: 61.6360790934
Ridge loss for alpha=1.5: 61.6496312133
Ridge loss for alpha=1.75: 61.6636076713
Ridge loss for alpha=2: 61.6776331539
Ridge loss for alpha=5: 61.8206621527
Ridge loss for alpha=10: 61.9883144732
Ridge loss for alpha=100: 63.9106882674
Ridge loss for alpha=1000: 69.3266510866
Ridge loss for alpha=10000: 82.0056669678
Ridge loss for alpha=100000: 88.4479064159
Ridge loss for alpha=1000000: 91.7235727543

Any idea why?

Thank!

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1 answer

An interesting problem. I can confirm that this is not a problem with the implementation of the algorithm, but the correct answer to your input.

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