Kernel Ridge and Hyperparameter Cross Validation in Scikit-learn

Kernel Ridge is a regression algorithm that combines ridge regression with the kernel trick. It is used to solve regression problems in which the data has complex relationships that are difficult to model linearly. In this article, we will discuss how to use Kernel Ridge in scikit-learn and how to optimize its hyperparameters through cross-validation.

Kernel Ridge in Scikit-learn

To use Kernel Ridge in scikit-learn, we first need to import the necessary modules:

from sklearn.kernel_ridge import KernelRidge
from sklearn.metrics import mean_squared_error

We can create an instance of Kernel Ridge with the desired kernel function and regularization parameter as follows:

kr = KernelRidge(kernel='rbf', alpha=0.1)

The kernel parameter specifies the kernel function to use, and the alpha parameter specifies the regularization strength. Other kernel functions that can be used include linear, polynomial, and sigmoid.

We can train our model on a set of training data and evaluate it on a set of test data as follows:

kr.fit(X_train, y_train)
y_pred = kr.predict(X_test)
mse = mean_squared_error(y_test, y_pred)

Hyperparameter Optimization with Cross Validation

To optimize the hyperparameters of our Kernel Ridge model, we can use cross-validation. Cross-validation involves training and evaluating our model on different subsets of the data to get a more accurate estimate of its performance.

We can use scikit-learn's GridSearchCV function to perform a grid search over a range of hyperparameters, and choose the ones that provide the best performance. For Kernel Ridge, we can optimize two hyperparameters: alpha and gamma (the kernel coefficient for rbf, poly and sigmoid).

Here's an example of how to use cross-validation to optimize these hyperparameters:

from sklearn.model_selection import GridSearchCV

param_grid = {'alpha': [0.1, 1, 10],
              'gamma': [0.1, 1, 10]}
grid_search = GridSearchCV(KernelRidge(kernel='rbf'), param_grid, cv=5)
grid_search.fit(X_train, y_train)
print(grid_search.best_params_)

In this example, we search over a range of values for alpha and gamma, and use 5-fold cross-validation to evaluate their performance. The best_params_ attribute of the GridSearchCV object gives us the hyperparameters that provide the best performance according to the cross-validation results.

We can then use the best hyperparameters to retrain our model and evaluate its performance:

best_params = grid_search.best_params_
kr_best = KernelRidge(kernel='rbf', alpha=best_params['alpha'], gamma=best_params['gamma'])
kr_best.fit(X_train, y_train)
y_pred = kr_best.predict(X_test)
mse = mean_squared_error(y_test, y_pred)

Conclusion

In this article, we discussed how to use Kernel Ridge in scikit-learn and how to optimize its hyperparameters through cross-validation. By carefully choosing the right kernel function and regularization strength, we can improve the accuracy of our regression model and better capture the complex relationships in our data.