Tikhonov regularization

Tikhonov regularization, also known as ridge regression or L2 regularization, is a regularization method used for linear regression. It controls the complexity and generalization ability of the model by adding an L2 norm penalty term to the objective function of the model. This penalty term penalizes the weight of the model by the sum of squares to avoid excessive weight, thereby mitigating the overfitting problem. This method introduces a regularization term into the loss function and adjusts the regularization coefficient to balance the fitting ability and generalization ability of the model. Tikhonov regularization has a wide range of applications in practical applications and can effectively improve the performance and stability of the model.

Before regularization, the objective function of linear regression can be expressed as:

J(w)=\frac{1}{2m }\sum_{i=1}^{m}(h_w(x^{(i)})-y^{(i)})^2

In this objective function , we can see that w is the weight vector of the model, h_w(x^{(i)}) is the model’s prediction result for the i-th sample x^{(i)}, y^{(i)} is the true label, m is the number of samples. In order to optimize this objective function, methods such as gradient descent are often used. These methods calculate the gradient of the objective function and update the weight vector w, thereby gradually reducing the value of the objective function, making the prediction results of the model closer to the real label. In this way, we can improve the performance of the model by optimizing the objective function.

In Tikhonov regularization, the objective function becomes:

J(w)=\frac{1}{ 2m}\sum_{i=1}^{m}(h_w(x^{(i)})-y^{(i)})^2 \frac{\lambda}{2}||w||_2 ^2

Among them, \lambda is the regularization parameter, used to control the intensity of the penalty term. ||w||_2^2 represents the L2 norm of the weight vector, which is the sum of the squares of all weights. This penalty term limits the values ​​of the weights so that they cannot be too large, thereby preventing the model from overfitting.

In practical applications, the value of the regularization parameter \lambda usually needs to be determined through cross-validation and other methods. If \lambda is too small, the regularization effect will become weak and the model will still be prone to overfitting; if \lambda is too large, the penalty term will overwhelm the original objective function, resulting in underfitting of the model.

Tikhonov regularization has some other characteristics and applications. For example, it can handle correlations between features better because it allows related feature weights to cancel each other out; it can also be used to handle high-dimensional data because it can reduce the number of features by penalizing unimportant features. .

The following is an example of linear regression using Tikhonov regularization.

Suppose there is a data set containing 2 features and a label. We use Python’s Scikit-learn library to achieve this:

from sklearn.linear_model import Ridge
from sklearn.model_selection import train_test_split
from sklearn.preprocessing import StandardScaler
from sklearn.datasets import make_regression

# 生成数据集
X, y = make_regression(n_samples=100, n_features=2, noise=0.5, random_state=42)

# 数据归一化
scaler = StandardScaler()
X = scaler.fit_transform(X)

# 划分训练集和测试集
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2, random_state=42)

# 构建模型
ridge = Ridge(alpha=1.0)  # alpha为正则化参数

# 模型训练
ridge.fit(X_train, y_train)

# 模型评估
print("Train score:", ridge.score(X_train, y_train))
print("Test score:", ridge.score(X_test, y_test))

In this example, we use the make_regression function of the Scikit-learn library to generate a dataset with 2 features and a label. We first normalized the data and then used the train_test_split function to divide the data set into a training set and a test set. Next, we used the Ridge function to build a Tikhonov regularized linear regression model, where the alpha parameter is the regularization parameter. Finally, we used the fit function to train the model, and used the score function to calculate the R2 scores on the training set and test set respectively.

It should be noted that the value of the regularization parameter alpha needs to be determined through cross-validation and other methods. In this example, we used the default value of alpha=1.0. If the alpha is too small, the model may not perform satisfactorily; if the alpha is too large, the model may be underfitted.

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