L2 regular term

loss = loss_func(prediction, labels) + l2_reg * reg_lambda

This code calculates the overall loss （loss）value, which consists of two parts: prediction loss and regularization term. The code is explained in detail below:

• Prediction loss calculation: Use a custom loss function loss_functo calculate the prediction loss. The function receives two parameters predictionand labels, which represent the predicted result of the model and the true label, respectively. The loss function measures the degree of difference between the predicted result and the true label.

• Regularization term calculation: Use the previously calculated regularization term l2_regand multiply it by hyperparameters reg_lambdato control the strength of regularization. The calculation method of the regularization term here is to L2apply the norm to the parameters of the model, and to prevent overfitting by punishing the excessive value of the parameter.

• Summation of loss values: Add the prediction loss and the regularization term to get the overall loss value. This overall loss value will be used as the optimization target in the training process, by minimizing it to adjust the model parameters, so that the model can better fit the training data and have better generalization ability.

The hyperparameter reg_lambdais the weight of the regularization term, which is used to balance the prediction loss and the influence of the regularization term in the overall loss. By reg_lambdaadjusting the value of , you can control how much the model attaches importance to prediction accuracy and complexity during training.

Why do you want to multiply the l2 norm of the parameter by a hyperparameter as a regular term, and what are the benefits of doing so?

The purpose of multiplying the norm of the parameter L2by a hyperparameter as a regular term is to control the complexity of the model and avoid overfitting.

Overfitting is the phenomenon in which a model performs well on training data but performs poorly on new, unseen data. Overfitting usually occurs when the model is too complex or the amount of training data is insufficient. When a model is too complex, it can overfit the noise and details in the training data, resulting in poor generalization to new data.

Regularization is one of the commonly used techniques to prevent overfitting. The regularization term restricts the model parameters by introducing a penalty term in the loss function to reduce the value range of the parameters. L2Regularization is one of the commonly used regularization methods, which uses L2the norm of the parameter as a penalty term.

The benefits of multiplying the norm of the parameter L2by a hyperparameter as a regularizer include:

• Control model complexity: L2The square of the norm constrains the value of the parameter, prompting the model to choose a smaller parameter value. This helps avoid overly complex models and reduces the risk of overfitting.

• Feature selection: L2Norm regularization tends to make the parameters corresponding to some unimportant features tend to zero. By penalizing the value of the parameter, the regularization term can constrain the importance of features, helping the model to select the most relevant and important features.

• Balance prediction accuracy and model complexity: By adjusting hyperparameters, you can control the weight of the regularization term in the overall loss, thus balancing the importance of prediction accuracy and complexity during the training process of the model. This allows a trade-off between accuracy and complexity to be chosen according to the needs of a specific problem.

Multiplying the norm of the parameter L2by a hyperparameter as a regular term can effectively control the complexity of the model, reduce the risk of overfitting, and balance the trade-off between the prediction accuracy and complexity of the model.

In addition to using the above methods to prevent overfitting, what are the regularization methods?

Besides L2regularization, there are other commonly used regularization methods that can be used to prevent overfitting. Here are some common regularization methods:

• L1Regularization ( Lassoregularization): L1Regularization uses L1the norm of the parameter as a penalty term. Unlike L2regularization, L1regularization tends to zero some parameters, enabling feature selection and sparsity. L1Regularization can reduce the complexity of the model by reducing the parameter values ​​​​corresponding to unimportant features.

• Elastic NetRegularization: Elastic NetRegularization is a linear combination of L1regularization and L2regularization. It combines the advantages of both and can preserve some highly correlated features while feature selection. Elastic NetRegularization provides a more flexible regularization method by adjusting L1and regularizing the weights.L2

• Dropout: Dropoutis a regularization technique commonly used in neural networks. During the training process, Dropoutthe output of some neurons is randomly set to zero, thereby forcing the model to learn multiple independent sub-models. By randomly discarding neurons, Dropoutthe dependence of the neural network on specific neurons can be reduced, and the generalization ability of the model can be improved.

• Data Augmentation ( Data Augmentation): Data augmentation is a method of increasing the number and diversity of training samples by transforming or adding noise to the original data. Common data augmentation techniques include image flipping, rotation, scaling, cropping, and adding random noise. By increasing the diversity of the data, data augmentation can help the model better capture the invariance and generalization ability of the data.

• Early Stopping ( Early Stopping): Early stopping is a regularization method based on validation set performance. It prevents the model from overfitting the training data by monitoring the performance of the model on the validation set during training and stopping the training after the performance on the validation set begins to decline.

Why can controlling the norm of the parameters prevent the model from overfitting? What is the underlying principle and mathematical logic?

Controlling the norm of a parameter, such as L2norm or L1norm, can help prevent model overfitting for reasons related to the complexity of the model and the optimization objective during optimization. Here are some explanations of the underlying principles and mathematical logic:

• Simplify model complexity: Overfitting usually occurs when the model is too complex. The complexity of the model can be measured by the value range of the parameters, and the norm of the parameters is one of the measurement methods. By limiting the norm of the parameters, the model can be forced to choose smaller parameter values, reducing the complexity of the model. Simplifying the complexity of the model can reduce the model's overfitting to the noise and details in the training data.

• Reduce the interaction between parameters: There are usually mutual influences and dependencies among the parameters of the model. When model parameters are highly correlated, they may jointly learn subtle differences in the training data, leading to overfitting. By controlling the norm of the parameters, especially using the L2 norm for regularization, the interaction between parameters can be reduced, making the learning of the model more independent and stable.

• Trade-off of predictive accuracy and model complexity: The goal of the optimization process is to improve the predictive accuracy of the model by minimizing the loss function. However, simply pursuing the best fit on the training data can lead to overfitting. By introducing the norm of the parameters as a regular term, it is possible to trade off prediction accuracy and model complexity during optimization. The regularization term adds a penalty term to the loss function, prompting the optimization process to not only consider fitting the training data, but also try to select smaller parameter values ​​when looking for the optimal solution, thus balancing the fitting ability and complexity of the model .

• In terms of mathematical logic, behind L2 norm regularization is the introduction of the sum of squares of parameters into the loss function as a penalty term. By minimizing a loss function that includes an L2-norm penalty, the optimization process tends to choose smaller parameter values, since larger parameter values ​​lead to larger penalties, which increase the overall loss. In this way, the model will find a balance between fitting the training data and controlling the complexity of the model during the training process to avoid overfitting.

Controlling the norm of parameters can help prevent model overfitting by simplifying model complexity, reducing the interaction between parameters, and trading off predictive accuracy and model complexity. These mechanisms are realized mathematically by introducing a norm penalty term, thereby constraining the value of the parameters during the optimization process