The squared L2 norm, often referred to as L2 regularization or weight decay, is preferred over the non-squared L2 norm (sometimes referred to as the L2 norm) for regularizing neural networks and other machine learning models for several reasons:
Mathematical Convenience: The squared L2 norm has a simple mathematical form that makes it easier to work with during optimization. When you compute gradients for weight updates, the gradient of the squared L2 norm with respect to a weight vector is just a linear multiple of the weight vector itself. This leads to straightforward weight update rules and efficient optimization.
Smoothing Effect: Squaring the L2 norm amplifies the effect of larger weight values while penalizing smaller weights. This has the practical effect of encouraging the model to distribute the error evenly among its parameters. It helps prevent the model from becoming too reliant on a few large weights, which can lead to overfitting.
Stronger Regularization: The squared L2 norm provides stronger regularization compared to the non-squared L2 norm. By squaring the weights, it penalizes large weights more severely. This results in a stronger push toward smaller weight values, reducing the risk of overfitting.
Convex Loss Function: The squared L2 norm regularization term, when added to a convex loss function, creates a convex regularized loss function. Convex optimization problems have desirable properties, such as having a unique global minimum. This can make the optimization process more stable and predictable.
Consistency with Weight Initialization: Many weight initialization techniques, such as Xavier/Glorot initialization and He initialization, are designed with squared L2 norm regularization in mind. The squared L2 norm encourages weight values to be close to zero, which aligns with the principles behind these initialization methods.
Parameter Scaling Invariance: Squared L2 norm regularization is invariant to changes in the scale of the parameters. In other words, if you multiply all the weights by a constant, the effect of regularization remains the same, making it more robust to weight scaling.
Convergence Properties: Squared L2 norm regularization tends to encourage quicker convergence during training. It can help stabilize the training process and make it less sensitive to the choice of hyperparameters.
Overall, the squared L2 norm is a well-established and widely used form of regularization in deep learning and machine learning because of its mathematical properties, regularization strength, and the beneficial effects it has on training stability and generalization performance. It strikes a balance between encouraging small weight values and avoiding overfitting, making it a valuable tool for improving the performance of neural networks.