Regularization L1 L2

This class uses empirical risk minimization (i.e., ERM) to formulate the optimization problem built upon collected data. Note that empirical risk is usually measured by applying a loss function on the model's predictions on collected data points. If the training data does not contain enough data points (for example, to train a linear model in $n$-dimensional space, we need at least $n$ data points), overfitting may happen so that the model produced by ERM is good at describing training data but may fail to predict correct results in unseen events. Regularization is a common technique to alleviate such a phenomenon by penalizing the magnitude (usually measured by the norm function) of model parameters. This trainer supports elastic net regularization, which penalizes a linear combination of L1-norm (LASSO), $|| \textbf{w}_c ||_1$, and L2-norm (ridge), $|| \textbf{w}_c ||_2^2$ regularizations for $c=1,\dots,m$. L1-norm and L2-norm regularizations have different effects and uses that are complementary in certain respects.

Together with the implemented optimization algorithm, L1-norm regularization can increase the sparsity of the model weights, $\textbf{w}_1,\dots,\textbf{w}_m$. For high-dimensional and sparse data sets, if users carefully select the coefficient of L1-norm, it is possible to achieve a good prediction quality with a model that has only a few non-zero weights (e.g., 1% of total model weights) without affecting its prediction power. In contrast, L2-norm cannot increase the sparsity of the trained model but can still prevent overfitting by avoiding large parameter values. Sometimes, using L2-norm leads to a better prediction quality, so users may still want to try it and fine tune the coefficients of L1-norm and L2-norm. Note that conceptually, using L1-norm implies that the distribution of all model parameters is a Laplace distribution while L2-norm implies a Gaussian distribution for them.

An aggressive regularization (that is, assigning large coefficients to L1-norm or L2-norm regularization terms) can harm predictive capacity by excluding important variables from the model. For example, a very large L1-norm coefficient may force all parameters to be zeros and lead to a trivial model. Therefore, choosing the right regularization coefficients is important in practice.