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A regularization term (or regularizer) () is added to a loss function: = ((),) + where is an underlying loss function that describes the cost of predicting () when the label is , such as the square loss or hinge loss; and is a parameter which controls the importance of the regularization term.
This regularization function, while attractive for the sparsity that it guarantees, is very difficult to solve because doing so requires optimization of a function that is not even weakly convex. Lasso regression is the minimal possible relaxation of penalization that yields a weakly convex optimization problem.
In statistics and machine learning, lasso (least absolute shrinkage and selection operator; also Lasso, LASSO or L1 regularization) [1] is a regression analysis method that performs both variable selection and regularization in order to enhance the prediction accuracy and interpretability of the resulting statistical model. The lasso method ...
Ridge regression is a method of estimating the coefficients of multiple-regression models in scenarios where the independent ... this is known as L 2 regularization. ...
In statistics and, in particular, in the fitting of linear or logistic regression models, the elastic net is a regularized regression method that linearly combines the L 1 and L 2 penalties of the lasso and ridge methods. Nevertheless, elastic net regularization is typically more accurate than both methods with regard to reconstruction. [1]
Spectral Regularization is also used to enforce a reduced rank coefficient matrix in multivariate regression. [4] In this setting, a reduced rank coefficient matrix can be found by keeping just the top singular values, but this can be extended to keep any reduced set of singular values and vectors.
Thus, when only a proper subset of all the principal components are selected for regression, the PCR estimator so obtained is based on a hard form of regularization that constrains the resulting solution to the column space of the selected principal component directions, and consequently restricts it to be orthogonal to the excluded directions.
Regularization can be accomplished by restricting the hypothesis space . A common example would be restricting H {\displaystyle {\mathcal {H}}} to linear functions: this can be seen as a reduction to the standard problem of linear regression .