Penalized regression

The important variations of OLS regression fall under the theme of penalized regression. In ordinary regression, the returned fit is the best fit on the training data, which can lead to overfitting. Penalizing means that we add a penalty for overconfidence in the parameter values.

Tip

Penalized regression is about tradeoffs

Penalized regression is another example of the bias-variance tradeoff. When using a penalty, we get a worse fit in the training data as we are adding bias. On the other hand, we reduce the variance and tend to avoid overfitting. Therefore, the overall result might be generalized in a better way.

L1 and L2 penalties

There are two types of penalties that are typically used for regression: L1 and L2 penalties. The L1 penalty means that we penalize the regression by the sum of the absolute values of the coefficients, and the L2 penalty penalizes by the sum of squares.

Let us now explore these ideas formally. The OLS optimization is given as follows:

In the...