# Why using L1 regularization over L2?

Conducting a linear regression model using a loss function, why should I use $L_1$ instead of $L_2$ regularization?

Is it better at preventing overfitting? Is it deterministic (so always a unique solution)? Is it better at feature selection (because producing sparse models)? Does it disperse the weights among the features?

• L2 does not do variable selection, so L1 is definitively better at this. Commented Oct 12, 2017 at 20:01
• stats.stackexchange.com/questions/866/… Commented Oct 13, 2017 at 3:13

## 2 Answers

Basically, we add a regularization term in order to prevent the coefficients to fit so perfectly to overfit.

The difference between L1 and L2 is L1 is the sum of weights and L2 is just the sum of the square of weights.

L1 cannot be used in gradient-based approaches since it is not-differentiable unlike L2

L1 helps perform feature selection in sparse feature spaces.Feature selection is to know which features are helpful and which are redundant.

The difference between their properties can be summarized as :

• It isn't true that "L1 cannot be used in gradient-based approaches". Keras supports it, for instance. Yes, the derivative is always constant so it makes it harder for gradient descent to find the minimum. But regularization is a small term within the loss function, so it's not very important in the grand scheme of things. Commented Oct 19, 2017 at 18:49

L2 has one very important advantage to L1, and that is invariance to rotation and scale.

This is especially important in geographical / physical application.

Say your technician accidentally installed your sensor in 45 degree angel, L1 would be affected, while L2 (Euclidean distance) would remain the same.

• This is not at all an answer to the question. Commented Oct 14, 2017 at 3:16
• Could you explain the invariance, please? Commented Oct 18, 2017 at 5:41
• @Chati, the question is about regularization. You are confusing it with other uses of 1-norm and 2-norm in loss functions. Commented Oct 19, 2017 at 18:47