# how Lasso regression helps to shrinks the coefficient to zero and why ridge regression dose not shrink the coefficient to zero?

How Lasso regression helps feature selection of model by making the coefficient to zero? ,

I could see few below with below diagram ,can any please explain in simple terms how to corelate below diagram with i.) how lasso shrinks the coefficient to zero ii.) how ridge dose not shrinks the coefficient to zero

## 2 Answers

These diagrams show the "constrained" version of lasso/ridge, in which you minimize the pure loss function subject to a constraint $$\|\beta\|_1\leq t$$ or $$\|\beta\|_2\leq t$$. (Another common version adds a penalty to the loss, and these are equivalent.)

The bluish solid shapes are the set of points with $$\|\beta\|\leq t$$, on the left with L1 norm and on the right with L2 norm. $$\hat{\beta}$$ represents the unpenalized optimum value of $$\beta$$, and the ovals around it are the level curves of the loss function.

We know that the optimum constrained solution will occur at a point of tangency of the level curves of the loss function and the constraint boundary, specifically with the "smallest" level curve that touches the constraint region. Then the point of the picture is that, with the pointy L1 constraint, you are much more likely for that point of tangency to occur at one of the corners, which correspond to one (or more, if we try to generalize to higher dimensions) of the coordinates in $$\beta$$ being zero. Compare to the L2 boundary, where the tangency being exactly along one of the axes is much less likely.

This StatQuest video does a fantastic job of explaining in simple terms why this is the case.