# If $\ell_0$ regularization can be done via the proximal operator, why are people still using LASSO?

I have just learned that a general framework in constrained optimization is called "proximal gradient optimization". It is interesting that the $$\ell_0$$ "norm" is also associated with a proximal operator. Hence, one can apply iterative hard thresholding algorithm to get the sparse solution of the following

$$\min \Vert Y-X\beta\Vert_F + \lambda \vert \beta \vert_0$$

If so, why people are still using $$\ell_1$$? If you can just get the result by non-convex optimization directly, why are people still using LASSO?

I want to know what's the downside of the proximal gradient approach for $$\ell_0$$ minimization. Is it because of the non-convexity and randomness associated with? That means the initial estimator is very important.

• Using in what context? – Neal Oct 15 at 18:24
• For example, at work, I don't really care what the scikit-learn or xgboost or statsmodels backends are doing as long as they fit a model correctly and in a reasonable amount of time. – Neal Oct 15 at 18:48
• – Rodrigo de Azevedo Oct 15 at 20:10
• @Neal, yes at work I don't care either. I am just curious.. – ArtificiallyIntelligence Oct 16 at 13:01