# Doubt on choice of $L_1$ regularization parameter for lasso

In this paper, on page $$14$$, I have some doubts in "How to choose $$\lambda$$" part. The author says

Let $$\beta^{(−i)}_{lasso}$$ be the LASSO solution obtained using $$(X^{(−i)},y^{(−i)})$$.

How can we solve the LASSO even after partitioning the data if we still don't have $$\lambda$$? What value of $$\lambda$$ do we use here?

• the author is just describing cross validation so that would be done for a range of $\lambda$s – jld Dec 26 '19 at 18:29
• Welcome to the community. As @jld said, the author describes (a bit too succinctly) a standard CV procedure. Please see my answer for a few more details. – usεr11852 Dec 26 '19 at 18:50

The optimal $$\lambda$$ is picked as "the one that minimizes the average cross validated mean squared error" (quote from the linked paper). In addition to that, through the CV partitioning we can still pick a "candidate" $$\lambda$$ (and the associated $$\beta^{(-i)}$$) such that it minimises the MSE within the particular $$i$$-th fold. The author could be a bit more specific as to the fact that multiple candidate $$\lambda$$'s are used sequentially and/or that $$\beta^{(-i)}$$ minimizes the error against $$y^{(-1)}$$ but the overall paragraph is not wrong. :)
• We will pick the $\lambda$ that minimises MSE with respect to this partitioning, then other $\lambda's$ which minimise MSE with respect to other partitionings, and then take the overall minimising $\lambda$. Is that the case?? – Martund Dec 27 '19 at 3:34