If I use the LARS algorithm to fit the LASSO path, is it sufficient to cross-validate using the values of $\lambda$ at each step in LARS or is it better to use a finer grid of $\lambda$ values? I guess I can ask this in two parts:
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Is the prediction optimal model found at one of the LARS steps?
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Is the correct subset of variables found at one of the LARS steps?
I would have done the cross-validation over the parameter $s = \left\Vert \beta \right\Vert _{1}\left/ \max \left\Vert \beta \right\Vert
_{1}\right.$ on a fine grid with $s\in \left( 0,1\right) $… but then I thought that defeats the purpose of using LARS, which is supposed to be attractive for its lower computational expense. Some clarification would be greatly appreciated.
Best Answer
I have a better understanding of the LASSO and LARS algorithm now so I thought I'd share what I've learned for the benefit of others. I am aware that the pathway between the LARS steps is linear so that, as Donbeo says, linear interpolation can be used to find all LASSO solutions. The point of my question was to ascertain whether this is even necessary. Apparently it is not.
For variable selection - Yes, the best subset occurs at a LARS step. Reasoning:
For prediction - Yes, the optimal $\lambda$ occurs at a LARS step. I've just worked through this paper by Zou, et al (2007), which has answered my question (at least when $X$ has full rank):