On tight bounds for the Lasso

Understand

We present upper and lower bounds for the prediction error of the Lasso.

• For the case of random Gaussian design, we show that under mild conditions the prediction error of the Lasso is up to smaller order terms dominated by the prediction error of its noiseless counterpart.
• We then provide exact expressions for the prediction error of the latter, in terms of compatibility constants.
• Here, we assume the active components of the underlying regression function satisfy some "betamin" condition.