Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 205 / 3 / a

With the normalization used here, the Lasso estimator minimizesOptimality at $\hat{\beta }$ relative to the feasible point ${\beta }^0$ givesThe columns of $X$ are centered, so $X^{T}1=0$ and the centered noise produces the same score function as $\epsilon$. Expanding the two squared norms and cancelling the noise norm yields the standard Basic inequality for the LassoThus the displayed inequality in the question has a factor-of-two typo: its left side should be $\Vert X(\hat{\beta }-{\beta }^0){\Vert }_2^2/(2n)$, or both terms on its right should be doubled. No scaling of the usual squared-error Lasso objective produces the three displayed coefficients simultaneously. Parts b and d explicitly ask us to use the stated inequality, so their requested constants follow from that stated version.