Prediction and coefficient error bound for compatible Lasso ($Q+\lambda \Vert \delta {\Vert }_1\le 16{\lambda }^{2}s/{\varphi }_0^2$)

With objective $\Vert Y-Xb{\Vert }_2^2/(2n)+\lambda \Vert b{\Vert }_1$, the Gaussian score event for the Lasso $\Vert X^Tϵ/n{\Vert }_{\infty }\le \lambda /2$ gives $Q+\lambda \Vert {\delta }_{S^c}{\Vert }_1\le 3\lambda \Vert {\delta }_S{\Vert }_1$, where $Q=\Vert X\delta {\Vert }_2^2/n$. The Lasso cone condition activates the Compatibility condition for the Lasso. Then $Q+\lambda \Vert \delta {\Vert }_1\le 4\lambda \sqrt{sQ}/{\varphi }_0\le Q/2+8{\lambda }^{2}s/{\varphi }_0^2$. Thus even $Q+2\lambda \Vert \delta {\Vert }_1\le 16{\lambda }^{2}s/{\varphi }_0^2$. This controls prediction and coefficient error together, including a design with correlated columns.