= Prediction and coefficient error bound for compatible Lasso
{title2=$Q+\lambda\|\delta\|_1\leq16\lambda^2s/\phi_0^2$}
With objective $\|Y-Xb\|_2^2/(2n)+\lambda\|b\|_1$, the <Gaussian score event for the Lasso> $\|X^T\epsilon/n\|_\infty\leq\lambda/2$ gives $Q+\lambda\|\delta_{S^c}\|_1\leq3\lambda\|\delta_S\|_1$, where $Q=\|X\delta\|_2^2/n$. The <Lasso cone condition> activates the <Compatibility condition for the Lasso>. Then $Q+\lambda\|\delta\|_1\leq4\lambda\sqrt{sQ}/\phi_0\leq Q/2+8\lambda^2s/\phi_0^2$. Thus even $Q+2\lambda\|\delta\|_1\leq16\lambda^2s/\phi_0^2$. This controls prediction and coefficient error together, including a design with correlated columns.
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