= Fast-rate Lasso prediction bound under compatibility
{title2=$Q+(1-c)\lambda\|\delta_{S^c}\|_1\leq(1+c)^2\lambda^2s/\phi^2$}
Let $\delta=\widehat\beta-\beta^0$, $s=|S|>0$ and $\phi>0$. On the score event $\|X^T\eta\|_\infty/n\leq c\lambda$ with $0<c<1$, the <prediction inequality from Lasso stationarity> gives $Q+(1-c)\lambda\|\delta_{S^c}\|_1\leq(1+c)\lambda\|\delta_S\|_1$. If the <Compatibility condition for the Lasso> with constant $\phi$ holds on this cone, then $Q+(1-c)\lambda\|\delta_{S^c}\|_1\leq(1+c)^2\lambda^2s/\phi^2$. Here $Q=\|X\delta\|_2^2/n$ and $S$ is the true <support of a vector>.
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