w}}}_I\|_2 .$$

Using $\lambda_0/(\lambda_G-2\|\hat{\epsilon}\|_\infty)
  \leq 3$, we obtain from Lemma \[lem:L1-nonsparse-dr\] $$\|\hat{{{\mathbf w}}}_G\|_1 \leq 3 \|\Delta \hat{{{\mathbf w}}} - \hat{{{\mathbf w}}}_G \|_1 .$$ Therefore $$\begin{aligned}
\|\Delta \hat{{{\mathbf w}}}-\Delta \hat{{{\mathbf w}}}_I\|_\infty \leq& \|\Delta \hat{{{\mathbf w}}}_J\|_1/s \\
=& s^{-1}
[\|\Delta \hat{{{\mathbf w}}}_G\|_1 - \|\Delta \hat{{{\mathbf w}}}-\Delta \hat{{{\mathbf w}}}_I\|_1] \\
\leq& 
s^{-1} [3\|\Delta \hat{{{\mathbf w}}} - \hat{{{\mathbf w}}}_G\|_1 - \|\Delta \hat{{{\mathbf w}}}-\Delta \hat{{{\mathbf w}}}_I\|_1],\end{aligned}$$ which implies that $$\begin{aligned}
\|\Delta \hat{{{\mathbf w}}}-\Delta \hat{{{\mathbf w}}}_I\|_2 \leq &
(\|\Delta \hat{{{\mathbf w}}}-\Delta \hat{{{\mathbf w}}}_I\|_1
\|\Delta \hat{{{\mathbf w}}}-\Delta \hat{{{\mathbf w}}}_I\|_\infty)^{1/2} \\
\leq& 
\left[\|\Delta \hat{{{\mathbf w}}}-\Delta \hat{{{\mathbf w}}}_I\|_1
(3\|\Delta \hat{{{\mathbf w}}} - \hat{{{\mathbf w}}}_G\|_1 - \|\Delta \hat{{{\mathbf w}}}-\Delta \hat{{{\mathbf w}}}_I\|_1)\right]^{1/2} s^{-1/2} \\
\leq& 
\left[
(3\|\Delta \hat{{{\mathbf w}}} - \hat{{{\mathbf w}}}_G\|_1/2)^2\right]^{1/2} s^{-1/2} \\
\leq& (3/2) s^{-1/2} \|\Delta \hat{{{\mathbf w}}} - \hat{{{\mathbf w}}}_G\|_1 \\
\leq& (3/2) s^{-1/2} \bar{k}^{1/2} \|\Delta \hat{{{\mathbf w}}} - \hat{{{\mathbf w}}}_G\|_2
\leq (3 k/s)^{1/2} \|\Delta \hat{{{\mathbf w}}}_I\|_2 .\end{aligned}$$ The third inequality uses the simple algebraic inequality $a(3b-a) \leq (3b/2)^2$. By rearranging this inequality, we obtain the desired bound. Note that in the above derivation, we have used the fact that $\bar{F} \cap G=\emptyset$, which implies that $\Delta \hat{{{\mathbf w}}}_G=\hat{{{\mathbf w}}}_G$, and thus $\Delta \hat{{{\mathbf w}}} - \hat{{{\mathbf w}}}_G=\Delta\hat{{{\mathbf w}}}_{G^c}$.

\[lem:L1-nonsparse1\] Let the conditions of Lemma \[lem:L1-nonsparse-dr\] and Lemma 