\lambda$ for various constants $\mu=0.5,1,2,4$.

The experiments are repeated for 100 times, and Table \[tab:performance\] reports the probability (percentage in the 100 runs) of exact support recovery for each configuration at various stages $\ell$. Note that $\ell=1$ corresponds to Lasso and $\ell=2$ is an adaptive Lasso like two stage method [@Zou06; @ZouLi08]. The main purpose of this study is to illustrate that it is beneficial to use more than two stages, as predicted by our theory. However, since only $O(\ln(\bar{k}))$ is sufficient, optimal results can be achieved with relatively small number of stages. These conclusions can be clearly seen from Table \[tab:performance\]. Specifically the results for $\ell=2$ are better than those of $\ell=1$ (standard Lasso), while results of $\ell=4$ are better than those of $\ell=2$. Although the performance of $\ell=8$ is even better, the improve over $\ell=4$ is small at the optimal configuration of $\lambda$ and $\theta$. This is consistent with our theory, which implies that a relatively small number of stages is needed to achieve good performance.

[|c|c|c|c|c|c|c|c|]{}\
$\lambda$ & $ 0.23$ & $ 0.47$ & $ 0.94$ & $ 1.9$ & $ 3.8$ & $ 7.5$\
$\ell=1$ & $  0$ & $  0$ & $  0$ & $  