exact recovery, such a procedure requires the condition that $$\min_{{{\mathbf w}}_j \in \bar{F}} |{{\mathbf w}}_j|  \geq c' \sigma \sqrt{\bar{k}\ln p/n} \label{eq:min-bias}$$ for some constant $c'$ (also depends on the RIP condition). This extra $\sqrt{\bar{k}}$ factor is referred to as the bias of the Lasso procedure in [@Zhang10-mc+]. Moreover, it is known that for exact recovery to hold, the requirement of $\min_{{{\mathbf w}}_j \in \bar{F}} |{{\mathbf w}}_j| \geq c \sigma \sqrt{\ln p/n}$ (up to a constant) is necessary for all statistical procedures, in the sense that if $\min_{{{\mathbf w}}_j \in \bar{F}} |{{\mathbf w}}_j| \leq c' \sigma \sqrt{\ln p/n}$ for a sufficiently small constant $c'$ (under appropriate RIP conditions), then no statistical procedure can achieve exact recovery with large probability. Therefore statistical procedures that can achieve exact support recovery under (\[eq:min-bias\]) are referred to as (nearly) unbiased feature selection methods in [@Zhang10-mc+]. Theorem \[thm:multi-stage-featsel\] shows that multi-stage convex relaxation with capped-$L_1$ regularization achieves unbiased feature selection.

Results most comparable to what we have obtained here are that of the FoBa procedure in [@Zhang08-foba] and that of the MC+ procedure in [@Zhang10-mc+]. Both can be regarded as (approximate) optimization methods for nonconvex formulations. The former is a forward backward greedy algorithm, which does 