M-estimation with the Trimmed l1 Penalty

We study high-dimensional M-estimators with the trimmed $\ell_1$ penalty. While standard $\ell_1$ penalty incurs bias (shrinkage), trimmed $\ell_1$ leaves the $h$ largest entries penalty-free. This family of estimators include the Trimmed Lasso for sparse linear regression and its counterpart for sparse graphical model estimation. The trimmed $\ell_1$ penalty is non-convex, but unlike other non-convex regularizers such as SCAD and MCP, it is not amenable and therefore prior analyzes cannot be applied. We characterize the support recovery of the estimates as a function of the trimming parameter $h$. Under certain conditions, we show that for any local optimum, (i) if the trimming parameter $h$ is smaller than the true support size, all zero entries of the true parameter vector are successfully estimated as zero, and (ii) if $h$ is larger than the true support size, the non-relevant parameters of the local optimum have smaller absolute values than relevant parameters and hence relevant parameters are not penalized. We then bound the $\ell_2$ error of any local optimum. These bounds are asymptotically comparable to those for non-convex amenable penalties such as SCAD or MCP, but enjoy better constants. We specialize our main results to linear regression and graphical model estimation. Finally, we develop a fast provably convergent optimization algorithm for the trimmed regularizer problem. The algorithm has the same rate of convergence as difference of convex (DC)-based approaches, but is faster in practice and finds better objective values than recently proposed algorithms for DC optimization. Empirical results further demonstrate the value of $\ell_1$ trimming.