M-estimation with the Trimmed l1 Penalty

TL;DR

This paper introduces the first statistical analysis of high-dimensional estimators with the trimmed  penalty, demonstrating its advantages in support recovery, error bounds, and applications in deep learning tasks.

Contribution

It provides the first statistical properties of the trimmed  penalty, develops a new efficient algorithm, and shows its benefits in various applications including deep learning.

Findings

Trimmed  outperforms vanilla  in sparse linear regression.

The proposed algorithm converges faster and finds lower objective values.

Trimmed  improves support recovery and error bounds in high-dimensional estimation.

Abstract

We study high-dimensional estimators with the trimmed $\ell_1$ penalty, which leaves the $h$ largest parameter entries penalty-free. While optimization techniques for this nonconvex penalty have been studied, the statistical properties have not yet been analyzed. We present the first statistical analyses for $M$-estimation and characterize support recovery, $\ell_\infty$ and $\ell_2$ error of the trimmed $\ell_1$ estimates as a function of the trimming parameter $h$. Our results show different regimes based on how $h$ compares to the true support size. Our second contribution is a new algorithm for the trimmed regularization problem, which has the same theoretical convergence rate as the difference of convex (DC) algorithms, but in practice is faster and finds lower objective values. Empirical evaluation of $\ell_1$ trimming for sparse linear regression and graphical model estimation indicate that trimmed $\ell_1$ can outperform vanilla $\ell_1$ and non-convex alternatives. Our last contribution is to show that the trimmed penalty is beneficial beyond $M$-estimation, and yields promising results for two deep learning tasks: input structures recovery and network sparsification.