Trimming the $\ell_1$ Regularizer: Statistical Analysis, Optimization, and Applications to Deep Learning
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Proceedings of the 36th International Conference on Machine Learning, PMLR 97:72427251, 2019.
Abstract
We study highdimensional estimators with the trimmed $\ell_1$ penalty, which leaves the h largest parameter entries penaltyfree. 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 Mestimation, 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 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 nonconvex alternatives. Our last contribution is to show that the trimmed penalty is beneficial beyond Mestimation, and yields promising results for two deep learning tasks: input structures recovery and network sparsification.
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