Group lasso is a commonly used regularization method in statistical learning in which parameters are eliminated from the model according to predefined groups. However, when the groups overlap, optimizing the group lasso penalized objective can be time-consuming on large-scale problems because of the non-separability induced by the overlapping groups. This bottleneck has seriously limited the application of overlapping group lasso regularization in many modern problems, such as gene pathway selection and graphical model estimation. In this paper, we propose a separable penalty as an approximation of the overlapping group lasso penalty. Thanks to the separability, the computation of regularization based on our penalty is substantially faster than that of the overlapping group lasso, especially for large-scale and high-dimensional problems. We show that the penalty is the tightest separable relaxation of the overlapping group lasso norm within the family of $\ell_{q_1}/\ell_{q_2}$ norms. Moreover, we show that the estimator based on the proposed separable penalty is statistically equivalent to the one based on the overlapping group lasso penalty with respect to their error bounds and the rate-optimal performance under the squared loss. We demonstrate the faster computational time and statistical equivalence of our method compared with the overlapping group lasso in simulation examples and a classification problem of cancer tumors based on gene expression and multiple gene pathways.

翻译：组套索是统计学习中常用的正则化方法，它根据预定义组别从模型中剔除参数。然而，当组间存在重叠时，由于重叠组导致的非可分性，在大规模问题中优化组套索罚目标会消耗大量时间。这一瓶颈严重限制了重叠组套索正则化在现代许多问题中的应用，例如基因通路选择和图形模型估计。在本文中，我们提出一种可分离惩罚项作为重叠组套索惩罚的近似。得益于其可分性，基于该惩罚的正则化计算速度显著快于重叠组套索，尤其对于大规模和高维问题。我们证明，该惩罚项是$\ell_{q_1}/\ell_{q_2}$范数族中最紧致的可分离松弛形式。此外，我们表明基于所提可分离惩罚项的估计量在误差界和平方损失下的速率最优性方面与基于重叠组套索惩罚的估计量统计等价。通过模拟实验及基于基因表达与多基因通路的癌症肿瘤分类问题，我们验证了所提方法相较于重叠组套索在计算速度和统计等价性上的优势。