We study the effect of norm based regularization on the size of coresets for the regularized regression problems. Specifically, given a matrix $ \mathbf{A} \in {\mathbb{R}}^{n \times d}$ with $n\gg d$ and a vector $\mathbf{b} \in \mathbb{R} ^ n $ and $\lambda > 0$, we analyze the size of coresets for regularized versions of regression of the form $\|\mathbf{Ax}-\mathbf{b}\|_p^r + \lambda\|{\mathbf{x}}\|_q^s$ . It has been shown for the case of ridge regression ($p,q,r,s=2$) that we can obtain a coreset smaller than the coreset for its unregularized counterpart i.e. least squares regression (Avron et al.). We show that when $r \neq s$, no coreset for some regularized regression can have size smaller than the optimal coreset of the unregularized version. The well known lasso problem falls under this category and hence does not allow a coreset smaller than the one for least squares regression. We propose a modified version of the lasso problem and obtain for it a coreset of size smaller than the least square regression. We empirically show that the modified version of lasso also induces sparsity in solution like the lasso. We also obtain smaller coresets for $\ell_p$ regression with $\ell_p$ regularization. We extend our methods to multi response regularized regression. Finally, we empirically demonstrate the coreset performance for the modified lasso and the $\ell_1$ regression with $\ell_1$ regularization.

本文研究基于规范化的正则化回归问题的簇核大小的影响，并在此基础上探讨了正则化回归的核心集比未正则化版本更小的情况，提出了一个修改后的 Lasso 问题，获得比最小二乘回归更小的核心集，并在多响应规则化回归中扩展了我们的方法，并通过实验展示了改进 Lasso 和 L1 回归的核心集表现。

关于正则化回归的核心集