  We study the effect of norm based regularization on the size of coresets for
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 $\|\mathbf{Ax}-\mathbf{b}\|_p^r
+ \lambda\|{\mathbf{x}}\|_q^s$ . Prior work has shown that for ridge regression
(where $p,q,r,s=2$) we can obtain a coreset that is smaller than the coreset
for the unregularized counterpart i.e. least squares regression (Avron et al).
We show that when $r \neq s$, no coreset for 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, similar to the original
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.
