  Imposition of a lasso penalty shrinks parameter estimates toward zero and
performs continuous model selection. Lasso penalized regression is capable of
handling linear regression problems where the number of predictors far exceeds
the number of cases. This paper tests two exceptionally fast algorithms for
estimating regression coefficients with a lasso penalty. The previously known
$\ell_2$ algorithm is based on cyclic coordinate descent. Our new $\ell_1$
algorithm is based on greedy coordinate descent and Edgeworth's algorithm for
ordinary $\ell_1$ regression. Each algorithm relies on a tuning constant that
can be chosen by cross-validation. In some regression problems it is natural to
group parameters and penalize parameters group by group rather than separately.
If the group penalty is proportional to the Euclidean norm of the parameters of
the group, then it is possible to majorize the norm and reduce parameter
estimation to $\ell_2$ regression with a lasso penalty. Thus, the existing
algorithm can be extended to novel settings. Each of the algorithms discussed
is tested via either simulated or real data or both. The Appendix proves that a
greedy form of the $\ell_2$ algorithm converges to the minimum value of the
objective function.
