  In optimization, it is known that when the objective functions are strictly
convex and well-conditioned, gradient-based approaches can be extremely
effective, e.g., achieving the exponential rate of convergence. On the other
hand, the existing Lasso-type estimator in general cannot achieve the optimal
rate due to the undesirable behavior of the absolute function at the origin. A
homotopic method is to use a sequence of surrogate functions to approximate the
$\ell_1$ penalty that is used in the Lasso-type of estimators. The surrogate
functions will converge to the $\ell_1$ penalty in the Lasso estimator. At the
same time, each surrogate function is strictly convex, which enables a provable
faster numerical rate of convergence. In this paper, we demonstrate that by
meticulously defining the surrogate functions, one can prove a faster numerical
convergence rate than any existing methods in computing for the Lasso-type of
estimators. Namely, the state-of-the-art algorithms can only guarantee
$O(1/\epsilon)$ or $O(1/\sqrt{\epsilon})$ convergence rates, while we can prove
an $O([\log(1/\epsilon)]^2)$ for the newly proposed algorithm. Our numerical
simulations show that the new algorithm also performs better empirically.
