In optimization of the least absolute shrinkage and selection operator (Lasso) problem, the fastest algorithm has a convergence rate of $O(1/\sqrt{\epsilon})$.
This polynomial order of $1/\epsilon$ is caused by the undesirable behavior of the absolute function at the origin.
In this paper, we propose an algorithm called homotopy shrinkage yielding (HOSKY), which helps expedite the warm-up stage of the existing algorithms.
With the acceleration by HOSKY in the warm-up stage, one can get a provable convergence rate lower than $O(1/\sqrt{\epsilon})$.
The main idea of the proposed HOSKY algorithm is to use a sequence of surrogate functions to approximate the $\ell_1$ penalty that is used in Lasso.
This sequence of surrogate functions, on the one hand, gets closer and closer to the $\ell_1$ penalty; on the other hand, they are strictly convex and well-conditioned, which enables a provable exponential rate of convergence by gradient-based approaches.
As we will prove in this paper, the convergence rate of the HOSKY algorithm is $O([\log(1/\epsilon_w)]^2)$, where $\epsilon_w$ is the precision used in the warm-up stage.
Our numerical simulations also show that HOSKY empirically performs better in the warm-up stage and accelerates the overall convergence rate.
Zhao, Yujie, and Xiaoming Huo. "A homotopic method to solve the lasso problems with an improved upper bound of convergence rate." arXiv preprint arXiv:2010.13934 (2020).