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.

翻译：在优化中，已知当目标函数严格凸且条件良好时，基于梯度的方法可以极其有效，例如实现指数收敛速度。另一方面，由于绝对值函数在原点的不良行为，现有的Lasso型估计通常无法达到最优收敛速度。同伦方法通过使用一系列代理函数来近似Lasso型估计中使用的ℓ1惩罚项。这些代理函数将收敛到Lasso估计中的ℓ1惩罚，同时每个代理函数是严格凸的，从而可实现可证明的更快数值收敛速度。本文证明，通过精心定义代理函数，我们可以在计算Lasso型估计时获得比现有方法更快的数值收敛速度。具体而言，现有最先进算法仅能保证O(1/ε)或O(1/√ε)的收敛速度，而我们对新提出的算法可证明O([log(1/ε)]^2)的收敛速度。数值模拟表明，新算法在实际中表现也更优。