Iteratively reweighted least square (IRLS) is a popular approach to solve
sparsity-enforcing regression problems in machine learning. State of the art
approaches are more efficient but typically rely on specific coordinate pruning
schemes. In this work, we show how a surprisingly simple reparametrization of
IRLS, coupled with a bilevel resolution (instead of an alternating scheme) is
able to achieve top performances on a wide range of sparsity (such as Lasso,
group Lasso and trace norm regularizations), regularization strength (including
hard constraints), and design matrices (ranging from correlated designs to
differential operators). Similarly to IRLS, our method only involves linear
systems resolutions, but in sharp contrast, corresponds to the minimization of
a smooth function. Despite being non-convex, we show that there is no spurious
minima and that saddle points are "ridable", so that there always exists a
descent direction. We thus advocate for the use of a BFGS quasi-Newton solver,
which makes our approach simple, robust and efficient. We perform a numerical
benchmark of the convergence speed of our algorithm against state of the art
solvers for Lasso, group Lasso, trace norm and linearly constrained problems.
These results highlight the versatility of our approach, removing the need to
use different solvers depending on the specificity of the ML problem under
study.

通过再参数化和双层解析度的相结合，我们提出了一种新的通过线性系统解决 Lasso 问题的方法，可以适用于各种疏松规则和设计矩阵，我们通过数值实验表明了该方法的高效性和鲁棒性。