  The lasso is the most famous sparse regression and feature selection method.
One reason for its popularity is the speed at which the underlying optimization
problem can be solved. Sorted L-One Penalized Estimation (SLOPE) is a
generalization of the lasso with appealing statistical properties. In spite of
this, the method has not yet reached widespread interest. A major reason for
this is that current software packages that fit SLOPE rely on algorithms that
perform poorly in high dimensions. To tackle this issue, we propose a new fast
algorithm to solve the SLOPE optimization problem, which combines proximal
gradient descent and proximal coordinate descent steps. We provide new results
on the directional derivative of the SLOPE penalty and its related SLOPE
thresholding operator, as well as provide convergence guarantees for our
proposed solver. In extensive benchmarks on simulated and real data, we show
that our method outperforms a long list of competing algorithms.
