  The pathwise coordinate optimization is one of the most important
computational frameworks for high dimensional convex and nonconvex sparse
learning problems. It differs from the classical coordinate optimization
algorithms in three salient features: {\it warm start initialization}, {\it
active set updating}, and {\it strong rule for coordinate preselection}. Such a
complex algorithmic structure grants superior empirical performance, but also
poses significant challenge to theoretical analysis. To tackle this long
lasting problem, we develop a new theory showing that these three features play
pivotal roles in guaranteeing the outstanding statistical and computational
performance of the pathwise coordinate optimization framework. Particularly, we
analyze the existing pathwise coordinate optimization algorithms and provide
new theoretical insights into them. The obtained insights further motivate the
development of several modifications to improve the pathwise coordinate
optimization framework, which guarantees linear convergence to a unique sparse
local optimum with optimal statistical properties in parameter estimation and
support recovery. This is the first result on the computational and statistical
guarantees of the pathwise coordinate optimization framework in high
dimensions. Thorough numerical experiments are provided to support our theory.
