  Many statistical learning problems can be posed as minimization of a sum of
two convex functions, one typically a composition of non-smooth and linear
functions. Examples include regression under structured sparsity assumptions.
Popular algorithms for solving such problems, e.g., ADMM, often involve
non-trivial optimization subproblems or smoothing approximation. We consider
two classes of primal-dual algorithms that do not incur these difficulties, and
unify them from a perspective of monotone operator theory. From this
unification we propose a continuum of preconditioned forward-backward operator
splitting algorithms amenable to parallel and distributed computing. For the
entire region of convergence of the whole continuum of algorithms, we establish
its rates of convergence. For some known instances of this continuum, our
analysis closes the gap in theory. We further exploit the unification to
propose a continuum of accelerated algorithms. We show that the whole continuum
attains the theoretically optimal rate of convergence. The scalability of the
proposed algorithms, as well as their convergence behavior, is demonstrated up
to 1.2 million variables with a distributed implementation.
