A Continuum of Optimal Primal-Dual Algorithms for Convex Composite Minimization Problems with Applications to Structured Sparsity
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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.
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