On the Convergence of Asynchronous Parallel Iteration with Arbitrary Delays

Recent years have witnessed the surge of asynchronous parallel (async-parallel) iterative algorithms due to problems involving very large-scale data and a large number of decision variables. Because of asynchrony, the iterates are computed with outdated information, and the age of the outdated information, which we call \emph{delay}, is the number of times it has been … Read more

Coordinate Friendly Structures, Algorithms and Applications

This paper focuses on coordinate update methods, which are useful for solving problems involving large or high-dimensional datasets. They decompose a problem into simple subproblems, where each updates one, or a small block of, variables while fixing others. These methods can deal with linear and nonlinear mappings, smooth and nonsmooth functions, as well as convex … Read more

ARock: an Algorithmic Framework for Asynchronous Parallel Coordinate Updates

We propose ARock, an asynchronous parallel algorithmic framework for finding a fixed point to a nonexpansive operator. In the framework, a set of agents (machines, processors, or cores) update a sequence of randomly selected coordinates of the unknown variable in an asynchronous parallel fashion. As special cases of ARock, novel algorithms for linear systems, convex … Read more

Self Equivalence of the Alternating Direction Method of Multipliers

The alternating direction method of multipliers (ADM or ADMM) breaks a complex optimization problem into much simpler subproblems. The ADM algorithms are typically short and easy to implement yet exhibit (nearly) state-of-the-art performance for large-scale optimization problems. To apply ADM, we first formulate a given problem into the “ADM-ready” form, so the final algorithm depends … Read more

One condition for all: solution uniqueness and robustness of l1-synthesis and l1-analysis minimizations

The l1-synthesis and l1-analysis models recover structured signals from their undersampled measurements. The solution of the former model is often a sparse sum of dictionary atoms, and that of the latter model often makes sparse correlations with dictionary atoms. This paper addresses the question: when can we trust these models to recover specific signals? We … Read more