Finding a fixed point to a nonexpansive operator, i.e., x = Tx, abstracts many problems in numerical linear algebra, optimization, and other areas of data sciences. To solve xed- point problems, we propose ARock, an algorithmic framework in which multiple agents (machines, processors, or cores) update x in an asynchronous parallel fashion. Asynchrony is crucial to parallel computing since it reduces synchronization wait, relaxes communication bottleneck, and thus speeds up computing significantly. At each step of ARock, an agent updates a randomly selected coordinate xi based on possibly out-of-date information on x. The agents share x through either global memory or communication. If writing xi is atomic, the agents can read and write x without memory locks. We prove that if the nonexpansive operator T has a fixed point, then with probability one, ARock generates a sequence that converges to a fixed point of T. Our conditions on T and step sizes are weaker than comparable work. Linear convergence is obtained under suitable assumptions. We propose special cases of ARock for linear systems, convex optimization, machine learning, as well as distributed and decentralized consensus problems. Numerical experiments of solving sparse logistic regression problems are presented.

No abstract uploaded!

Frequentists' inference often delivers point estimators associated with condence intervals or sets for parameters of interest. Constructing the condence intervals or sets requires understanding the sampling distributions of the point estimators, which, in many but not all cases, are related to asymptotic Normal distributions ensured by central limit theorems. Although previous literature has established various forms of central limit theorems for statistical inference in super population models, we still need general and convenient forms of central limit theorems for some randomization-based causal analysis of experimental data, where the parameters of interests are functions of a nite population and randomness comes solely from the treatment assignment. We use central limit theorems for sample surveys and rank statistics to establish general forms of the nite population central limit theorems that are particularly useful for proving asymptotic distributions of randomization tests under the sharp null hypothesis of zero individual causal eects, and for obtaining the asymptotic repeated sampling distributions of the causal eect estimators. The new central limit theorems hold for general experimental designs with multiple treatment levels, multiple treatment factors and vector outcomes, and are immediately applicable for studying the asymptotic properties of many methods in causal inference, including instrumental variable, regression adjustment, rerandomization, clustered randomized experiments, and so on. Previously, the asymptotic properties of these problems are often based on heuristic arguments, which in fact rely on general forms of nite population central limit theorems that have not been established before. Our new theorems ll in this gap by providing more solid theoretical foundation for asymptotic randomization-based causal inference.

The nuclear norm is widely used to induce low-rank solutions for many optimization problems with matrix variables. Recently, it has been shown that the augmented Lagrangian method (ALM) and the alternating direction method (ADM) are very efficient for many convex programming problems arising from various applications, provided that the resulting subproblems are sufficiently simple to have closed-form solutions. In this paper, we are interested in the application of the ALM and the ADM for some nuclear norm involved minimization problems. When the resulting subproblems do not have closed-form solutions, we propose to linearize these subproblems such that closed-form solutions of these linearized subproblems can be easily derived. Global convergence results of these linearized ALM and ADM are established under standard assumptions. Finally, we verify the effectiveness and efficiency of these new methods by some numerical experiments.

It has been an open question whether the family of merit functions $\psi _p\(p> 1) $, the generalized Fischer-Burmeister (FB) merit function, associated to the second-order cone is smooth or not. In this paper we answer it partly, and show that \psi _p is smooth for \psi _p , and we provide the condition for its coerciveness. Numerical results are reported to illustrate the influence of \psi _p on the performance of the merit function method based on \psi _p .