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.

We adapt the Douglas–Rachford (DR) splitting method to solve nonconvex feasibility problems by studying this method for a class of nonconvex optimization problem. While the convergence properties of the method for convex problems have been well studied, far less is known in the nonconvex setting. In this paper, for the direct adaptation of the method to minimize the sum of a proper closed function g and a smooth function f with a Lipschitz continuous gradient, we show that if the step-size parameter is smaller than a computable threshold and the sequence generated has a cluster point, then it gives a stationary point of the optimization problem. Convergence of the whole sequence and a local convergence rate are also established under the additional assumption that f and g are semi-algebraic. We also give simple sufficient conditions guaranteeing the boundedness of the sequence generated. We then apply our nonconvex DR splitting method to finding a point in the intersection of a closed convex set C and a general closed set D by minimizing the squared distance to C subject to D. We show that if either set is bounded and the step-size parameter is smaller than a computable threshold, then the sequence generated from the DR splitting method is actually bounded. Consequently, the sequence generated will have cluster points that are stationary for an optimization problem, and the whole sequence is convergent under an additional assumption that C and D are semi-algebraic. We achieve these results based on a new merit function constructed particularly for the DR splitting method. Our numerical results indicate that our DR splitting method usually outperforms the usual DR splitting method and the alternating projection method in finding a sparse solution of a linear system, in terms of both the solution quality and the number of iterations taken.

We develop a generic game platform that can be used to model various real-world systems with multiple intelligent cloud-computing pools and parallel-queues for resources-competing users. Inside the platform, the software structure is modelled as Blockchain. All the users are associated with Big Data arrival streams whose random dynamics is modelled by triply stochastic renewal reward processes (TSRRPs). Each user may be served simultaneously by multiple pools while each pool with parallel- servers may also serve multi-users at the same time via smart policies in the Blockchain, e.g. a Nash equilibrium point myopically at each fixed time to a game-theoretic scheduling problem. To illustrate the effectiveness of our game platform, we model the performance measures of its internal data flow dynamics (queue length and workload processes) as reflecting diffusion with regime-switchings (RDRSs) under our scheduling policies. By RDRS models, we can prove our myopic game-theoretic policy to be an asymptotic Pareto minimal-dual-cost Nash equilibrium one globally over the whole time horizon to a randomly evolving dynamic game problem. Iterative schemes for simulating our multi-dimensional RDRS models are also developed with the support of numerical comparisons.

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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 and nonconvex problems. In addition, they are easy to parallelize. The great performance of coordinate update methods depends on solving simple sub-problems. To derive simple subproblems for several new classes of applications, this paper systematically studies coordinate-friendly operators that perform low-cost coordinate updates. Based on the discovered coordinate friendly operators, as well as operator splitting techniques, we obtain new coordinate update algorithms for a variety of problems in machine learning, image processing, as well as sub-areas of optimization. Several problems are treated with coordinate update for the first time in history. The obtained algorithms are scalable to large instances through parallel and even asynchronous computing. We present numerical examples to illustrate how effective these algorithms are.