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There has been a growing interest in solving consensus optimization problems in a multi-agent system. It is known that there is an exactness-speed dilemma for decentralized optimization for the naive gradient method, where either we have fast linear convergence to a neighborhood of size O (η) using a fixed constant step size η , or slow convergence to the exact optimizer with a decreasing step size. We reinvestigate a special case of this problem from a statistical point of view, where each agent's local function is a sample estimate of a common population function. It is shown that when considering the convergence to the population target instead of to the exact consensual solution, the convergence is linear up to the statistical accuracy of the problem, using a step size only depending on the network characteristics instead of the sample size N. The objective function is not required to be strongly convex (but their statistical limit is), which allows for example the absolute deviation function which is used in median regression. Such results are in stark contrast to the optimization literature, where strong convexity of the objective function is necessary for linear convergence. [ABSTRACT FROM AUTHOR]

Testing for homogeneity of two random vectors is a fundamental problem in statistics. In the past two decades, numerous efforts have been made to detect heterogeneity when the random vectors are multivariate or even high dimensional. Due to the "curse of dimensionality", existing tests based on Euclidean distance may fail to capture the overall homogeneity in high-dimensional settings while can only capture the moment discrepancy. To address this issue, we propose a fully nonparametric test for homogeneity of two random vectors. Our method involves randomly selecting two subspaces consisting of components of the vectors, projecting the subspaces onto one-dimensional spaces, respectively, and constructing the test statistic using the Cramér–von Mises distance of the projections. To enhance the performance, we repeatedly implement this procedure to construct the final test statistic. Theoretically, if the replication time tends to infinity, we can avoid potential power loss caused by lousy directions. Owing to the U -statistic theory, the asymptotic null distribution of our proposed test is standard normal, regardless of the parent distributions of the random samples and the relationship between data dimensions and sample sizes. As a result, no re-sampling procedure is needed to determine critical values. The empirical size and power of the proposed test are demonstrated through numerical simulations. [ABSTRACT FROM AUTHOR]