This paper investigates the influence of directed networks on the convergence of smooth non-convex stochastic decentralized optimization associated with column-stochastic mixing matrices. We find that the canonical spectral gap, a widely-used metric in undirected networks, alone fails to adequately characterize the impact of directed networks. Through a new analysis of the Push-Sum strategy, a fundamental building block for decentralized algorithms over directed graphs, we identify another novel metric called the equilibrium skewness. Next, we establish the first convergence lower bound for non-convex stochastic decentralized algorithms over directed networks, which explicitly manifests the impact of both the spectral gap and equilibrium skewness and justifies the imperative need for both metrics in analysis.Moreover, by jointly considering the spectral gap and equilibrium skewness, we present the state-of-theart convergence rate for the Push-DIGing algorithm. Our findings reveal that these two metrics exert a more pronounced negative impact on Push-DIGing than suggested by our lower bound. We further integrate the technique of multi-round gossip to Push-DIGing to obtain MG-Push-DIGing, which nearly achieves the established lower bound, demonstrating its convergence optimality, best-possible resilience to directed networks, and the tightness of our lower bound. Experiments verify our theoretical findings.
Space arrangement of basic functions of the natural scenes decomposed by basic ICA which simulates visual perception is chaotic. The result is contradicted with physiological mechanisms of vision. So, we put up with a new model to solve the problem which based on the information integrate mechanism in visual cortex receptive fields. And, to solve the problem of train image fault area detection, a novel algorithm is proposed by using this new model. Experimental results show that the new algorithm can increase fault detection rate with high efficiency and little samples compared with traditional methods which absence of the visual information integrate mechanisms.
Keywords-Visual information integrate; topology basic functions; neuronal response; fault detectionI.
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