  Stochastic optimization is a vital field in the realm of mathematical
optimization, finding applications in diverse areas ranging from operations
research to machine learning. In this paper, we introduce a novel first-order
optimization algorithm designed for scenarios where Markovian noise is present,
incorporating Nesterov acceleration for enhanced efficiency. The convergence
analysis is performed using an assumption on noise depending on the distance to
the solution. We also delve into the consensus problem over Markov-varying
networks, exploring how this algorithm can be applied to achieve agreement
among multiple agents with differing objectives during changes in the
communication system. To show the performance of our method on the problem
above, we conduct experiments to demonstrate the superiority over the classic
approach.
