Adaptive Momentum Algorithms With Application In High-order Markov Chain

High-order Markov chain is a stochastic model to describe a series of random events,in which the probability of each event depends on the state reached by several consecutive events ahead.As a statistical model of real-world process,Markov chain is applied widely in vehicle cruise control system,currency exchange rate and animal population dynamics.In which Markov chain limiting probability distribution vector plays an important role.The high-order is considered to be the common method to solve the limiting probability distribution vector of high-order Markov.However,the convergence rate of the high-order power method is very slow when the spectral gap of probability transfer tensor is relatively small what's more the convergence of the algorithm depends on the selection of the initial probability distribution vector.Based on above,this paper focuses on the fast and effective algorithm for solving the limiting probability distribution vector of high-order Markov chains.The specific content is as follows:Chapter ? concisely introduces introduce the background,significance and research status of the limiting probability distribution problem of higher order Markov chains and briefly states the research status of the traditional momentum algorithms.Chapter ? proposes a deformation scheme of adding momentum in the iteration process of power method on the basis of the calculation framework of higher-order power method,it combines with the idea of momentum algorithm in machine learning.That is,the high-order power method with momentum term.According to the rule of momentum parameter selection in traditional momentum method,the formula of momentum parameter selection is derived in this paper.Numerical experiments show that the convergence speed of higher-order power method is faster than that of high-order power method,and the iteration steps are less.Chapter ?,a high-order quadratic extrapolation algorithm is put forward.The highorder power method of driving quantity mentioned in the second chapter involves the selection of free parameters.On the basis of momentum formula,different momentum parameters need to be adjusted for different tensor examples and the generalization degree of the algorithm needs to be strengthened.Thus,a high-order quadratic extrapolation algorithm is raised without considering free parameters.Numerical experiments show the effectiveness and superiority of the high-order quadratic extrapolation method.The fourth chapter summarizes the full text and prospects the work.