In machine learning and AI, the MCMC method is a classical model-free method for sampling target distributions. A fact is that the optimal transport first-order method (gradient flow) forms the MCMC scheme, known as Langevin dynamics. A natural question arises: Can we propose more advanced optimization techniques for MCMC methods? We positively answer this question by considering the high order optimization method using transport information geometry. Here we introduce a theoretical framework for accelerated gradient flow and Newton’s flows in probability space with information metrics. Several numerical examples in Gaussian families and Bayesian logistic regression are shown to demonstrate the effectiveness of the proposed method.
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