Approximation Analysis of Stochastic Gradient Langevin Dynamics by using Fokker-Planck Equation and Ito Process


Issei Sato, Hiroshi Nakagawa ;
Proceedings of the 31st International Conference on Machine Learning, PMLR 32(2):982-990, 2014.


The stochastic gradient Langevin dynamics (SGLD) algorithm is appealing for large scale Bayesian learning. The SGLD algorithm seamlessly transit stochastic optimization and Bayesian posterior sampling. However, solid theories, such as convergence proof, have not been developed. We theoretically analyze the SGLD algorithm with constant stepsize in two ways. First, we show by using the Fokker-Planck equation that the probability distribution of random variables generated by the SGLD algorithm converges to the Bayesian posterior. Second, we analyze the convergence of the SGLD algorithm by using the Ito process, which reveals that the SGLD algorithm does not strongly but weakly converges. This result indicates that the SGLD algorithm can be an approximation method for posterior averaging.

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