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.

Abstract

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.

Cite this Paper

BibTeX
@InProceedings{pmlr-v32-satoa14, title = {Approximation Analysis of Stochastic Gradient Langevin Dynamics by using Fokker-Planck Equation and Ito Process }, author = {Sato, Issei and Nakagawa, Hiroshi}, booktitle = {Proceedings of the 31st International Conference on Machine Learning}, pages = {982--990}, year = {2014}, editor = {Xing, Eric P. and Jebara, Tony}, volume = {32}, number = {2}, series = {Proceedings of Machine Learning Research}, address = {Bejing, China}, month = {22--24 Jun}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v32/satoa14.pdf}, url = {https://proceedings.mlr.press/v32/satoa14.html}, abstract = {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.} }
Endnote
%0 Conference Paper %T Approximation Analysis of Stochastic Gradient Langevin Dynamics by using Fokker-Planck Equation and Ito Process %A Issei Sato %A Hiroshi Nakagawa %B Proceedings of the 31st International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2014 %E Eric P. Xing %E Tony Jebara %F pmlr-v32-satoa14 %I PMLR %P 982--990 %U https://proceedings.mlr.press/v32/satoa14.html %V 32 %N 2 %X 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.
RIS
TY - CPAPER TI - Approximation Analysis of Stochastic Gradient Langevin Dynamics by using Fokker-Planck Equation and Ito Process AU - Issei Sato AU - Hiroshi Nakagawa BT - Proceedings of the 31st International Conference on Machine Learning DA - 2014/06/18 ED - Eric P. Xing ED - Tony Jebara ID - pmlr-v32-satoa14 PB - PMLR DP - Proceedings of Machine Learning Research VL - 32 IS - 2 SP - 982 EP - 990 L1 - http://proceedings.mlr.press/v32/satoa14.pdf UR - https://proceedings.mlr.press/v32/satoa14.html AB - 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. ER -
APA
Sato, I. & Nakagawa, H.. (2014). Approximation Analysis of Stochastic Gradient Langevin Dynamics by using Fokker-Planck Equation and Ito Process . Proceedings of the 31st International Conference on Machine Learning, in Proceedings of Machine Learning Research 32(2):982-990 Available from https://proceedings.mlr.press/v32/satoa14.html.

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