Random Coordinate Underdamped Langevin Monte Carlo

Zhiyan Ding, Qin Li, Jianfeng Lu, Stephen Wright
Proceedings of The 24th International Conference on Artificial Intelligence and Statistics, PMLR 130:2701-2709, 2021.

Abstract

The Underdamped Langevin Monte Carlo (ULMC) is a popular Markov chain Monte Carlo sampling method. It requires the computation of the full gradient of the log-density at each iteration, an expensive operation if the dimension of the problem is high. We propose a sampling method called Random Coordinate ULMC (RC-ULMC), which selects a single coordinate at each iteration to be updated and leaves the other coordinates untouched. We investigate the computational complexity of RC-ULMC and compare it with the classical ULMC for strongly log-concave probability distributions. We show that RC-ULMC is always cheaper than the classical ULMC, with a significant cost reduction when the problem is highly skewed and high dimensional. Our complexity bound for RC-ULMC is also tight in terms of dimension dependence.

Cite this Paper

BibTeX
@InProceedings{pmlr-v130-ding21b, title = { Random Coordinate Underdamped Langevin Monte Carlo }, author = {Ding, Zhiyan and Li, Qin and Lu, Jianfeng and Wright, Stephen}, booktitle = {Proceedings of The 24th International Conference on Artificial Intelligence and Statistics}, pages = {2701--2709}, year = {2021}, editor = {Banerjee, Arindam and Fukumizu, Kenji}, volume = {130}, series = {Proceedings of Machine Learning Research}, month = {13--15 Apr}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v130/ding21b/ding21b.pdf}, url = {https://proceedings.mlr.press/v130/ding21b.html}, abstract = { The Underdamped Langevin Monte Carlo (ULMC) is a popular Markov chain Monte Carlo sampling method. It requires the computation of the full gradient of the log-density at each iteration, an expensive operation if the dimension of the problem is high. We propose a sampling method called Random Coordinate ULMC (RC-ULMC), which selects a single coordinate at each iteration to be updated and leaves the other coordinates untouched. We investigate the computational complexity of RC-ULMC and compare it with the classical ULMC for strongly log-concave probability distributions. We show that RC-ULMC is always cheaper than the classical ULMC, with a significant cost reduction when the problem is highly skewed and high dimensional. Our complexity bound for RC-ULMC is also tight in terms of dimension dependence. } }
Endnote
%0 Conference Paper %T Random Coordinate Underdamped Langevin Monte Carlo %A Zhiyan Ding %A Qin Li %A Jianfeng Lu %A Stephen Wright %B Proceedings of The 24th International Conference on Artificial Intelligence and Statistics %C Proceedings of Machine Learning Research %D 2021 %E Arindam Banerjee %E Kenji Fukumizu %F pmlr-v130-ding21b %I PMLR %P 2701--2709 %U https://proceedings.mlr.press/v130/ding21b.html %V 130 %X The Underdamped Langevin Monte Carlo (ULMC) is a popular Markov chain Monte Carlo sampling method. It requires the computation of the full gradient of the log-density at each iteration, an expensive operation if the dimension of the problem is high. We propose a sampling method called Random Coordinate ULMC (RC-ULMC), which selects a single coordinate at each iteration to be updated and leaves the other coordinates untouched. We investigate the computational complexity of RC-ULMC and compare it with the classical ULMC for strongly log-concave probability distributions. We show that RC-ULMC is always cheaper than the classical ULMC, with a significant cost reduction when the problem is highly skewed and high dimensional. Our complexity bound for RC-ULMC is also tight in terms of dimension dependence.
APA
Ding, Z., Li, Q., Lu, J. & Wright, S.. (2021). Random Coordinate Underdamped Langevin Monte Carlo . Proceedings of The 24th International Conference on Artificial Intelligence and Statistics, in Proceedings of Machine Learning Research 130:2701-2709 Available from https://proceedings.mlr.press/v130/ding21b.html.

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