Stochastic Particle-Optimization Sampling and the Non-Asymptotic Convergence Theory

Jianyi Zhang, Ruiyi Zhang, Lawrence Carin, Changyou Chen
Proceedings of the Twenty Third International Conference on Artificial Intelligence and Statistics, PMLR 108:1877-1887, 2020.

Abstract

Particle-optimization-based sampling (POS) is a recently developed effective sampling technique that interactively updates a set of particles. A representative algorithm is the Stein variational gradient descent (SVGD). We prove, under certain conditions, SVGD experiences a theoretical pitfall, {\it i.e.}, particles tend to collapse. As a remedy, we generalize POS to a stochastic setting by injecting random noise into particle updates, thus termed stochastic particle-optimization sampling (SPOS). Notably, for the first time, we develop non-asymptotic convergence theory for the SPOS framework (related to SVGD), characterizing algorithm convergence in terms of the 1-Wasserstein distance w.r.t. the numbers of particles and iterations. Somewhat surprisingly, with the same number of updates (not too large) for each particle, our theory suggests adopting more particles does not necessarily lead to a better approximation of a target distribution, due to limited computational budget and numerical errors. This phenomenon is also observed in SVGD and verified via a synthetic experiment. Extensive experimental results verify our theory and demonstrate the effectiveness of our proposed framework.

Cite this Paper

BibTeX
@InProceedings{pmlr-v108-zhang20d, title = {Stochastic Particle-Optimization Sampling and the Non-Asymptotic Convergence Theory}, author = {Zhang, Jianyi and Zhang, Ruiyi and Carin, Lawrence and Chen, Changyou}, booktitle = {Proceedings of the Twenty Third International Conference on Artificial Intelligence and Statistics}, pages = {1877--1887}, year = {2020}, editor = {Chiappa, Silvia and Calandra, Roberto}, volume = {108}, series = {Proceedings of Machine Learning Research}, month = {26--28 Aug}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v108/zhang20d/zhang20d.pdf}, url = {https://proceedings.mlr.press/v108/zhang20d.html}, abstract = {Particle-optimization-based sampling (POS) is a recently developed effective sampling technique that interactively updates a set of particles. A representative algorithm is the Stein variational gradient descent (SVGD). We prove, under certain conditions, SVGD experiences a theoretical pitfall, {\it i.e.}, particles tend to collapse. As a remedy, we generalize POS to a stochastic setting by injecting random noise into particle updates, thus termed stochastic particle-optimization sampling (SPOS). Notably, for the first time, we develop non-asymptotic convergence theory for the SPOS framework (related to SVGD), characterizing algorithm convergence in terms of the 1-Wasserstein distance w.r.t. the numbers of particles and iterations. Somewhat surprisingly, with the same number of updates (not too large) for each particle, our theory suggests adopting more particles does not necessarily lead to a better approximation of a target distribution, due to limited computational budget and numerical errors. This phenomenon is also observed in SVGD and verified via a synthetic experiment. Extensive experimental results verify our theory and demonstrate the effectiveness of our proposed framework.} }
Endnote
%0 Conference Paper %T Stochastic Particle-Optimization Sampling and the Non-Asymptotic Convergence Theory %A Jianyi Zhang %A Ruiyi Zhang %A Lawrence Carin %A Changyou Chen %B Proceedings of the Twenty Third International Conference on Artificial Intelligence and Statistics %C Proceedings of Machine Learning Research %D 2020 %E Silvia Chiappa %E Roberto Calandra %F pmlr-v108-zhang20d %I PMLR %P 1877--1887 %U https://proceedings.mlr.press/v108/zhang20d.html %V 108 %X Particle-optimization-based sampling (POS) is a recently developed effective sampling technique that interactively updates a set of particles. A representative algorithm is the Stein variational gradient descent (SVGD). We prove, under certain conditions, SVGD experiences a theoretical pitfall, {\it i.e.}, particles tend to collapse. As a remedy, we generalize POS to a stochastic setting by injecting random noise into particle updates, thus termed stochastic particle-optimization sampling (SPOS). Notably, for the first time, we develop non-asymptotic convergence theory for the SPOS framework (related to SVGD), characterizing algorithm convergence in terms of the 1-Wasserstein distance w.r.t. the numbers of particles and iterations. Somewhat surprisingly, with the same number of updates (not too large) for each particle, our theory suggests adopting more particles does not necessarily lead to a better approximation of a target distribution, due to limited computational budget and numerical errors. This phenomenon is also observed in SVGD and verified via a synthetic experiment. Extensive experimental results verify our theory and demonstrate the effectiveness of our proposed framework.
APA
Zhang, J., Zhang, R., Carin, L. & Chen, C.. (2020). Stochastic Particle-Optimization Sampling and the Non-Asymptotic Convergence Theory. Proceedings of the Twenty Third International Conference on Artificial Intelligence and Statistics, in Proceedings of Machine Learning Research 108:1877-1887 Available from https://proceedings.mlr.press/v108/zhang20d.html.

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