Accelerated Flow for Probability Distributions

Amirhossein Taghvaei, Prashant Mehta
Proceedings of the 36th International Conference on Machine Learning, PMLR 97:6076-6085, 2019.

Abstract

This paper presents a methodology and numerical algorithms for constructing accelerated gradient flows on the space of probability distributions. In particular, we extend the recent variational formulation of accelerated methods in (Wibisono et al., 2016) from vector valued variables to probability distributions. The variational problem is modeled as a mean-field optimal control problem. A quantitative estimate on the asymptotic convergence rate is provided based on a Lyapunov function construction, when the objective functional is displacement convex. An important special case is considered where the objective functional is the relative entropy. For this case, two numerical approximations are presented to implement the Hamilton’s equations as a system of N interacting particles. The algorithm is numerically illustrated and compared with the MCMC and Hamiltonian MCMC algorithms.

Cite this Paper

BibTeX
@InProceedings{pmlr-v97-taghvaei19a, title = {Accelerated Flow for Probability Distributions}, author = {Taghvaei, Amirhossein and Mehta, Prashant}, booktitle = {Proceedings of the 36th International Conference on Machine Learning}, pages = {6076--6085}, year = {2019}, editor = {Chaudhuri, Kamalika and Salakhutdinov, Ruslan}, volume = {97}, series = {Proceedings of Machine Learning Research}, month = {09--15 Jun}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v97/taghvaei19a/taghvaei19a.pdf}, url = {https://proceedings.mlr.press/v97/taghvaei19a.html}, abstract = {This paper presents a methodology and numerical algorithms for constructing accelerated gradient flows on the space of probability distributions. In particular, we extend the recent variational formulation of accelerated methods in (Wibisono et al., 2016) from vector valued variables to probability distributions. The variational problem is modeled as a mean-field optimal control problem. A quantitative estimate on the asymptotic convergence rate is provided based on a Lyapunov function construction, when the objective functional is displacement convex. An important special case is considered where the objective functional is the relative entropy. For this case, two numerical approximations are presented to implement the Hamilton’s equations as a system of N interacting particles. The algorithm is numerically illustrated and compared with the MCMC and Hamiltonian MCMC algorithms.} }
Endnote
%0 Conference Paper %T Accelerated Flow for Probability Distributions %A Amirhossein Taghvaei %A Prashant Mehta %B Proceedings of the 36th International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2019 %E Kamalika Chaudhuri %E Ruslan Salakhutdinov %F pmlr-v97-taghvaei19a %I PMLR %P 6076--6085 %U https://proceedings.mlr.press/v97/taghvaei19a.html %V 97 %X This paper presents a methodology and numerical algorithms for constructing accelerated gradient flows on the space of probability distributions. In particular, we extend the recent variational formulation of accelerated methods in (Wibisono et al., 2016) from vector valued variables to probability distributions. The variational problem is modeled as a mean-field optimal control problem. A quantitative estimate on the asymptotic convergence rate is provided based on a Lyapunov function construction, when the objective functional is displacement convex. An important special case is considered where the objective functional is the relative entropy. For this case, two numerical approximations are presented to implement the Hamilton’s equations as a system of N interacting particles. The algorithm is numerically illustrated and compared with the MCMC and Hamiltonian MCMC algorithms.
APA
Taghvaei, A. & Mehta, P.. (2019). Accelerated Flow for Probability Distributions. Proceedings of the 36th International Conference on Machine Learning, in Proceedings of Machine Learning Research 97:6076-6085 Available from https://proceedings.mlr.press/v97/taghvaei19a.html.

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