Minimizing Convex Functionals over Space of Probability Measures via KL Divergence Gradient Flow

Rentian Yao, Linjun Huang, Yun Yang
Proceedings of The 27th International Conference on Artificial Intelligence and Statistics, PMLR 238:2530-2538, 2024.

Abstract

Motivated by the computation of the non-parametric maximum likelihood estimator (NPMLE) and the Bayesian posterior in statistics, this paper explores the problem of convex optimization over the space of all probability distributions. We introduce an implicit scheme, called the implicit KL proximal descent (IKLPD) algorithm, for discretizing a continuous-time gradient flow relative to the Kullback–Leibler (KL) divergence for minimizing a convex target functional. We show that IKLPD converges to a global optimum at a polynomial rate from any initialization; moreover, if the objective functional is strongly convex relative to the KL divergence, for example, when the target functional itself is a KL divergence as in the context of Bayesian posterior computation, IKLPD exhibits globally exponential convergence. Computationally, we propose a numerical method based on normalizing flow to realize IKLPD. Conversely, our numerical method can also be viewed as a new approach that sequentially trains a normalizing flow for minimizing a convex functional with a strong theoretical guarantee.

Cite this Paper

BibTeX
@InProceedings{pmlr-v238-yao24a, title = {Minimizing Convex Functionals over Space of Probability Measures via {KL} Divergence Gradient Flow}, author = {Yao, Rentian and Huang, Linjun and Yang, Yun}, booktitle = {Proceedings of The 27th International Conference on Artificial Intelligence and Statistics}, pages = {2530--2538}, year = {2024}, editor = {Dasgupta, Sanjoy and Mandt, Stephan and Li, Yingzhen}, volume = {238}, series = {Proceedings of Machine Learning Research}, month = {02--04 May}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v238/yao24a/yao24a.pdf}, url = {https://proceedings.mlr.press/v238/yao24a.html}, abstract = {Motivated by the computation of the non-parametric maximum likelihood estimator (NPMLE) and the Bayesian posterior in statistics, this paper explores the problem of convex optimization over the space of all probability distributions. We introduce an implicit scheme, called the implicit KL proximal descent (IKLPD) algorithm, for discretizing a continuous-time gradient flow relative to the Kullback–Leibler (KL) divergence for minimizing a convex target functional. We show that IKLPD converges to a global optimum at a polynomial rate from any initialization; moreover, if the objective functional is strongly convex relative to the KL divergence, for example, when the target functional itself is a KL divergence as in the context of Bayesian posterior computation, IKLPD exhibits globally exponential convergence. Computationally, we propose a numerical method based on normalizing flow to realize IKLPD. Conversely, our numerical method can also be viewed as a new approach that sequentially trains a normalizing flow for minimizing a convex functional with a strong theoretical guarantee.} }
Endnote
%0 Conference Paper %T Minimizing Convex Functionals over Space of Probability Measures via KL Divergence Gradient Flow %A Rentian Yao %A Linjun Huang %A Yun Yang %B Proceedings of The 27th International Conference on Artificial Intelligence and Statistics %C Proceedings of Machine Learning Research %D 2024 %E Sanjoy Dasgupta %E Stephan Mandt %E Yingzhen Li %F pmlr-v238-yao24a %I PMLR %P 2530--2538 %U https://proceedings.mlr.press/v238/yao24a.html %V 238 %X Motivated by the computation of the non-parametric maximum likelihood estimator (NPMLE) and the Bayesian posterior in statistics, this paper explores the problem of convex optimization over the space of all probability distributions. We introduce an implicit scheme, called the implicit KL proximal descent (IKLPD) algorithm, for discretizing a continuous-time gradient flow relative to the Kullback–Leibler (KL) divergence for minimizing a convex target functional. We show that IKLPD converges to a global optimum at a polynomial rate from any initialization; moreover, if the objective functional is strongly convex relative to the KL divergence, for example, when the target functional itself is a KL divergence as in the context of Bayesian posterior computation, IKLPD exhibits globally exponential convergence. Computationally, we propose a numerical method based on normalizing flow to realize IKLPD. Conversely, our numerical method can also be viewed as a new approach that sequentially trains a normalizing flow for minimizing a convex functional with a strong theoretical guarantee.
APA
Yao, R., Huang, L. & Yang, Y.. (2024). Minimizing Convex Functionals over Space of Probability Measures via KL Divergence Gradient Flow. Proceedings of The 27th International Conference on Artificial Intelligence and Statistics, in Proceedings of Machine Learning Research 238:2530-2538 Available from https://proceedings.mlr.press/v238/yao24a.html.

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