q-Paths: Generalizing the geometric annealing path using power means

Vaden Masrani, Rob Brekelmans, Thang Bui, Frank Nielsen, Aram Galstyan, Greg Ver Steeg, Frank Wood
Proceedings of the Thirty-Seventh Conference on Uncertainty in Artificial Intelligence, PMLR 161:1938-1947, 2021.

Abstract

Many common machine learning methods involve the geometric annealing path, a sequence of intermediate densities between two distributions of interest constructed using the geometric average. While alternatives such as the moment-averaging path have demonstrated performance gains in some settings, their practical applicability remains limited by exponential family endpoint assumptions and a lack of closed form energy function. In this work, we introduce $q$-paths, a family of paths which is derived from a generalized notion of the mean, includes the geometric and arithmetic mixtures as special cases, and admits a simple closed form involving the deformed logarithm function from nonextensive thermodynamics. Following previous analysis of the geometric path, we interpret our $q$-paths as corresponding to a $q$-exponential family of distributions, and provide a variational representation of intermediate densities as minimizing a mixture of $\alpha$-divergences to the endpoints. We show that small deviations away from the geometric path yield empirical gains for Bayesian inference using Sequential Monte Carlo and generative model evaluation using Annealed Importance Sampling.

Cite this Paper

BibTeX
@InProceedings{pmlr-v161-masrani21a, title = {q-Paths: Generalizing the geometric annealing path using power means}, author = {Masrani, Vaden and Brekelmans, Rob and Bui, Thang and Nielsen, Frank and Galstyan, Aram and Ver Steeg, Greg and Wood, Frank}, booktitle = {Proceedings of the Thirty-Seventh Conference on Uncertainty in Artificial Intelligence}, pages = {1938--1947}, year = {2021}, editor = {de Campos, Cassio and Maathuis, Marloes H.}, volume = {161}, series = {Proceedings of Machine Learning Research}, month = {27--30 Jul}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v161/masrani21a/masrani21a.pdf}, url = {https://proceedings.mlr.press/v161/masrani21a.html}, abstract = {Many common machine learning methods involve the geometric annealing path, a sequence of intermediate densities between two distributions of interest constructed using the geometric average. While alternatives such as the moment-averaging path have demonstrated performance gains in some settings, their practical applicability remains limited by exponential family endpoint assumptions and a lack of closed form energy function. In this work, we introduce $q$-paths, a family of paths which is derived from a generalized notion of the mean, includes the geometric and arithmetic mixtures as special cases, and admits a simple closed form involving the deformed logarithm function from nonextensive thermodynamics. Following previous analysis of the geometric path, we interpret our $q$-paths as corresponding to a $q$-exponential family of distributions, and provide a variational representation of intermediate densities as minimizing a mixture of $\alpha$-divergences to the endpoints. We show that small deviations away from the geometric path yield empirical gains for Bayesian inference using Sequential Monte Carlo and generative model evaluation using Annealed Importance Sampling.} }
Endnote
%0 Conference Paper %T q-Paths: Generalizing the geometric annealing path using power means %A Vaden Masrani %A Rob Brekelmans %A Thang Bui %A Frank Nielsen %A Aram Galstyan %A Greg Ver Steeg %A Frank Wood %B Proceedings of the Thirty-Seventh Conference on Uncertainty in Artificial Intelligence %C Proceedings of Machine Learning Research %D 2021 %E Cassio de Campos %E Marloes H. Maathuis %F pmlr-v161-masrani21a %I PMLR %P 1938--1947 %U https://proceedings.mlr.press/v161/masrani21a.html %V 161 %X Many common machine learning methods involve the geometric annealing path, a sequence of intermediate densities between two distributions of interest constructed using the geometric average. While alternatives such as the moment-averaging path have demonstrated performance gains in some settings, their practical applicability remains limited by exponential family endpoint assumptions and a lack of closed form energy function. In this work, we introduce $q$-paths, a family of paths which is derived from a generalized notion of the mean, includes the geometric and arithmetic mixtures as special cases, and admits a simple closed form involving the deformed logarithm function from nonextensive thermodynamics. Following previous analysis of the geometric path, we interpret our $q$-paths as corresponding to a $q$-exponential family of distributions, and provide a variational representation of intermediate densities as minimizing a mixture of $\alpha$-divergences to the endpoints. We show that small deviations away from the geometric path yield empirical gains for Bayesian inference using Sequential Monte Carlo and generative model evaluation using Annealed Importance Sampling.
APA
Masrani, V., Brekelmans, R., Bui, T., Nielsen, F., Galstyan, A., Ver Steeg, G. & Wood, F.. (2021). q-Paths: Generalizing the geometric annealing path using power means. Proceedings of the Thirty-Seventh Conference on Uncertainty in Artificial Intelligence, in Proceedings of Machine Learning Research 161:1938-1947 Available from https://proceedings.mlr.press/v161/masrani21a.html.

Related Material