The continuous categorical: a novel simplex-valued exponential family

Elliott Gordon-Rodriguez, Gabriel Loaiza-Ganem, John Cunningham
Proceedings of the 37th International Conference on Machine Learning, PMLR 119:3637-3647, 2020.

Abstract

Simplex-valued data appear throughout statistics and machine learning, for example in the context of transfer learning and compression of deep networks. Existing models for this class of data rely on the Dirichlet distribution or other related loss functions; here we show these standard choices suffer systematically from a number of limitations, including bias and numerical issues that frustrate the use of flexible network models upstream of these distributions. We resolve these limitations by introducing a novel exponential family of distributions for modeling simplex-valued data {–} the continuous categorical, which arises as a nontrivial multivariate generalization of the recently discovered continuous Bernoulli. Unlike the Dirichlet and other typical choices, the continuous categorical results in a well-behaved probabilistic loss function that produces unbiased estimators, while preserving the mathematical simplicity of the Dirichlet. As well as exploring its theoretical properties, we introduce sampling methods for this distribution that are amenable to the reparameterization trick, and evaluate their performance. Lastly, we demonstrate that the continuous categorical outperforms standard choices empirically, across a simulation study, an applied example on multi-party elections, and a neural network compression task.

Cite this Paper

BibTeX
@InProceedings{pmlr-v119-gordon-rodriguez20a, title = {The continuous categorical: a novel simplex-valued exponential family}, author = {Gordon-Rodriguez, Elliott and Loaiza-Ganem, Gabriel and Cunningham, John}, booktitle = {Proceedings of the 37th International Conference on Machine Learning}, pages = {3637--3647}, year = {2020}, editor = {III, Hal Daumé and Singh, Aarti}, volume = {119}, series = {Proceedings of Machine Learning Research}, month = {13--18 Jul}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v119/gordon-rodriguez20a/gordon-rodriguez20a.pdf}, url = {https://proceedings.mlr.press/v119/gordon-rodriguez20a.html}, abstract = {Simplex-valued data appear throughout statistics and machine learning, for example in the context of transfer learning and compression of deep networks. Existing models for this class of data rely on the Dirichlet distribution or other related loss functions; here we show these standard choices suffer systematically from a number of limitations, including bias and numerical issues that frustrate the use of flexible network models upstream of these distributions. We resolve these limitations by introducing a novel exponential family of distributions for modeling simplex-valued data {–} the continuous categorical, which arises as a nontrivial multivariate generalization of the recently discovered continuous Bernoulli. Unlike the Dirichlet and other typical choices, the continuous categorical results in a well-behaved probabilistic loss function that produces unbiased estimators, while preserving the mathematical simplicity of the Dirichlet. As well as exploring its theoretical properties, we introduce sampling methods for this distribution that are amenable to the reparameterization trick, and evaluate their performance. Lastly, we demonstrate that the continuous categorical outperforms standard choices empirically, across a simulation study, an applied example on multi-party elections, and a neural network compression task.} }
Endnote
%0 Conference Paper %T The continuous categorical: a novel simplex-valued exponential family %A Elliott Gordon-Rodriguez %A Gabriel Loaiza-Ganem %A John Cunningham %B Proceedings of the 37th International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2020 %E Hal Daumé III %E Aarti Singh %F pmlr-v119-gordon-rodriguez20a %I PMLR %P 3637--3647 %U https://proceedings.mlr.press/v119/gordon-rodriguez20a.html %V 119 %X Simplex-valued data appear throughout statistics and machine learning, for example in the context of transfer learning and compression of deep networks. Existing models for this class of data rely on the Dirichlet distribution or other related loss functions; here we show these standard choices suffer systematically from a number of limitations, including bias and numerical issues that frustrate the use of flexible network models upstream of these distributions. We resolve these limitations by introducing a novel exponential family of distributions for modeling simplex-valued data {–} the continuous categorical, which arises as a nontrivial multivariate generalization of the recently discovered continuous Bernoulli. Unlike the Dirichlet and other typical choices, the continuous categorical results in a well-behaved probabilistic loss function that produces unbiased estimators, while preserving the mathematical simplicity of the Dirichlet. As well as exploring its theoretical properties, we introduce sampling methods for this distribution that are amenable to the reparameterization trick, and evaluate their performance. Lastly, we demonstrate that the continuous categorical outperforms standard choices empirically, across a simulation study, an applied example on multi-party elections, and a neural network compression task.
APA
Gordon-Rodriguez, E., Loaiza-Ganem, G. & Cunningham, J.. (2020). The continuous categorical: a novel simplex-valued exponential family. Proceedings of the 37th International Conference on Machine Learning, in Proceedings of Machine Learning Research 119:3637-3647 Available from https://proceedings.mlr.press/v119/gordon-rodriguez20a.html.

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