Convergence rate of Bayesian tensor estimator and its minimax optimality

Taiji Suzuki
Proceedings of the 32nd International Conference on Machine Learning, PMLR 37:1273-1282, 2015.

Abstract

We investigate the statistical convergence rate of a Bayesian low-rank tensor estimator, and derive the minimax optimal rate for learning a low-rank tensor. Our problem setting is the regression problem where the regression coefficient forms a tensor structure. This problem setting occurs in many practical applications, such as collaborative filtering, multi-task learning, and spatio-temporal data analysis. The convergence rate of the Bayes tensor estimator is analyzed in terms of both in-sample and out-of-sample predictive accuracies. It is shown that a fast learning rate is achieved without any strong convexity of the observation. Moreover, we show that the method has adaptivity to the unknown rank of the true tensor, that is, the near optimal rate depending on the true rank is achieved even if it is not known a priori. Finally, we show the minimax optimal learning rate for the tensor estimation problem, and thus show that the derived bound of the Bayes estimator is tight and actually near minimax optimal.

Cite this Paper

BibTeX
@InProceedings{pmlr-v37-suzuki15, title = {Convergence rate of Bayesian tensor estimator and its minimax optimality}, author = {Suzuki, Taiji}, booktitle = {Proceedings of the 32nd International Conference on Machine Learning}, pages = {1273--1282}, year = {2015}, editor = {Bach, Francis and Blei, David}, volume = {37}, series = {Proceedings of Machine Learning Research}, address = {Lille, France}, month = {07--09 Jul}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v37/suzuki15.pdf}, url = {https://proceedings.mlr.press/v37/suzuki15.html}, abstract = {We investigate the statistical convergence rate of a Bayesian low-rank tensor estimator, and derive the minimax optimal rate for learning a low-rank tensor. Our problem setting is the regression problem where the regression coefficient forms a tensor structure. This problem setting occurs in many practical applications, such as collaborative filtering, multi-task learning, and spatio-temporal data analysis. The convergence rate of the Bayes tensor estimator is analyzed in terms of both in-sample and out-of-sample predictive accuracies. It is shown that a fast learning rate is achieved without any strong convexity of the observation. Moreover, we show that the method has adaptivity to the unknown rank of the true tensor, that is, the near optimal rate depending on the true rank is achieved even if it is not known a priori. Finally, we show the minimax optimal learning rate for the tensor estimation problem, and thus show that the derived bound of the Bayes estimator is tight and actually near minimax optimal.} }
Endnote
%0 Conference Paper %T Convergence rate of Bayesian tensor estimator and its minimax optimality %A Taiji Suzuki %B Proceedings of the 32nd International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2015 %E Francis Bach %E David Blei %F pmlr-v37-suzuki15 %I PMLR %P 1273--1282 %U https://proceedings.mlr.press/v37/suzuki15.html %V 37 %X We investigate the statistical convergence rate of a Bayesian low-rank tensor estimator, and derive the minimax optimal rate for learning a low-rank tensor. Our problem setting is the regression problem where the regression coefficient forms a tensor structure. This problem setting occurs in many practical applications, such as collaborative filtering, multi-task learning, and spatio-temporal data analysis. The convergence rate of the Bayes tensor estimator is analyzed in terms of both in-sample and out-of-sample predictive accuracies. It is shown that a fast learning rate is achieved without any strong convexity of the observation. Moreover, we show that the method has adaptivity to the unknown rank of the true tensor, that is, the near optimal rate depending on the true rank is achieved even if it is not known a priori. Finally, we show the minimax optimal learning rate for the tensor estimation problem, and thus show that the derived bound of the Bayes estimator is tight and actually near minimax optimal.
RIS
TY - CPAPER TI - Convergence rate of Bayesian tensor estimator and its minimax optimality AU - Taiji Suzuki BT - Proceedings of the 32nd International Conference on Machine Learning DA - 2015/06/01 ED - Francis Bach ED - David Blei ID - pmlr-v37-suzuki15 PB - PMLR DP - Proceedings of Machine Learning Research VL - 37 SP - 1273 EP - 1282 L1 - http://proceedings.mlr.press/v37/suzuki15.pdf UR - https://proceedings.mlr.press/v37/suzuki15.html AB - We investigate the statistical convergence rate of a Bayesian low-rank tensor estimator, and derive the minimax optimal rate for learning a low-rank tensor. Our problem setting is the regression problem where the regression coefficient forms a tensor structure. This problem setting occurs in many practical applications, such as collaborative filtering, multi-task learning, and spatio-temporal data analysis. The convergence rate of the Bayes tensor estimator is analyzed in terms of both in-sample and out-of-sample predictive accuracies. It is shown that a fast learning rate is achieved without any strong convexity of the observation. Moreover, we show that the method has adaptivity to the unknown rank of the true tensor, that is, the near optimal rate depending on the true rank is achieved even if it is not known a priori. Finally, we show the minimax optimal learning rate for the tensor estimation problem, and thus show that the derived bound of the Bayes estimator is tight and actually near minimax optimal. ER -
APA
Suzuki, T.. (2015). Convergence rate of Bayesian tensor estimator and its minimax optimality. Proceedings of the 32nd International Conference on Machine Learning, in Proceedings of Machine Learning Research 37:1273-1282 Available from https://proceedings.mlr.press/v37/suzuki15.html.

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