Online Learning of Eigenvectors

Dan Garber, Elad Hazan, Tengyu Ma
Proceedings of the 32nd International Conference on Machine Learning, PMLR 37:560-568, 2015.

Abstract

Computing the leading eigenvector of a symmetric real matrix is a fundamental primitive of numerical linear algebra with numerous applications. We consider a natural online extension of the leading eigenvector problem: a sequence of matrices is presented and the goal is to predict for each matrix a unit vector, with the overall goal of competing with the leading eigenvector of the cumulative matrix. Existing regret-minimization algorithms for this problem either require to compute an \textiteigen decompostion every iteration, or suffer from a large dependency of the regret bound on the dimension. In both cases the algorithms are not practical for large scale applications. In this paper we present new algorithms that avoid both issues. On one hand they do not require any expensive matrix decompositions and on the other, they guarantee regret rates with a mild dependence on the dimension at most. In contrast to previous algorithms, our algorithms also admit implementations that enable to leverage sparsity in the data to further reduce computation. We extend our results to also handle non-symmetric matrices.

Cite this Paper


BibTeX
@InProceedings{pmlr-v37-garberb15, title = {Online Learning of Eigenvectors}, author = {Garber, Dan and Hazan, Elad and Ma, Tengyu}, booktitle = {Proceedings of the 32nd International Conference on Machine Learning}, pages = {560--568}, 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/garberb15.pdf}, url = { http://proceedings.mlr.press/v37/garberb15.html }, abstract = {Computing the leading eigenvector of a symmetric real matrix is a fundamental primitive of numerical linear algebra with numerous applications. We consider a natural online extension of the leading eigenvector problem: a sequence of matrices is presented and the goal is to predict for each matrix a unit vector, with the overall goal of competing with the leading eigenvector of the cumulative matrix. Existing regret-minimization algorithms for this problem either require to compute an \textiteigen decompostion every iteration, or suffer from a large dependency of the regret bound on the dimension. In both cases the algorithms are not practical for large scale applications. In this paper we present new algorithms that avoid both issues. On one hand they do not require any expensive matrix decompositions and on the other, they guarantee regret rates with a mild dependence on the dimension at most. In contrast to previous algorithms, our algorithms also admit implementations that enable to leverage sparsity in the data to further reduce computation. We extend our results to also handle non-symmetric matrices.} }
Endnote
%0 Conference Paper %T Online Learning of Eigenvectors %A Dan Garber %A Elad Hazan %A Tengyu Ma %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-garberb15 %I PMLR %P 560--568 %U http://proceedings.mlr.press/v37/garberb15.html %V 37 %X Computing the leading eigenvector of a symmetric real matrix is a fundamental primitive of numerical linear algebra with numerous applications. We consider a natural online extension of the leading eigenvector problem: a sequence of matrices is presented and the goal is to predict for each matrix a unit vector, with the overall goal of competing with the leading eigenvector of the cumulative matrix. Existing regret-minimization algorithms for this problem either require to compute an \textiteigen decompostion every iteration, or suffer from a large dependency of the regret bound on the dimension. In both cases the algorithms are not practical for large scale applications. In this paper we present new algorithms that avoid both issues. On one hand they do not require any expensive matrix decompositions and on the other, they guarantee regret rates with a mild dependence on the dimension at most. In contrast to previous algorithms, our algorithms also admit implementations that enable to leverage sparsity in the data to further reduce computation. We extend our results to also handle non-symmetric matrices.
RIS
TY - CPAPER TI - Online Learning of Eigenvectors AU - Dan Garber AU - Elad Hazan AU - Tengyu Ma BT - Proceedings of the 32nd International Conference on Machine Learning DA - 2015/06/01 ED - Francis Bach ED - David Blei ID - pmlr-v37-garberb15 PB - PMLR DP - Proceedings of Machine Learning Research VL - 37 SP - 560 EP - 568 L1 - http://proceedings.mlr.press/v37/garberb15.pdf UR - http://proceedings.mlr.press/v37/garberb15.html AB - Computing the leading eigenvector of a symmetric real matrix is a fundamental primitive of numerical linear algebra with numerous applications. We consider a natural online extension of the leading eigenvector problem: a sequence of matrices is presented and the goal is to predict for each matrix a unit vector, with the overall goal of competing with the leading eigenvector of the cumulative matrix. Existing regret-minimization algorithms for this problem either require to compute an \textiteigen decompostion every iteration, or suffer from a large dependency of the regret bound on the dimension. In both cases the algorithms are not practical for large scale applications. In this paper we present new algorithms that avoid both issues. On one hand they do not require any expensive matrix decompositions and on the other, they guarantee regret rates with a mild dependence on the dimension at most. In contrast to previous algorithms, our algorithms also admit implementations that enable to leverage sparsity in the data to further reduce computation. We extend our results to also handle non-symmetric matrices. ER -
APA
Garber, D., Hazan, E. & Ma, T.. (2015). Online Learning of Eigenvectors. Proceedings of the 32nd International Conference on Machine Learning, in Proceedings of Machine Learning Research 37:560-568 Available from http://proceedings.mlr.press/v37/garberb15.html .

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