Near-Optimal Algorithms for Online Matrix Prediction

Elad Hazan, Satyen Kale, Shai Shalev-Shwartz
Proceedings of the 25th Annual Conference on Learning Theory, PMLR 23:38.1-38.13, 2012.

Abstract

In several online prediction problems of recent interest the comparison class is composed of matrices with bounded entries. For example, in the online max-cut problem, the comparison class is matrices which represent cuts of a given graph and in online gambling the comparison class is matrices which represent permutations over n teams. Another important example is online collaborative filtering in which a widely used comparison class is the set of matrices with a small trace norm. In this paper we isolate a property of matrices, which we call (β,τ)-decomposability, and derive an efficient online learning algorithm, that enjoys a regret bound of \emphÕ(√βτT ) for all problems in which the comparison class is composed of (β,τ)-decomposable matrices. By analyzing the decomposability of cut matrices, low trace-norm matrices and triangular matrices, we derive near optimal regret bounds for online max-cut, online collaborative filtering and online gambling. In particular, this resolves (in the affirmative) an open problem posed by Abernethy (2010); Kleinberg et al. (2010). Finally, we derive lower bounds for the three problems and show that our upper bounds are optimal up to logarithmic factors. In particular, our lower bound for the online collaborative filtering problem resolves another open problem posed by Shamir and Srebro (2011).

Cite this Paper

BibTeX
@InProceedings{pmlr-v23-hazan12b, title = {Near-Optimal Algorithms for Online Matrix Prediction}, author = {Hazan, Elad and Kale, Satyen and Shalev-Shwartz, Shai}, booktitle = {Proceedings of the 25th Annual Conference on Learning Theory}, pages = {38.1--38.13}, year = {2012}, editor = {Mannor, Shie and Srebro, Nathan and Williamson, Robert C.}, volume = {23}, series = {Proceedings of Machine Learning Research}, address = {Edinburgh, Scotland}, month = {25--27 Jun}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v23/hazan12b/hazan12b.pdf}, url = {https://proceedings.mlr.press/v23/hazan12b.html}, abstract = {In several online prediction problems of recent interest the comparison class is composed of matrices with bounded entries. For example, in the online max-cut problem, the comparison class is matrices which represent cuts of a given graph and in online gambling the comparison class is matrices which represent permutations over n teams. Another important example is online collaborative filtering in which a widely used comparison class is the set of matrices with a small trace norm. In this paper we isolate a property of matrices, which we call (β,τ)-decomposability, and derive an efficient online learning algorithm, that enjoys a regret bound of \emphÕ(√βτT ) for all problems in which the comparison class is composed of (β,τ)-decomposable matrices. By analyzing the decomposability of cut matrices, low trace-norm matrices and triangular matrices, we derive near optimal regret bounds for online max-cut, online collaborative filtering and online gambling. In particular, this resolves (in the affirmative) an open problem posed by Abernethy (2010); Kleinberg et al. (2010). Finally, we derive lower bounds for the three problems and show that our upper bounds are optimal up to logarithmic factors. In particular, our lower bound for the online collaborative filtering problem resolves another open problem posed by Shamir and Srebro (2011).} }
Endnote
%0 Conference Paper %T Near-Optimal Algorithms for Online Matrix Prediction %A Elad Hazan %A Satyen Kale %A Shai Shalev-Shwartz %B Proceedings of the 25th Annual Conference on Learning Theory %C Proceedings of Machine Learning Research %D 2012 %E Shie Mannor %E Nathan Srebro %E Robert C. Williamson %F pmlr-v23-hazan12b %I PMLR %P 38.1--38.13 %U https://proceedings.mlr.press/v23/hazan12b.html %V 23 %X In several online prediction problems of recent interest the comparison class is composed of matrices with bounded entries. For example, in the online max-cut problem, the comparison class is matrices which represent cuts of a given graph and in online gambling the comparison class is matrices which represent permutations over n teams. Another important example is online collaborative filtering in which a widely used comparison class is the set of matrices with a small trace norm. In this paper we isolate a property of matrices, which we call (β,τ)-decomposability, and derive an efficient online learning algorithm, that enjoys a regret bound of \emphÕ(√βτT ) for all problems in which the comparison class is composed of (β,τ)-decomposable matrices. By analyzing the decomposability of cut matrices, low trace-norm matrices and triangular matrices, we derive near optimal regret bounds for online max-cut, online collaborative filtering and online gambling. In particular, this resolves (in the affirmative) an open problem posed by Abernethy (2010); Kleinberg et al. (2010). Finally, we derive lower bounds for the three problems and show that our upper bounds are optimal up to logarithmic factors. In particular, our lower bound for the online collaborative filtering problem resolves another open problem posed by Shamir and Srebro (2011).
RIS
TY - CPAPER TI - Near-Optimal Algorithms for Online Matrix Prediction AU - Elad Hazan AU - Satyen Kale AU - Shai Shalev-Shwartz BT - Proceedings of the 25th Annual Conference on Learning Theory DA - 2012/06/16 ED - Shie Mannor ED - Nathan Srebro ED - Robert C. Williamson ID - pmlr-v23-hazan12b PB - PMLR DP - Proceedings of Machine Learning Research VL - 23 SP - 38.1 EP - 38.13 L1 - http://proceedings.mlr.press/v23/hazan12b/hazan12b.pdf UR - https://proceedings.mlr.press/v23/hazan12b.html AB - In several online prediction problems of recent interest the comparison class is composed of matrices with bounded entries. For example, in the online max-cut problem, the comparison class is matrices which represent cuts of a given graph and in online gambling the comparison class is matrices which represent permutations over n teams. Another important example is online collaborative filtering in which a widely used comparison class is the set of matrices with a small trace norm. In this paper we isolate a property of matrices, which we call (β,τ)-decomposability, and derive an efficient online learning algorithm, that enjoys a regret bound of \emphÕ(√βτT ) for all problems in which the comparison class is composed of (β,τ)-decomposable matrices. By analyzing the decomposability of cut matrices, low trace-norm matrices and triangular matrices, we derive near optimal regret bounds for online max-cut, online collaborative filtering and online gambling. In particular, this resolves (in the affirmative) an open problem posed by Abernethy (2010); Kleinberg et al. (2010). Finally, we derive lower bounds for the three problems and show that our upper bounds are optimal up to logarithmic factors. In particular, our lower bound for the online collaborative filtering problem resolves another open problem posed by Shamir and Srebro (2011). ER -
APA
Hazan, E., Kale, S. & Shalev-Shwartz, S.. (2012). Near-Optimal Algorithms for Online Matrix Prediction. Proceedings of the 25th Annual Conference on Learning Theory, in Proceedings of Machine Learning Research 23:38.1-38.13 Available from https://proceedings.mlr.press/v23/hazan12b.html.

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